% The Logarithmic Super Divergence and its use in % Statistical Inference % Asymptotic Properties of Minimum S-Divergence Estimator for Discrete Models % Asymptotic Distribution of Logarithmic Super Divergence % for Both Estimating and Testing Problem under Discrete Set-up The Logarithmic Super Divergence and Statistical Inference : Asymptotic Properties % Logarithmic Super Divergence -- % A New Family Of Divergences % This Is The Title For A Template \\ Latex File For Writing Papers
Avijit Maji
Abhik Ghosh and Ayanendranath Basu
% Indian Statistical Institute % Author Name
% Author Address Line 1
% Author Address Line 2
% Author E-mail Address
Statistical inference based on divergence measures have a long history. % been used % for a long time now. Recently, Maji, Ghosh and Basu (2014) have introduced a general family of divergences called the % viz. logarithmic super divergence (LSD) family. This family acts as a superfamily for both of the logarithmic power divergence (LPD) family (eg. Renyi, 1961) and the logarithmic density power divergence (LDPD) family introduced by Jones et al.~(2001). In this paper we describe the asymptotic properties of the inference procedures resulting from this divergence in discrete models. The properties are well supported by real data examples. % of % introduced by Maji et al. (2014) % for both estimation and hypotheses % of testing problem under discrete models only. % The abstract summarises the context of the work and its major contributions. % This paper introduces a new superfamily of divergences % that is similar in spirit to the % % closely % % resembles with % $S$-divergence family introduced by Ghosh et al.~(2013). % This new family % % provides % % serves as % % a super family of % % which % serves as an umbrella % that % % which % contains the % logarithmic power divergence family (Renyi, 1961, Maji et al., 2014) and % the logarithmic density power divergence family (Jones et al., 2001) % as special cases. % % This family has strong connection with % % existing family of divergences. % Various properties of this new family % are theoretically % % have been % established; % % and % these properties are demonstrated through % simulation studies and real data examples. % usage in simulation studies and real data examples % have also been studied.
1406.2112.tex
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\title{
% The Logarithmic Super Divergence and its use in
% Statistical Inference
% Asymptotic Properties of Minimum S-Divergence Estimator for Discrete Models 
% Asymptotic Distribution of Logarithmic Super Divergence 
% for Both Estimating and Testing Problem under Discrete Set-up 
The Logarithmic Super Divergence and Statistical Inference 
: Asymptotic Properties
% Logarithmic Super Divergence --
% A New Family Of Divergences
% This Is The Title For A Template \\ Latex File For Writing Papers
}
\date{}

\author{
Avijit Maji, Abhik Ghosh and Ayanendranath Basu \\
% Indian Statistical Institute
%    Author Name \\
%    Author Address Line 1 \\
%    Author Address Line 2 \\
%    Author E-mail Address \\
}
\institute {Indian Statistical Institute
\at 203, B.T. Road, Kolkata-700108, India.\\
\email{avijit.maji$@$hotmail.com, abhianik@gmail.com, ayanendranath.basu@gmail.com.}
}
% \institute{J.B. Doe
% \at Doe Institute, 281 Prime Street, Daisy Town, NA 02467,
% USA\\Tel.: +127-47-678901, Fax: +127-47-678907
% \and
% J.B. Doe \and S.Q. Public
% \at Public-Enterprises
% \and
% J.A. Smith
% \at Smith University,\\\email{smith@smith.edu}
% }
\begin{document}
\maketitle

\begin{abstract}
Statistical inference based on divergence measures have 
a long history.
% been used 
% for a long time now. 
Recently, Maji, Ghosh and Basu (2014) 
have introduced a general family of divergences 
called the
% viz. 
logarithmic super divergence (LSD) family. This family acts 
as a superfamily for both of the logarithmic power 
divergence (LPD) family (eg. Renyi, 1961) and the logarithmic density
power divergence (LDPD) family introduced by 
Jones et al.~(2001). 
In this paper we describe the asymptotic properties 
of the inference procedures resulting from this 
divergence in discrete models. The properties are well supported 
by real data examples.
% of 
% introduced by Maji et al. (2014) 
% for both estimation and hypotheses
% of testing problem under discrete models only.
% The abstract summarises the context of the work and its major contributions.
% This paper introduces a new superfamily of divergences 
% that is similar in spirit to the 
% % closely
% % resembles with 
% $S$-divergence family introduced by Ghosh et al.~(2013).
% This new family 
% % provides 
% % serves as 
% % a super family of 
% % which 
% serves as an umbrella 
% that
% % which 
% contains the 
% logarithmic power divergence family (Renyi, 1961, Maji et al., 2014) and
% the logarithmic density power divergence family (Jones et al., 2001)
% as special cases.
% % This family has strong connection with 
% % existing family of divergences. 
% Various properties of this new family 
% are theoretically
% % have been 
% established;
% % and 
% these properties are demonstrated through
% simulation studies and real data examples.
% usage in simulation studies and real data examples 
% have also been studied.
\end{abstract}
\vspace{2 mm}

{\em \bf Key words}: asymptotic properties, logarithmic density power divergence, 
logarithmic power divergence, logarithmic super divergence, $S$-divergence,
statistical inference. 
\section{Introduction}
\label{Sec_Introduction}

% \paragraph{}
% This is the Introduction Section.
The 
% Parametric inference uses the 
density-based minimum divergence approach 
has long been an important parametric inference tool.
% from long time ago. 
In this approach the 
% The 
closeness between the data and the model is measured by
% by 
a density-based divergence between the data density and the
model density, such as 
a $\chi^2$ type divergence or a $\phi$-divergence
(Csisz\'{a}r 1963, 1967a,b; Ali and Silvey 1966)
 or a Bregman divergence (Bregman, 1967). 
Apart from their natural appeal, most
% Most 
of these 
methods are very useful for their inherent robustness properties.
% nature.
A prominent member of the class of density-based divergences 
is the Pearson's $\chi^2$ (Pearson, 1900)
which started its journey 
from the very early days
% as old as early time 
of formal research
in statistics. 
From the robustness perspective, however, Beran's 1977 
work is the first useful reference in the literature
of density-based minimum divergence inference. 
% However Beran's 1977 work is the most useful early reference in the 
% literature of density-based minimum divergence inference
% from the robustness perspective. 
In the present paper
we focus on some variants of the power divergence (PD) measure of 
Cressie and Read (1984) and the density power divergence (DPD) of Basu et al.~(1998)
and discuss various properties related to statistical inference based on a
generalized superfamilies of these measures. 
The primary divergence class 
of logarithmic super divergences (LSDs)
which is of interest to us in this paper 
has been proposed recently by Maji, Ghosh and Basu (2014). 
In the present paper we establish the theoretical asymptotic
properties of the resulting statistical procedures. 
\paragraph{}
The rest of the paper is organized as follows. Section \ref{SEC:LPD_LDPD}
describes the logarithmic power divergence family and the 
logarithmic density power divergence family. Section \ref{SEC:lsd_defn}
gives the form and the estimating equation of the 
logarithmic super divergence (LSD) family whereas Section \ref{SEC:lsd_properties}
establishes the asymptotic distribution for LSD estimator. Section \ref{SEC:est_example}
gives some illustration of the proposed estimation procedure with real data. 
% shows estimation example. 
Section \ref{SEC:testing} gives the asymptotic
distribution of the test statistic using LSD measure for both one sample 
and two sample problem and Section \ref{SEC:testing:example} 
provides a hypothesis testing example.
% gives
% example under testing set-up. 
Concluding remarks are in Section \ref{Sec_Conclusion}.
% Among many other 
% things, our analysis highlights the limitation of the first 
% order influence function analysis as an indicator of robustness. 
%  but  minimum distance estimation has been started by Wolfowitz in
% the $1950$s ($1952, 1953, 1954, 1957$). After that Beran's $1977$ paper is a useful
% milestone in this regard and onwards more recently monographs by
% Vajda ($1989$), Pardo ($2006$) and Basu et al. ($2011$) act as useful resources
% in this domain. Our discussion mainly focuses on power divergence family 
% (Cressie and Read ($1984$)) and density power divergence family (Basu et al. ($1998$))
% and close relatives of these families such as logarithmic power
% divergence family and logarithmic density power divergence family. 
% \section{Background}
% \label{Sec_Background}
% 
% \subsection{The Power Divergence (PD) Family}
% 
% % \paragraph{}
% The class of $\chi^2$ type distances, which are generally referred to 
% as $\phi$-divergences, $f$-divergences, or disparities 
% (Csisz\'{a}r 1963, 1967a,b; Ali and Silvey 1966; Lindsay 1994),  
% dominate the literature in the area of 
% density-based minimum divergence inference. 
% A prominent subclass of this family is 
% % used to be  
% the Cressie-Read (Cressie and Read, 1984)
% family of power divergences. 
% This family was originally developed by these authors with the 
% primary aim of multinomial goodness-of-fit testing. 
% Read and Cressie, in their 1988 monograph on the same subject, briefly 
% elaborated on the problem of parametric estimation,
% but their focus remained on goodness-of-fit testing. 
% % Though it was originally 
% % developed with the primary aim of multinomial goodness-of-fit testing
% % later Read and Cressie (1988) briefly elaborated the 
% % problem of parametric estimation, but the focus remained 
% % on goodness-of-fit testing. 
% However, several later authors have successfully used it for 
% robust parametric inference; see e.g. Basu et al.~(2011). 
% % Later some books including
% % Basu et al. (2011) have shown successful use of 
% % this family in producing robust and asymptotically fully efficient estimators under 
% % parametric settings. 
% % In discrete models this
% % This 
% % family has this form
% Given two probability density functions $f$ and $g$ 
% with respect to the same measure,
% the power divergence between them is defined as 
% \begin{equation}\label{EQ:cressie-read}
% {\rm PD}_\lambda(g,f)=\frac{1}
% {\lambda(\lambda + 1)} \int g \left[\left(\frac{g}{f}\right)^\lambda-1\right],
% %     {\rm PD}_\lambda( r_n, f_\theta) = \frac{1}
% % {\lambda(\lambda + 1)} \sum  r_n \left[ \left
% % ( \frac{ r_n}{f_\theta} \right)^\lambda - 1 \right],
% \end{equation}
% which is indexed by a single real valued tuning parameter $\lambda$. It 
% generates several well-known divergences for different values of $\lambda$. 
% For example 
% % includes 
% % various other known families by putting different values of $\lambda$ as
% $\lambda = 1, 0,-1/2,-1 \mbox{ and } -2$ generate the
% % respectively to get 
% Pearson's chi-square (PCS), the likelihood disparity (LD),
% the Hellinger distance (HD), the Kullback-Leibler divergence (KLD) and the
% Neyman's chi-square (NCS) respectively. 
% However the LD (PD$_0$) and the KLD (PD$_{-1}$) are the limiting expressions 
% of the form in (\ref{EQ:cressie-read}) as $\lambda \rightarrow 0$
% and $-1$ respectively, since the direct substitution of 
% these tuning parameter in (\ref{EQ:cressie-read}) make the 
% divergence undefined.
% % Though to get LD and KLD we need to
% % use the limiting values. 
% % \begin{definition}
% %  
% % 	Let $C$ be a thrice differentiable, strictly 
% % convex function on $[-1,\infty)$, satisfying $C(0) = 0$. Let 
% % the Pearson residual at the value $x$ be defined by
% % $$
% % % \delta(x) = 
% % \delta_n(x) = \frac{r_n(x)}{f_\theta(x)} - 1.
% % $$
% % Then the disparity between $\utilde{r}$ and $\utilde{f_\theta}$ 
% % generated by $C$ is defined by
% % % \end{linenumbers}
% % \begin{equation}\label{EQ:disparity}
% %     \rho_C( r_n, f_\theta) = \sum_{x=0}^\infty C(\delta(x)) f_\theta(x).
% % \end{equation}
% % \end{definition}
% % 	Specific forms of the function $C$ generate many well 
% % known disparities. For
% % example, $C(\delta) = (\delta + 1) \log(\delta + 1) - \delta $ 
% % generates the well known likelihood disparity (LD) given by
% % \begin{equation}\label{EQ:LD}
% % {\rm LD}( r_n, f_\theta) = \sum \left[r_n \log( r_n/f_\theta) + 
% % (f_\theta -  r_n)\right] = \sum  r_n \log( r_n/f_\theta).
% % \end{equation}
% % It is a form of the Kullback-Leibler divergence (Kullback and Leibler,
% % 1951). By the Kullback-Leibler divergence we will usually refer to the 
% % symmetric opposite of the likelihood disparity which has the form
% % \begin{equation}
% % {\rm KLD}( r_n, f_\theta) = \sum [f_\theta \log(f_\theta/ r_n) + 
% % ( r_n - f_\theta)] = \sum f_\theta \log(f_\theta/ r_n).
% % \end{equation}
% % This distance corresponds to $C(\delta) = \delta - \log(\delta + 1)$. The (twice, squared)
% % Hellinger distance (HD) has the form
% % \begin{equation}
% % {\rm HD}( r_n, f_\theta) = 2 \sum \left[ r_n^{1/2} - f_\theta^{1/2}\right]^2
% % \end{equation}
% % which corresponds to $C(\delta) = 2((\delta + 1)^{1/2} - 1)^2$. The Pearson's chi-square
% % (divided by 2) is defined as
% % \begin{equation}\label{EQ:PCS}
% % {\rm PCS}(r_n, f_\theta) = \sum \frac{(r_n - f_\theta)^2}{2 f_\theta},
% % \end{equation}
% % where $C(\delta) = \delta^2/2$, and the Neyman's chi-square (divided by 2) is defined as
% % \begin{equation}\label{EQ:NCS}
% % {\rm NCS}( r_n, f_\theta) = \sum \frac{(r_n - f_\theta)^2}{2  r_n},
% % \end{equation}
% % where $C(\delta) = \delta^2/[2(\delta+1)]$. 
% % This section covers the background of the topic.
% 
% 
% % \section{Other Sections}
% % \label{Sec_Other Sections}
% % 
% % \paragraph{}
% \subsection{The Density Power Divergence (DPD) Family}
% Minimum disparity estimation (Lindsay, 1994) based on the PD family 
% provides a very useful robust technique for 
% discrete parametric models. It is also easily extended 
% to continuous models conceptually. However the practical 
% difficulty in doing so is that the ordinary empirical 
% does not automatically provide a continuous density estimate 
% of the unknown distribution and constructing the divergence 
% becomes a problem for continuous models. 
% % gives a very 
% % suitable method under discrete set-up and it can be easily
% % extended to continuous set-up. The main problem using it under continuous
% % set-up is to use kernel density estimation. To overcome this problem,
% As an alternative
% Basu et al.~(1998) 
% considered minimum divergence estimators based on the 
% % introduced 
% Density Power Divergence family which 
% is defined as 
% % follows:
% \begin{equation}\label{EQ:dpd}
%     {\rm DPD}_\alpha(g,f) = 
% % \displaystyle \left\{
% % \begin{array}{ll}
% %     \displaystyle 
% \int  \left[f^{1+\alpha} - \left(1 + 
% \frac{1}{\alpha}\right)  f^\alpha g + \frac{1}{\alpha} g^{1+\alpha}\right] ~
% % & 
% {\rm for} ~\alpha \geq 0.
% % \\
% % 	\displaystyle \int g \log(g/f) & {\rm for} ~\alpha = 0.
% % \end{array}
% % \right.
% \end{equation}
% Note that ${\rm DPD}_0(g, f) = \lim 
% \limits_{\alpha\rightarrow 0}{\rm DPD}_\alpha(g, f)
% = \int g \log(g/f)$
% equals the 
% % gives 
% likelihood disparity and $\alpha=1$ 
% generates the 
% % gives
% $L_2$ distance. 
% In minimum divergence estimation of the unknown 
% parameter based on the density power divergence,
% one minimizes the measure ${\rm DPD}_\alpha(\hat{g},f_\theta)$ 
% over $\theta \in \Theta$ to get an estimate of $\theta$. 
% Here $\hat{g}$ is a nonparametric density estimate of the 
% true density obtained from sample data, $f_\theta$ is 
% the parametric model density, and $\Theta$ 
% is the relevant parameter space. 
% In such an estimation scheme, the 
% last term in the expression of the divergence in (\ref{EQ:dpd})
% is independent of $\theta$. 
% % In the expression of the divergence in (\ref{EQ:dpd}), the 
% % term $\int g^{1+\alpha}$ is independent 
% % of $\theta$; 
% Thus we essentially need to minimize
% \begin{equation}\label{EQ:essential_equation_dpd}
%  \int  f_\theta^{1+\alpha} - \left(1 + \frac{1}{\alpha}\right) 
% \int f_\theta^\alpha \hat{g} =
%  \int  f_\theta^{1+\alpha} - \left(1 + \frac{1}{\alpha}\right) 
% \int f_\theta^\alpha~d\hat{G},
% \end{equation}
% where $\hat{G}$ is the distribution function 
% corresponding to $\hat{g}$. 
% Note that in expression (\ref{EQ:essential_equation_dpd}) the 
% density $\hat{g}$ appears only as a linear term (unlike, say, the objective 
% function of the minimum power divergence functional for any $\lambda$ 
% other than $\lambda = 0$). Thus, given a random sample $X_1,\ldots,X_n$ 
% from the true distribution $G$ we can approximate the expression (\ref{EQ:essential_equation_dpd}) 
% by replacing $\hat{G}$ with its empirical estimate $G_n$. For given $\alpha$, therefore, 
% the minimum density power divergence estimator $\hat{\theta}_\alpha$ of $\theta$ 
% can be obtained by minimizing
% \begin{equation}\label{EQ:dpd_adjusted}
% 	\int  f_\theta^{1+\alpha} - \left(1 + \frac{1}{\alpha}\right) 
% \int f_\theta^\alpha dG_n = \int  f_\theta^{1+\alpha} - \left(1 + 
% \frac{1}{\alpha}\right) \frac{1}{n} \sum_{i=1}^n f_\theta^\alpha 
% (X_i) = \frac{1}{n} \sum_{i=1}^n V_\theta(X_i)
% \end{equation}
% over $\theta \in \Theta$, where $V_\theta(x) = \int  f_\theta^{1+\alpha} - 
% \left(1 + \frac{1}{\alpha}\right) f_\theta^\alpha(x) $. The minimization of 
% the above expression over $\theta$ does not require the use of any nonparametric 
% density estimate. The structure of the objective function in (\ref{EQ:dpd_adjusted})
% also demonstrates that the minimum DPD estimator is an $M$-estimator.
% \section{The $S$-Divergence Family}
% % \subsection{A Generalized Form: The $S$-Divergence}
% In the very recent past
% % Very recently 
% Ghosh et al.~(2013) have developed a generalized family of density-based divergence
% measures as a function of two parameters $\alpha$ and $\lambda$, that serves
% as a superfamily which 
% has been called 
% % is named as 
% the family
% of $S$-divergences.
% % , and it contains both the PD and the DPD families as subfamilies. 
% This family is
% defined as
% \begin{eqnarray}
% 		S_{\alpha, \lambda}(g,f) & = &
% \frac{1}{1+\lambda (1-\alpha)} \int   \left[\left( f^{1+\alpha} - g^{1+\alpha} \right)
% - \frac{(1+\alpha)}{(\alpha-\lambda (1-\alpha))} g^{1 + \lambda
% (1-\alpha)} \left(f^{\alpha-\lambda (1-\alpha)} - g^{\alpha-\lambda 
% (1-\alpha)}\right)\right]  \nonumber
% \\ \nonumber \\
%  &=&  \frac{1}{A} ~ \int ~ f^{1+\alpha}  -   \frac{1+\alpha}{A B}
% ~ \int ~~ f^{B} g^{A}  + \frac{1}{B} ~ \int ~~ g^{1+\alpha},
% ~~~~ \alpha > 0, ~-\infty < \lambda < \infty,\label{EQ:general_form}
% \end{eqnarray}
% where
% \begin{eqnarray}
%     A = 1+\lambda (1-\alpha) ~~ \mbox{and} ~~   
% B = \alpha - \lambda (1-\alpha). \nonumber
% \end{eqnarray}
% Note that, $ A+B=1+\alpha $. When $A$ equals zero, the 
% corresponding divergence measure is defined as the limit of 
% the form in (\ref{EQ:general_form}) as $A \rightarrow 0$ and is
% given by
% \begin{eqnarray}
%     S_{(\alpha,\lambda: A = 0)}(g,f) &=& \lim_{A \rightarrow 0} ~ 
% S_{\alpha, \lambda}(g,f) =  \int f^{1+\alpha} \log\left(\frac{f}{g}\right) - 
% \int \frac{(f^{1+\alpha} - g^{1+\alpha})}{{1+\alpha}}.
% \end{eqnarray}
% Similarly, if $B=0$ then the divergence measure is defined to be
% \begin{eqnarray}
%     S_{(\alpha,\lambda: B = 0)}(g,f) = \lim_{B \rightarrow 0} ~ 
% S_{\alpha, \lambda}(g,f) =  \int g^{1+\alpha} \log\left(\frac{g}{f}\right) - 
% \int \frac{(g^{1+\alpha} - f^{1+\alpha})}{{1+\alpha}}.
% \end{eqnarray}
% 	Note that for $\alpha = 0$, the class of $S$-divergences 
% reduces to the power divergence (PD) family  with parameter $\lambda$
% which is defined as 
% the power divergence between them is defined as 
% \begin{equation}\label{EQ:cressie-read}
% {\rm PD}_\lambda(g,f)=\frac{1}
% {\lambda(\lambda + 1)} \int g \left[\left(\frac{g}{f}\right)^\lambda-1\right],
% %     {\rm PD}_\lambda( r_n, f_\theta) = \frac{1}
% % {\lambda(\lambda + 1)} \sum  r_n \left[ \left
% % ( \frac{ r_n}{f_\theta} \right)^\lambda - 1 \right],
% \end{equation}
% which is indexed by a single real valued tuning parameter $\lambda$.
% This family is a prominent subclass of the $\chi^2$ type distances.
% The class of $\chi^2$ type distances, which are generally referred to 
% as $\phi$-divergences, $f$-divergences, or disparities 
% (Csisz\'{a}r 1963, 1967a,b; Ali and Silvey 1966; Lindsay 1994),  
% dominate the literature in the area of 
% density-based minimum divergence inference. PD 
% family was originally developed by 
% % the Cressie-Read
%  Cressie and Read (1984)
% % family of power divergences. 
% % these authors
%  with the 
% primary aim of multinomial goodness-of-fit testing. 
% Read and Cressie, in their 1988 monograph on the same subject, briefly 
% elaborated on the problem of parametric estimation,
% but their focus remained on goodness-of-fit testing. 
% % Though it was originally 
% % developed with the primary aim of multinomial goodness-of-fit testing
% % later Read and Cressie (1988) briefly elaborated the 
% % problem of parametric estimation, but the focus remained 
% % on goodness-of-fit testing. 
% However, several later authors have successfully used it for 
% robust parametric inference; see e.g. Basu et al.~(2011). 
%  It 
% generates several well-known divergences for different values of $\lambda$. 
% For example 
% % includes 
% % various other known families by putting different values of $\lambda$ as
% $\lambda = 1, 0,-1/2,-1 \mbox{ and } -2$ generate the
% % respectively to get 
% Pearson's chi-square (PCS), the likelihood disparity (LD),
% the Hellinger distance (HD), the Kullback-Leibler divergence (KLD) and the
% Neyman's chi-square (NCS) respectively. 
% However the LD (PD$_0$) and the KLD (PD$_{-1}$) are the limiting expressions 
% of the form in (\ref{EQ:cressie-read}) as $\lambda \rightarrow 0$
% and $-1$ respectively, since the direct substitution of 
% these tuning parameter in (\ref{EQ:cressie-read}) make the 
% divergence undefined.
% On the other hand, $\lambda = 0$ generates the 
% density power divergence (DPD) family
% as a function of $\alpha$
% which was introduced by Basu et al.~(1998)
% to overcome a significant difficulty often experienced by PD family
% under continuous set-up.
% Minimum disparity estimation (Lindsay, 1994) based on the PD family 
% provides a very useful robust technique for 
% discrete parametric models. It is also easily extended 
% to continuous models conceptually. However the practical 
% difficulty in doing so is that the ordinary empirical 
% does not automatically provide a continuous density estimate 
% of the unknown distribution and constructing the divergence 
% becomes a problem for continuous models. 
% % gives a very 
% % suitable method under discrete set-up and it can be easily
% % extended to continuous set-up. The main problem using it under continuous
% % set-up is to use kernel density estimation. To overcome this problem,
% As an alternative
% Basu et al.~(1998) 
% considered minimum divergence estimators based on the 
% % introduced 
% Density Power Divergence family which 
% is defined as 
% % follows:
% \begin{equation}\label{EQ:dpd}
%     {\rm DPD}_\alpha(g,f) = 
% % \displaystyle \left\{
% % \begin{array}{ll}
% %     \displaystyle 
% \int  \left[f^{1+\alpha} - \left(1 + 
% \frac{1}{\alpha}\right)  f^\alpha g + \frac{1}{\alpha} g^{1+\alpha}\right] ~
% % & 
% {\rm for} ~\alpha \geq 0.
% % \\
% % 	\displaystyle \int g \log(g/f) & {\rm for} ~\alpha = 0.
% % \end{array}
% % \right.
% \end{equation}
% Note that ${\rm DPD}_0(g, f) = \lim 
% \limits_{\alpha\rightarrow 0}{\rm DPD}_\alpha(g, f)
% = \int g \log(g/f)$
% equals the 
% % gives 
% likelihood disparity and $\alpha=1$ 
% generates the 
% % gives
% $L_2$ distance. 
% In minimum divergence estimation of the unknown 
% parameter based on the density power divergence,
% one minimizes the measure ${\rm DPD}_\alpha(\hat{g},f_\theta)$ 
% over $\theta \in \Theta$ to get an estimate of $\theta$. 
% Here $\hat{g}$ is a nonparametric density estimate of the 
% true density obtained from sample data, $f_\theta$ is 
% the parametric model density, and $\Theta$ 
% is the relevant parameter space. 
% In such an estimation scheme, the 
% last term in the expression of the divergence in (\ref{EQ:dpd})
% is independent of $\theta$. 
% % In the expression of the divergence in (\ref{EQ:dpd}), the 
% % term $\int g^{1+\alpha}$ is independent 
% % of $\theta$; 
% Thus we essentially need to minimize
% \begin{equation}\label{EQ:essential_equation_dpd}
%  \int  f_\theta^{1+\alpha} - \left(1 + \frac{1}{\alpha}\right) 
% \int f_\theta^\alpha \hat{g} =
%  \int  f_\theta^{1+\alpha} - \left(1 + \frac{1}{\alpha}\right) 
% \int f_\theta^\alpha~d\hat{G},
% \end{equation}
% where $\hat{G}$ is the distribution function 
% corresponding to $\hat{g}$. 
% Note that in expression (\ref{EQ:essential_equation_dpd}) the 
% density $\hat{g}$ appears only as a linear term (unlike, say, the objective 
% function of the minimum power divergence functional for any $\lambda$ 
% other than $\lambda = 0$). Thus, given a random sample $X_1,\ldots,X_n$ 
% from the true distribution $G$ we can approximate the expression (\ref{EQ:essential_equation_dpd}) 
% by replacing $\hat{G}$ with its empirical estimate $G_n$. For given $\alpha$, therefore, 
% the minimum density power divergence estimator $\hat{\theta}_\alpha$ of $\theta$ 
% can be obtained by minimizing
% \begin{equation}\label{EQ:dpd_adjusted}
% 	\int  f_\theta^{1+\alpha} - \left(1 + \frac{1}{\alpha}\right) 
% \int f_\theta^\alpha dG_n = \int  f_\theta^{1+\alpha} - \left(1 + 
% \frac{1}{\alpha}\right) \frac{1}{n} \sum_{i=1}^n f_\theta^\alpha 
% (X_i) = \frac{1}{n} \sum_{i=1}^n V_\theta(X_i)
% \end{equation}
% over $\theta \in \Theta$, where $V_\theta(x) = \int  f_\theta^{1+\alpha} - 
% \left(1 + \frac{1}{\alpha}\right) f_\theta^\alpha(x) $. The minimization of 
% the above expression over $\theta$ does not require the use of any nonparametric 
% density estimate. The structure of the objective function in (\ref{EQ:dpd_adjusted})
% also demonstrates that the minimum DPD estimator is an $M$-estimator.
% For $\alpha =1$,
% $S_{1, \lambda}$ equals the $L_2$ divergence irrespective of the value 
% of $\lambda$. 
% % , while $\lambda = -1/2$ generates 
% % the family in (\ref{EQ:generalized_hellinger})
% % .
\section{The Logarithmic Power Divergence (LPD) and the Logarithmic 
Density Power Divergence (LDPD) Families}\label{SEC:LPD_LDPD}
Jones et al.~(2001) described 
a class of divergence measures
% several related versions of 
% the DPD measure, 
which 
% also 
do not require 
any nonparametric smoothing techniques for their construction. 
% A 
% class of divergences of this 
This 
% type 
% is the 
family of divergences is given by 
\begin{equation}\label{EQ:LDPD}
{\rm LDPD}_\beta(g,f) = \displaystyle \log \int  f^{1+\beta} - 
\left(1 + \frac{1}{\beta}\right)  
\log \int f^\beta g + \frac{1}{\beta} \log \int g^{1+\beta},
\end{equation}
$\beta\geq0$,
where LDPD stands for logarithmic density power divergence. 
% Once again 
For the case $\beta=0$ we define 
$${\rm LDPD}_0(g,f)=
\lim_{\beta\rightarrow0} {\rm LDPD}_\beta(g,f)=
\lim_{\lambda\rightarrow0} {\rm PD}_\lambda(g,f)={\rm PD}_0(g,f),$$
where the power divergence (PD) measure has the form
% with PD has the form
\begin{equation}\label{EQ:cressie-read}
{\rm PD}_\lambda(g,f)=\frac{1}
{\lambda(\lambda + 1)} \int g \left[\left(\frac{g}{f}\right)^\lambda-1\right],
~~~~ -\infty<\lambda<\infty.
%     {\rm PD}_\lambda( r_n, f_\theta) = \frac{1}
% {\lambda(\lambda + 1)} \sum  r_n \left[ \left
% ( \frac{ r_n}{f_\theta} \right)^\lambda - 1 \right],
\end{equation}
% \paragraph{}
The family of divergences in (\ref{EQ:LDPD})
% This divergence 
is similar to the density power divergence family given by
% in that it may be recovered from the latter by replacing 
% the identity function of the DPD by the logarithmic function,
% since DPD has the form
 \begin{equation}\label{EQ:dpd}
    {\rm DPD}_\alpha(g,f) = 
% \displaystyle \left\{
% \begin{array}{ll}
%     \displaystyle 
\int  \left[f^{1+\alpha} - \left(1 + 
\frac{1}{\alpha}\right)  f^\alpha g + \frac{1}{\alpha} g^{1+\alpha}\right] ~
% & 
{\rm for} ~\alpha \geq 0.
% \\
% 	\displaystyle \int g \log(g/f) & {\rm for} ~\alpha = 0.
% \end{array}
% \right.
\end{equation}
The LDPD family may be recovered from the DPD by replacing the 
identity function with the logarithm function. 
In spite of the similarity between the forms of the 
DPD and the LDPD families, Jones et al.~(2001) had originally 
developed the latter as a special case of an estimation 
method proposed by Windham (1995). 
% This divergence is very close to density power divergence
% where only difference is the logarithmic function is 
% interchanged with identity function. 
Following the connection shown
between PD and DPD by Patra et al.~(2013), 
% very recently 
Maji, Chakraborty and Basu (2014) have recently shown that the same sort of connection exists
between LDPD and the logarithmic power divergence (LPD) family.
To show this connection, the LDPD measure can be written as 
\begin{eqnarray}\label{EQ:LDPD_log}
{\rm LDPD}_\beta(g,f) &=& \displaystyle \log \int  f^{1+\beta} - \left(1 + \frac{1}{\beta}\right)  
\log \int \left(\frac{g}{f}\right) f^{1+\beta} \\ \nonumber
& & + \frac{1}{\beta} \log \int \left(\frac{g}{f}\right)^{1+\beta} f^{1+\beta}.
\end{eqnarray}
Replacing the $f^{1+\beta}$ term with $f$ 
in each of the three terms on the right hand side of (\ref{EQ:LDPD_log}) 
leads
to the density-based divergence $\frac{1}{\beta} \log \int \frac{g^{1+\beta}}{f^\beta}.$
After standardizing this further we express this divergence as 
 $\frac{1}{\beta(\beta+1)} \log \int \frac{g^{1+\beta}}{f^\beta}.$ This family of
divergences will be called the logarithmic power divergence family 
by Maji, Chakraborty and Basu (2014). Using a
different symbol for the tuning parameter, this
family has the form
\begin{equation}
\label{EQ:LPD}
{\rm LPD}_{\gamma}(g,f)=\frac{1}{\gamma(\gamma+1)} \log \int 
\frac{g^{1+\gamma}}{f^\gamma} , \gamma \in \mathbb{R}.
\end{equation}
The limiting forms as $\gamma \rightarrow 0$ and $\gamma \rightarrow -1$
generate, respectively, the likelihood disparity LD (or PD$_{0}$) and the 
Kullback-Leibler divergence KLD (or PD$_{-1}$). 
These divergences have the form 
\begin{equation}
 {\rm LD}(g,f)=  \int g \log \left(\frac{g}{f}\right),
\end{equation}
\begin{equation}
 {\rm KLD}(g,f)=  \int f \log \left(\frac{f}{g}\right).
\end{equation}
For any other value 
of $\gamma$ the LPD measure can be seen to be a function of the PD 
measure at the same value of $\gamma$. Specifically,
\begin{equation}
 \label{EQ:pd_lpd}
{\rm LPD}_\gamma(g,f)=\frac{1}{\gamma(\gamma+1)} \log 
\left[\gamma(\gamma+1){\rm PD}_\gamma(g,f)+1\right].
\end{equation}
Apart from being 
briefly considered by
% the 
Renyi as a measure of the amount of information
% divergence 
(Renyi, 1961), 
the LPD family
% it 
is a member 
of the ($h,\phi$) divergence family (eg. Pardo, $2006$),
where 
$h(x)=\frac{1}{r(r+1)} [r(r+1)\log x+1]$. 
% \paragraph{}
% Just as Ghosh et al.~(2013) have shown that there exists a superfamily 
% which contains both the PD and DPD families, 
% one can develop a similar superfamily encompassing the 
% % the same idea is developed here for 
% LPD and the LDPD divergences. 
% This superfamily is defined in the next section.
% % The next section generates this superfamily.
% % Thus we get a family called logarithmic super 
% % divergence family which is described in next section. 
\section{The Logarithmic Super Divergence}
\label{SEC:lsd_defn}
% \paragraph{}
We now define the Logarithmic Super Divergence 
(or Logarithmic \\ $S$-Divergence)
introduced in Maji, Ghosh and Basu (2014)
and establish 
the asymptotic properties of the procedures resulting from it. 
% its properties. 
The
% and discuss its properties and uses in statistical inference. The
Logarithmic $S$-Divergence (LSD) is defined (Maji, Ghosh and Basu, 2014) as
\begin{eqnarray}
\label{lsd_form}
		{\rm LSD}_{\beta, \gamma}(g,f) 
% &=& \frac{1}{1+\lambda (1-\beta)} ~ \int ~  
% \left[\left( f^{1+\beta} - g^{1+\beta} \right) - 
% \frac{(1+\beta)}{(\beta-\lambda (1-\beta))} 
% g^{1 + \lambda (1-\beta)} \left(f^{\beta-\lambda (1-\beta)} - 
% g^{\beta-\lambda (1-\beta)}\right)\right]  \nonumber
% \\
% \nonumber \\
% \nonumber \\
 &=&  \frac{1}{A} ~ \log \int f^{1+\beta}  -   \frac{1+\beta}{A B} ~ 
\log \int f^{B} g^{A} \nonumber \\ 
& & + \frac{1}{B} ~ \log \int g^{1+\beta},
~~~~ \beta > 0, ~-\infty < \gamma < \infty,
\end{eqnarray}
where
\begin{eqnarray}
    A &=& 1+\gamma (1-\beta) \nonumber \\ 
\mbox{           and               }
    B &=& \beta - \gamma (1-\beta). \nonumber
\end{eqnarray}
% It has to be noted that, $ A+B=1+\beta $.
For $\beta=0~(A=1+\gamma,B=-\gamma),$ this family 
coincides with the 
% becomes same as 
logarithmic power divergence family of (\ref{EQ:LPD}) with parameter $\gamma$, 
while $\gamma=0$ gives the logarithmic density power divergence family
in (\ref{EQ:LDPD}) with parameter $\beta$. 
\paragraph{}
Replacing the logarithmic function with the identity function in (\ref{lsd_form})
generates the divergence
% we get 
\begin{eqnarray}
		S_{\alpha, \lambda}(g,f)
%  & = &
% \frac{1}{1+\lambda (1-\alpha)} \int   \left[\left( f^{1+\alpha} - g^{1+\alpha} \right)
% - \frac{(1+\alpha)}{(\alpha-\lambda (1-\alpha))} g^{1 + \lambda
% (1-\alpha)} \left(f^{\alpha-\lambda (1-\alpha)} - g^{\alpha-\lambda 
% (1-\alpha)}\right)\right]  \nonumber
% \\ \nonumber \\
 &=&  \frac{1}{A} ~ \int ~ f^{1+\alpha}  -   \frac{1+\alpha}{A B}
~ \int ~~ f^{B} g^{A}  + \frac{1}{B} ~ \int ~~ g^{1+\alpha},
\label{EQ:general_form}
\end{eqnarray}
with $\beta$ and $\gamma$ being replaced by $\alpha$ and $\lambda$
respectively. 
This divergence is known as the $S$-divergence and has been introduced by Ghosh et al.~(2013).
% \begin{theorem} 
% 	Given two densities $g$ and $f$, the measure 
% ${\rm LSD}_{\beta, \gamma}(g,f) $ represents a genuine 
% statistical divergence for all $\beta \ge 0$ and $\gamma \in \mathbb{R}$.
% 
% \begin{proof}
% 
%  A simple application of Holder's inequality establishes the above result. 
% \end{proof}
% \end{theorem}
% 
\subsection{Estimating Equation}
\label{SEC:lsd_est_eqn}
Consider a parametric class of model densities 
$\{ f_\theta : \theta \in \Theta \subseteq {\mathbb{R}}^p \}$
and 
suppose that our interest is in estimating 
% we want to estimate 
$\theta$. Let $G$ denote the distribution function corresponding
to the true density $g$. The minimum 
LSD
% logarithmic super divergence 
functional
$T_{\beta,\gamma}(G)$ at $G$ is defined as 
\begin{equation}
 {\rm LSD}_{\beta, \gamma}\left(g,f_{T_{\beta,\gamma}(G)}\right)
= \min \limits_{\theta \in \Theta } {\rm LSD}_{\beta, \gamma}(g,f_\theta).
\end{equation}
It takes the value $\theta$ when the true density $g = f_\theta$ is in the model; when 
it does not, $\theta_{\beta,\gamma}^g = T_{\beta,\gamma}(G)$ represents 
the best fitting parameter, and $f_{\theta^g}$ is the model element closest 
to $g$ in terms of logarithmic super divergence. For simplicity of 
 notation, we suppress the subscript $\beta,\gamma$ in $\theta_{\beta,\gamma}^g$.
% 
% $$ $$
A simple differentiation gives us the estimating equation 
for $\theta$, which is 
% as
\begin{equation}
\label{EQ:lsd_est_eqn}
 \frac{\int f_\theta^{1+\beta}  u_\theta}{\int f_\theta^{1+\beta}} = 
\frac{\int f_\theta^{B} g^A  u_\theta}{\int f_\theta^{B} g^A }.
\end{equation}
% which is also helpful to estimate $\theta$. 
For $\beta=0~(A=1+\gamma,B=-\gamma)$, the equation thus becomes the same as the estimating 
equation of the logarithmic power divergence family with parameter $\gamma$. 
For $\gamma=0~(A=1, B=\beta)$, on the other hand, it is the estimating 
equation for the LDPD measure. 
Here $u_\theta(x)=\frac{\partial}{\partial \theta} \log f_\theta(x)$ 
is the likelihood score function. 
% \subsection{Influence Function}
% Consider the minimum LSD functional $T_{\beta,\gamma}(G)$. The 
% value $\theta=T_{\beta,\gamma}(G)$ solves the equation (\ref{EQ:lsd_est_eqn}).
% Consider the estimating equation 
% at the mixture contamination 
% % corresponding to the true 
% density 
% $(1-\epsilon)~g(x)+\epsilon~I(x=y)$ where $I(x=y)$ is the indicator function at $y$. 
% Let $\theta_\epsilon$ be the corresponding functional which solves the estimating 
% equation in this case. Taking a derivative of both sides of this estimating 
% equation and evaluating at $\epsilon=0$, 
% % one gets 
% the 
% influence function is found to be:
% % as follows:
% % Influence function is found as follows:
% \begin{equation}
% IF(y,T,G)=A J^{-1}(\theta) {b}(\theta),
% %  IF=\frac{A\left[\int f_\theta^{1+\alpha}u_\theta~\int f_\theta^B g^A - 
% % f_\theta^B(y) g^{A-1}(y)~\int f_\theta^{1+\alpha}u_\theta
% % -\int f_\theta^{1+\alpha} \int f_\theta^B g^A u_\theta + 
% % f_\theta^B(y) g^{A-1}(y) u_\theta(y) \int f_\theta^{1+\alpha} \right]}{J(\theta)},
% \end{equation}
% where 
% $\theta=T(G)$,
% % \begin{dmath}
% \begin{eqnarray}
%  J(\theta)&=&(1+\beta) \int f_\theta^{1+\beta} u_\theta u_\theta^T 
% \int f_\theta^B g^A -\int f_\theta^{1+\beta} i_\theta \int f_\theta^B g^A 
% -A \int f_\theta^{1+\beta}u_\theta \int f_\theta^B g^A u_\theta \nonumber \\
% & &-B \int f_\theta^{1+\beta} \int f_\theta^B g^A u_\theta u_\theta^T  
% + \int f_\theta^{1+\beta} \int f_\theta^B g^A i_\theta, 
% \end{eqnarray}    
% %       \end{dmath}
% where $i_\theta=-\nabla u_\theta$
% and  
% % \begin{dmath}
% \begin{eqnarray}
%      {b}(\theta) &=& \left(\int f_\theta^{1+\beta}u_\theta~\int f_\theta^B g^A - 
% f_\theta^B(y) g^{A-1}(y)~\int f_\theta^{1+\beta}u_\theta \right) \nonumber \\
% & &-\left(\int f_\theta^{1+\beta} \int f_\theta^B g^A u_\theta + 
% f_\theta^B(y) g^{A-1}(y) u_\theta(y) \int f_\theta^{1+\beta} \right),
% \end{eqnarray}
% %      \end{dmath}
% when the model holds, so that $g=f_\theta$ for some $\theta$, 
% % At the model $g=f_\theta$, 
% the influence function becomes,
% \begin{eqnarray}
%  IF(y,T,F_\theta)&=& \left({\int f_\theta^{1+\beta}
% u_\theta u_\theta^T ~\int f_\theta^{1+\beta} -
% \left[\int f_\theta^{1+\beta}~u_\theta\right] 
% \left[\int f_\theta^{1+\beta}~u_\theta\right]^T}\right)^{-1} \nonumber \\
% & &\left({f_\theta^\beta(y)~\left[u_\theta(y) \int f_\theta^{1+\beta} 
% -\int f_\theta^{1+\beta} u_\theta\right]}\right).
% \end{eqnarray}
% The remarkable observation here is that the influence function at the model 
% is independent of $\gamma$ and depends only on $\beta$. 
% % From the Figure \ref{fig:lsd_if_colour}
% it is clear that the first order influence function is unbounded 
% for $\beta=0$ whereas for other values of $\beta$ the function supposed 
% to be bounded which is a limitation of first order influence function 
% and this fact will be more clearer through analysis of higher 
% order influence function described in the next section. 
% \begin{figure}
% \centering
% \includegraphics[width=130mm, height=100mm] {if_lsd.jpg}
% \caption{Influence function for $N(\theta,1)$ Distribution}
%  %{fig:p3}
% \label{fig:lsd_if1}
% \end{figure}
% \begin{figure}
% \centering
% \includegraphics[width=130mm, height=100mm] {if_lsd.png}
% \caption{Influence function for $N(\theta,1)$ Distribution}
% %  {fig:p3}
% \label{fig:lsd_if_colour}
% \end{figure}

\section{Asymptotic Properties of the Minimum LSD Estimators in 
Discrete Models}
\label{SEC:lsd_properties}
Under the parametric set-up of Section {\ref{SEC:lsd_est_eqn}},
consider a discrete family of distributions. We will use 
the term ``density function'' generally for the sake of a 
unified notation, irrespective of whether the distribution is discrete
or continuous. Let $X_1,\ldots,X_n$ be a random sample from the 
true distribution having density function $g$. Representing 
the logarithmic $S$-divergence in terms of the parameter $\beta$ and $\gamma$
(as given in Section \ref{SEC:lsd_defn}), let 
$\hat{\theta}_{\beta,\gamma}$ be the estimator obtained by minimizing 
${\rm LSD}_{\beta,\gamma}(\hat{g},f_\theta)$ over $\theta \in \Theta$, where 
$\hat{g}$ is a suitable nonparametric density estimate of $g$; 
in the discrete case the vector of relative frequencies 
based on the sample data is the canonical choice for $\hat{g}$. 
For similar theoretical properties of the minimum $S$-divergence estimator, see Ghosh (2014).
% \paragraph{}
% The 
% proofs of the mathematical results considered in this paper are 
% available in Maji, Ghosh and Basu (2014).
% similar
% in spirit to those of Ghosh et al.~(2013) and Ghosh (2013). 
% In order to keep this paper focused on applications, we skip the proofs
% wherever they are substantially similar to the above references.  
% \subsection{Asymptotic Properties}
\subsection{Asymptotic Properties} 
	Suppose $X_1,X_2,\ldots,X_n$ are $n$ independent 
and identically distributed observations from a discrete 
distribution $G$ modeled by 
$\mathcal{F} = \{ F_{\theta} : \theta \in \Theta \subseteq \mathbb{R}^p \}$ and 
let the distribution have support $\mathcal{X} = \{ 0,1,2,\ldots \}$ without loss
of generality. Denote the 
relative frequency at $x$ as obtained from the data by $r_n(x) = \frac{1}{n} 
\sum \limits_{i=1}^n \chi(X_i = x)$ where $\chi(A)$ is the 
indicator of $A$. 
% Note 
% that, the 
The minimum LSD estimator 
is then obtained 
% has to be obtained 
by minimizing the LSD 
measure between the data 
density $r_n$ and the model density $f_\theta$
with respect to $\theta$.
% and the model distribution. However, in the discrete set-up, 
% both the data-generating true distribution and the model distribution are 
% characterized by the probability vectors 
% $\mathbf{r}_n = ( r_n(0), r_n(1), \cdots )^T$ and 
% $\mathbf{f}_\theta = ( f_\theta(0), f_\theta(1), \cdots )^T$ respectively. Thus 
% in this discrete set-up, the minimum LSD estimator of $\theta$ can be obtained 
% by just minimizing ${\rm LSD}_{\beta,\gamma}(r_n,f_{\theta})$, the LSD measure 
% between $\mathbf{r}_n$ and $\mathbf{f}_\theta$, with respect to $\theta$.

% 	Note that we need to minimize 
Minimizing
${\rm LSD}_{\beta,\gamma}(r_n,f_{\theta})$ 
with respect to $\theta$ is equivalent to
minimizing
% or equivalently 
$H_n(\theta)$ where
\begin{eqnarray}
    H_n(\theta) = \frac{1}{1+\beta} \left[ \frac{1}{A} \log \sum_x f_{\theta}^{1+\beta}(x) 
- \frac{1+\beta}{AB} \log \sum_x f_{\theta}^B(x) r_n^A(x) \right].
\end{eqnarray}
	Now,
\begin{eqnarray}
    \nabla H_n(\theta) &=& \frac{1}{A} 
\left[\frac{\sum_x f_{\theta}^{1+\beta}(x) u_\theta(x)}{\sum_x f_{\theta}^{1+\beta}(x)} -
\frac{\sum_x f_{\theta}^B(x) r_n^A(x) u_\theta(x)}{\sum_x f_{\theta}^B(x) r_n^A(x)}\right],
% \nonumber \\
% &=&  -\frac{1}{A} \sum_x K(\delta_n(x)) f_{\theta}^{1+\beta}(x) w_{\theta}(x)
% \left[ \sum_x 
% f_{\theta}^{1+\beta}(x)u_{\theta}(x) \sum_x f_{\theta}^{B}(x) r_n^A(x)-  \sum_x f_{\theta}^B(x) 
% u_{\theta}(x) r_n^A(x) \sum_x f_{\theta}^{1+\beta}(x) \right]  \nonumber \\
\end{eqnarray}
where $\nabla$ is 
% defined as 
the gradient with respect to $\theta$.
Equating 
the above
% this 
to zero, the
% and the 
% where $\delta_n(x) = \frac{r_n(x)}{f_{\theta}(x)}$. 
% The 
estimating equation becomes 
\begin{eqnarray}
    \sum_x K(\delta_n(x)) f_{\theta}^{1+\beta}(x) w_{\theta}(x) = 0,
\end{eqnarray}
where $\delta_n(x) = \frac{r_n(x)}{f_{\theta}(x)},~~
\delta_g(x) = \frac{g(x)}{f_{\theta}(x)},~~
w_{\theta}(x)=[B(\theta)u_\theta(x)-A(\theta)],~~
K(\delta)=\delta^A-1,~~ A(\theta)=\sum_x 
f_{\theta}^{1+\beta}(x)u_{\theta}(x) \\ \mbox{ and } 
B(\theta)=\sum_x 
f_{\theta}^{1+\beta}(x).$

	Define 
% \begin{eqnarray}
%     J_g &=& E_g \left[ w_{\theta^g}(X)w_{\theta^g}^T(X) K'(\delta_g^g(X))
%  f_{\theta^g}^\beta(X) \right] - \sum_x  K(\delta_g^g(X)) \nabla_2 f_{\theta^g}(x). \\
%     V_g &=& V_g \left[ K'(\delta_g^g(X)) 
% f_{\theta^g}^\beta(X) w_{\theta^g}(X) \right].
% \end{eqnarray}
\begin{eqnarray}
J_g &=& E_g \left[ w_{\theta^g}(X)u_{\theta^g}^T(X) K'(\delta_g^g(X))
 f_{\theta^g}^{\beta}(X) \right] \nonumber \\
 & & -  \sum_x  K(\delta_g^g(x)) 
f_{\theta}^{1+\beta}(x) \nabla w_{\theta^g}(x) \nonumber \\
& & - (1+\beta) \sum_x  K(\delta_g^g(X)) 
f_{\theta}^{1+\beta}(x) w_{\theta^g}(x) u_{\theta^g}(x) . \\
    V_g &=& V_g \left[ K'(\delta_g^g(X)) 
f_{\theta^g}^\beta(X) w_{\theta^g}(X) \right].
\end{eqnarray}
% \\
We now present the necessary assumptions for our asymptotic results.  

% \textbf{Assumptions:}
\begin{enumerate}
    \item The model family ${F_\theta}$ is identifiable which 
means that different values of the parameter must generate 
different probability distributions of the observable variables.
	 \item The probability density function $f_\theta$ of 
the model distribution have common support so that the 
set $\mathcal{X} = \{ x : f_{\theta}(x) >0 \}$ is independent of $\theta$.
			Also the true distribution $g$ is 
compatible with the model family.
	 \item There exists an open subset $\omega \subset \Theta$
 for which the best fitting parameter $\theta^g$ is an interior 
point and for almost all $x$, the density $f_{\theta}(x)$ admits 
all third derivatives of the type $\nabla_{jkl} f_{\theta}(x)$   $\mbox{ for all } \theta \in \omega$. 
Here the subscripts $j,k,l$ of $\nabla$ represent the indicated 
partial derivatives. 
		\item The matrix $\frac{1+\beta}{A} J_g$ is positive definite.
		\item The quantities 
% \\ 
$\sum_x g^{1/2}(x) 
f_{\theta}^{\beta}(x) |u_{j\theta}(x)|$, 
$\sum_x g^{1/2}(x) 
f_{\theta}^{\beta}(x) |w_{j\theta}(x)|$, \\
$\sum_x g^{1/2}(x) 
f_{\theta}^{\beta}(x) |u_{j\theta}(x)| |u_{k\theta}(x)|$ and 

$\sum_x g^{1/2}(x) f_{\theta}^{\beta}(x) |u_{jk\theta}(x)|$  
are bounded $\mbox{ for all } j, k$ and $\mbox{ for all } \theta \in \omega$.

		\item For almost all $x$, there exists functions 
$M_{jkl}(x)$, $M_{jk,l}(x)$, $M_{j,k,l}(x)$,
possibly depending on $\beta$ that dominate, in absolute 
value, 
% \\ 
$f_{\theta}^{\beta}(x) u_{jkl\theta}(x)$, $f_{\theta}^{\beta}(x) 
u_{jk\theta}(x) u_{l\theta}(x)$ and $f_{\theta}^{\beta}(x) 
u_{j\theta}(x) u_{k\theta}(x) u_{l\theta}(x)$ respectively $ \mbox{ for all } j, k, l$ and 
that are uniformly bounded in expectation with respect to $g$ and $f_{\theta} 
~~ \mbox{for all } \theta \in \omega$, where 
the subscripts of $u$ denote the indicated partial 
derivatives of the score function.
		\item The function $\left( \frac{g(x)}{f_{\theta}(x)} \right)^{A-1}$ is 
uniformly bounded (by, say, $C$) $\mbox{ for all } \theta \in \omega$.
%  \\

\end{enumerate}
	To prove the consistency and asymptotic normality of 
the minimum LSD estimator, we will assume, for the rest of the paper, that the 
seven conditions stated above are satisfied.
% above $7$ 
% conditions hold. 
% We will first consider some Lemmas. 
\paragraph{}
Based on the above assumptions we now 
start the proofs of the required results. 
The proofs are primarily along the 
lines of Lindsay (1994) and Basu et al.~(2011).
Define $ \eta_n(x) = \sqrt n (\sqrt
\delta_n-\sqrt\delta_g)^2$.
% \\
We then have the following lemmas. 
\begin{lemma}
\label{lem:lamma1}
  	 For any $k \in [0,2]$, we have
\begin{enumerate}
    \item $E[\eta_n^k(x)] \le n^{\frac{k}{2}}E[|\delta_n(X) - 
\delta_g(X)|]^k \le \left[ \frac{g(x)(1-g(x))}{f_{\theta}^2(x)}\right]^{\frac{k}{2}}.$ 
% \\
	 \item   $E[|\delta_n(X) - \delta_g(X)|] \le \frac{2g(x)(1-g(x))}{f_{\theta}(x)}.$
\end{enumerate}
\begin{proof}
%  \textbf{Proof :} 
For $a,b \ge 0$, we have the inequality $(\sqrt a - \sqrt b)^2 \le |a-b|$. So we get
\begin{eqnarray}
    E[\eta_n^k(x)] &=& n^{\frac{k}{2}} E[(\sqrt
\delta_n-\sqrt\delta_g)^2]^k \nonumber \\
		& \le & n^{\frac{k}{2}} E[|
\delta_n-\delta_g |]^k. \nonumber 
\end{eqnarray}

	For the next part see that, $ n r_n(x) \sim Bin( n, g(x) ) ~~ \mbox{ for all } x$. Now, 
for any  $k \in [0,2]$, we get by the Lyapunov's inequality that 

\begin{eqnarray}
    E[|\delta_n(X) - \delta_g(X)|]^k &\le & \left[ E(\delta_n(X) - \delta_g(X))^2 \right]^{\frac{k}{2}} \nonumber \\					&=& \frac{1}{f_{\theta}^k(x)} \left[ E(r_n(X) - g(X))^2 \right]^{\frac{k}{2}} \nonumber 
\\		&=& \frac{1}{f_{\theta}^k(x)} \left[ \frac{g(x)(1-g(x))}{n}\right]^{\frac{k}{2}}. \nonumber 
\\ \nonumber
\end{eqnarray}

	For the second part, note that
\begin{eqnarray}
    E[|\delta_n(X) - \delta_g(X)|] &=& \frac{1}{f_{\theta}^k(x)} \left[ E|r_n(X) - g(X)| \right]^{\frac{k}{2}} \nonumber 
\\		&\le& \frac{2g(x)(1-g(x))}{f_{\theta}(x)}. \nonumber 
\\ \nonumber
\end{eqnarray}
where the last inequality follows from the result about the mean-deviation of a Binomial random variable.
\end{proof}

\end{lemma}

\begin{lemma}
\label{lem:lamma2}
	$E[\eta_n^k(x)] \rightarrow 0$, as $n \rightarrow \infty$,  for $k \in [0,2).$
\begin{proof}
  This follows from Theorem 4.5.2 of Chung (1974) by noting that \\ 
$n^{1/4} (r_n^{1/2}(x) - g^{1/2}(x)) \rightarrow 0$ 
with probability one for each $x \in \mathcal{X}$ and by the Lemma \ref{lem:lamma1}(1), $sup_n E[\eta_n^k(x)] $ is bounded.
\end{proof}

\end{lemma}
% \noindent
% \textbf{Proof :} 
% \begin{flushright}$\square$\end{flushright} 
% 
	Let us now define, $a_n(x) = K(\delta_n(x)) - K(\delta_g(x))$ and 
$b_n(x) = (\delta_n(x)-\delta_g(x))K'(\delta_g(x))$.
% \\

 We will need the limiting distributions of 
% \\ 
% \begin{center}
$$S_{1n} = \sqrt n \sum_x a_n(x)f_{\theta}^{1+\beta}(x)w_{\theta}(x)$$
and $$S_{2n} = \sqrt n \sum_x 
b_n(x)f_{\theta}^{1+\beta}(x)w_{\theta}(x).$$
% \end{center}

 Define $\tau_n(x) = \sqrt n |a_n(x) - b_n(x)|$. 
% \\

\begin{lemma}
\label{lem:lamma3}
	Suppose Assumption 5 holds. Then $E|S_{1n}-S_{2n}| \rightarrow 0 $ as $n \rightarrow \infty$,
%  \\
and hence $S_{1n}-S_{2n} \xrightarrow{P} 0 $ as $n \rightarrow \infty.$
\begin{proof}
 	By Lemma 2.15 of Basu et ~al.~(2011) [or, Lindsay(1994), Lemma 25], there 
exists some positive constant $\beta$ such that 
$$
\tau_n(x) \le \beta \sqrt n (\sqrt\delta_n-\sqrt\delta_g)^2 = \beta \eta_n(x).
$$ 
Also, by Lemma \ref{lem:lamma1}, $ E[\tau_n(x)] \le \beta \frac{g^{1/2}(x)}{f_{\theta}(x)}$.\\
And by Lemma \ref{lem:lamma2}, $E[\tau_n(x)] = \beta E[\eta_n(x)] \rightarrow 0$ as $n \rightarrow \infty$. Thus we get, 
\begin{eqnarray}
    E|S_{1n}-S_{2n}| &\le& \sum_x E[\tau_n(x)] f_{\theta}^{1+\beta}(x)|w_{\theta}(x)| \nonumber \\
                     &\le&  \beta \sum_x g^{1/2}(x) f_{\theta}^{\beta}(x)|w_{\theta}(x)| \nonumber \\
							& < &   
\infty ~~~~~~ \mbox{ (by Assumption $5$)}. \nonumber \\ \nonumber
\end{eqnarray}
So, by the Dominated Convergence Theorem (DCT),  $E|S_{1n}-S_{2n}| \rightarrow 0 $ as $n \rightarrow \infty$.
% \\
Hence, by Markov's inequality, $S_{1n}-S_{2n} \xrightarrow{P} 0 $ as $n \rightarrow \infty$.
\end{proof}

\end{lemma}
% \noindent
% \textbf{Proof :}
% \begin{flushright}$\square$\end{flushright} 
% 
% 
\begin{lemma}
\label{lem:lamma4}
	Suppose $V_g =  V_g \left[ K'(\delta_g(X)) f_{\theta}^\beta(X) w_{\theta}(X) \right] $ is finite. Then 
$$
S_{1n} \rightarrow N(0, V_g).
$$
\begin{proof}
 	By Lemma \ref{lem:lamma3}, the asymptotic distribution of 
$S_{1n}$ and $S_{2n}$ are the same. Now, we have
\begin{eqnarray}
    S_{2n} &=& \sqrt n \sum_x (\delta_n(x) - \delta_g(x)) 
K'(\delta_g(x)) f_{\theta}^{1+\beta}(x) w_{\theta}(x) \nonumber \\
				&=&  \sqrt n \sum_x (r_n(x) - 
g(x)) K'(\delta_g(x)) f_{\theta}^{\beta}(x) w_{\theta}(x) \nonumber \\
			&=&  \sqrt n \left( \frac{1}{n} 
\sum_{i=1}^n \left[ K'(\delta_g(X_i)) f_{\theta}^{\beta}(X_i) w_{\theta}(X_i) - 
E_g\{ K'(\delta_g(X))f_{\theta}^{\beta}(X)w_{\theta}(X)\} \right]\right) \nonumber \\
			&\rightarrow & N( 0, V_g) ~~~~~ \mbox{(by the
Central Limit Theorem)}. \nonumber \\ \nonumber 
\end{eqnarray}
% Hence Proved.
\end{proof}

\end{lemma}
% \noindent
% \textbf{Proof :} \begin{flushright}$\square$\end{flushright} 
% 
	We will now consider the final theorem of this section 
about the consistency and asymptotic normality of the minimum LSD estimator.
% \\
% \begin{theopargself}
\begin{theorem}
\label{th:est_th}
	Under Assumptions $1-7$, there exists a consistent 
sequence $\theta_n$ of roots to the minimum LSD estimating equation (\ref{EQ:lsd_est_eqn}).
% \\
	Also, the asymptotic distribution of 
$\sqrt n (\theta_n - \theta_g)$ is $p-$dimensional normal 
with mean $0$ and variance $J_g^{-1}V_g J_g^{-1}$.
\begin{proof}
Because of the lengthy and somewhat messy calculations, the 
proof of consistency has been put in Appendix.
\paragraph{}
{\bf Proof of the asymptotic Normality :}		For the Asymptotic normality, we expand 
$$\sum_x K(\delta_n(x)) f_{\theta}^{1+\beta}(x)w_{\theta}(x)$$
 in Taylor series about $\theta = \theta^g$ to get
\begin{eqnarray}
\label{EQ:kfw_exapnd}
    & & \sum_x K(\delta_n(x)) f_{\theta}^{1+\beta}(x)w_{\theta}(x)  \nonumber \\
& = & \sum_x K(\delta_n^g(x)) f_{\theta^g}^{1+\beta}(x)w_{\theta^g}(x) 
+ \sum_k (\theta_k - \theta_k^g) \nabla_k \left( \sum_x K(\delta_n(x)) 
f_{\theta}^{1+\beta}(x)w_{\theta}(x) \right)|_{\theta = \theta^g} \nonumber \\
 & & ~~~~~~~~~ + \frac{1}{2} \sum_{k,l} (\theta_k - \theta_k^g)
(\theta_l - \theta_l^g)\nabla_{kl} \left(\sum_x K(\delta_n(x)) 
f_{\theta}^{1+\beta}(x)w_{\theta}(x)\right)|_{\theta = \theta '} \\ \nonumber
\end{eqnarray}
where, $\theta '$ lies in between $\theta$ and $\theta^g$.
% \\
	Now, let $\theta_n$ be the solution of the minimum 
LSD estimating equation, which can be assumed to be consistent. 
Replace $\theta$ by $\theta_n$ in above (\ref{EQ:kfw_exapnd}) 
so that the LHS of the equation becomes zero and hence we get
\begin{eqnarray}
\label{EQ:kfw_times}
  &  & - \sqrt n \sum_x K(\delta_n^g(x)) f_{\theta^g}^{1+\beta}(x)
w_{\theta^g}(x)  =  \sqrt n \sum_k (\theta_{nk} - 
\theta_k^g) \times \nonumber \\
 & & ~~~~~~~~~~~~ \{ \nabla_k \left( \sum_x K(\delta_n(x)) 
f_{\theta}^{1+\beta}(x)w_{\theta}(x) \right)|_{\theta = \theta^g} 
+ \nonumber \\
& & ~~~~~~~~~~~~~~~~~~~~~~~\frac{1}{2} \sum_{l} 
(\theta_{nl} - \theta_l^g)\nabla_{kl} \left(\sum_x K(\delta_n(x)) 
f_{\theta}^{1+\beta}(x)w_{\theta}(x)\right)|_{\theta = \theta '}  
\}. \nonumber \\
\end{eqnarray}
	Note that, the first term within the bracketed quantity 
in the RHS of above (\ref{EQ:kfw_times}) converges to $J_g$ with probability 
tending to one, while the second bracketed term is an 
$o_p(1)$ term (as proved in the proof of consistency part). Also, 
by using the Lemma \ref{lem:lamma4}, we get that	
\begin{eqnarray}
   & & \sqrt n \sum_x K(\delta_n^g(x)) 
f_{\theta^g}^{1+\beta}(x)w_{\theta^g}(x) \nonumber \\
 ~~~~~~~~~~~ & = & \sqrt n \sum_x  [K(\delta_n^g(x)) - 
K(\delta_g^g(x))]  f_{\theta^g}^{1+\beta}(x)w_{\theta^g}(x)  (\mbox{Using estimating equation)} \nonumber \\
 ~~~~~~~~~~~ & = & S_{1n}|_{\theta=\theta^g} 
\rightarrow N_p(0, V_g). \nonumber \\ 
\end{eqnarray}
Therefore, by Lehmann(1983, Lemma 4.1), 
% it follows 
% that 
$\sqrt n(\theta_n - \theta^g)$ has asymptotic 
distribution as $ N_p(0,J_g^{-1} V_g J_g^{-1})$. 
\end{proof}
\end{theorem}
% \end{theopargself}
% \noindent
A very interesting observation that follows from the 
asymptotic distribution just established is that the 
asymptotic distribution of the estimator is independent 
of the parameter $\gamma$. Yet the behavior of the 
estimator varies widely with $\gamma$ under the presence of
outliers. Here we briefly report the findings 
reported by Maji, Ghosh and Basu (2014) in this connection, 
which is at least partially indicated by the results 
of the current and the subsequent section. Maji, Ghosh and Basu (2014) 
have observed that the first order influence function of the minimum 
LSD estimator is independent of $\gamma$, predicting that the 
robustness properties of the minimum LSD estimators are 
similar for each value of $\gamma$. This is immediately 
contradicted by the other results of Maji, Ghosh and Basu (2014)
as well as the next section of the current article. 
Further exploration by Maji, Ghosh and Basu (2014) indicate 
that the second order influence function 
% indicate that the second order 
% influence function 
gives a much more accurate picture of the 
robustness of these estimators. This clearly indicates the 
limitation of the first order influence function in quantifying 
the robustness of the estimators in this case. In fact 
the second order influence analysis (see Maji, Ghosh and Basu, 2014 for details)
shows that the limitation of the first order influence function 
can go both ways -- it can fail to indicate the stability of a robust 
estimator, and can also describe a highly unstable estimator as 
a robust one.
% stable.   
% \begin{flushright}$\square$\end{flushright} 
% This class of estimator considered in this paper follows closely estimator
% described in Windham (1995). 
% Let $\hat{\theta}_{\beta,\gamma}$ be the 
% minimizer of LSD$_{\beta,\gamma}$($g,f_\theta$) over $\theta \in \Theta$. 
% % parameter value
% % obtained by minimizing the divergence between $g$ and $f_\theta$. 
% Under standard regularity conditions 
% (see Jones et al., 2001 and Maji, Ghosh and Basu, 2014),
% % (see Ghosh 2013), 
% it 
% follows 
% % is shown 
% that $\hat{\theta}_{\beta,\gamma}$
% is consistent for ${\theta_{\beta,\gamma}}$ and 
% has the asymptotic distribution given by
% % the following result is 
% % also easy to establish (for details see Jones et al. (2001)):
% \begin{equation}
%  n^{\frac{1}{2}}(\hat{\theta}_{\beta,\gamma}-
% \theta_{\beta,\gamma})\rightarrow N(0,J_{\beta,\gamma}^{-1}K_{\beta,\gamma} J_{\beta,\gamma}^{-1}).
% \end{equation}
% The matrices $J_{\beta,\gamma}$ and $K_{\beta,\gamma}$ 
% as defined in the following subsection. 
% % are defined by the following relations:
% \subsubsection{Expression for the Variance}
% Define,
% \begin{equation}
% % L_{j,i}=\int g^j f_\theta^i,
% % M_{j,i}=\int g^j f_\theta^i u_\theta,
% % N_{j,i}=\int g^j f_\theta^i u_\theta u_\theta^T,
% L_{j,i}=\sum g^j f_\theta^i,
% M_{j,i}=\sum g^j f_\theta^i u_\theta,
% N_{j,i}=\sum g^j f_\theta^i u_\theta u_\theta^T,
% \end{equation}
% for $j=0,1$ and positive $i$. Under model conditions,
% % \begin{equation}
% \begin{eqnarray}
%  \label{k_j_exp}
% %  K_{\alpha,\lambda}&=&\frac{1}{L_{0,\alpha+\lambda}^2}\left[N_{0,\alpha+2\lambda}-\xi_{\alpha,\lambda}
% % M_{0,\alpha+2\lambda}^T-
% % M_{0,\alpha+2\lambda}\xi_{\alpha,\lambda}^T
% % +\xi_{\alpha,\lambda} \xi_{\alpha,\lambda}^T L_{0,\alpha+2\lambda}\right], \nonumber \\
% % & &J_{\alpha,\lambda}=\frac{N_{0,\alpha+\lambda}}{L_{0,\alpha+\lambda}}-\xi_{\alpha,\lambda}\xi_{\alpha,\lambda}^T,
% K_{\beta,\gamma}(=K_{\beta})&=&
% \frac{1}{L_{0,1+\beta}^2}
% \left[N_{0,1+2 \beta}-\xi_{\beta}
% M_{0,1+2\beta}^T-
% M_{0,1+2\beta}\xi_{\beta}^T
% +\xi_{\beta} \xi_{\beta}^T L_{0,1+2\beta}\right], \nonumber \\
% & & J_{\beta,\gamma}(=J_{\beta})=
% % \frac
% % {N_{0,1+\beta}}{L_{0,1+\beta}}-\xi_{\beta}\xi_{\beta}^T,
% \frac{N_{0,1+\beta}}{L_{0,1+\beta}}-\xi_{\beta}\xi_{\beta}^T,
% \end{eqnarray}
% where $\xi_{\beta}=
% \frac
% {M_{0,1+\beta}}
% {L_{0,1+\beta}}$
% since under model ($g=f_\theta$) both $J$ and $K$ depend only on $\beta$.
% 
% $$$$
% $$$${}
\section{Examples}
% {Estimation Example Using LSD}
\label{SEC:est_example}
\subsection{An Example with a Geometric Model}
The data set given in Basu et al.~(2011),
Table $2.4$, represent the cases of peritonitis
for $390$ kidney patients.
% This data set (Basu et al., 2011) gives details of peritonitis
% for $390$ kidney patients. The observed frequencies 
% getting from the number of cases of peritonitis are reported at the top
% of the table. 
Following Basu et al.~(2011) we 
use a 
% are using a 
geometric model with 
% 
% distribution with
parameter $\theta$ (success probability) 
% around $0.5$ 
as our target distribution. 
A quick look at the data reveals that a geometric model
with $\theta=0.5$ may fit the data well. 
We fit a geometric model with parameter $\theta$ using the 
LSD measure for various combinations of $\beta$ and $\gamma$. 
% Though this data does not have 
% any distant outlier, we 
We can treat the two observations at 10 
and 12 as mild to moderate outliers. We have 
evaluated the minimum
% used 
% the 
LSD estimator 
in this case for the full data as well as the outlier 
deleted data.
% to this data
% both with and without those outliers. 
The estimates are 
presented 
shown in
Tables \ref{tab:kidney_w_out} and \ref{tab:kidney_wo_out} respectively. 
The estimates highlight an interesting point; 
% that though 
for 
$\gamma<0$ or for larger values of $\beta$ with $\gamma\geq0$ the 
parameter estimates are close for full data and outlier deleted data.
However for 
% values are 
% very close in both cases but for 
$\gamma>0$ and 
% with smaller values of 
$\beta$ small, the presence or absence 
of the outliers do not lead to a substantially
larger difference. 
This gives a clear indication about which 
combinations of the $(\beta,\gamma)$ values keep 
the estimators stable and which are the ones that are easily 
affected. 
% the estimates show distant values under two set-up.

 \begin{table}

	\centering 
	\caption{The estimates of the parameter 
of the geometric model for 
% Geometric parameter values for 
different values of $\gamma$ and $\beta$ for the 
Peritonitis Incidence Data with outlier
}
		\begin{tabular}{c c c c c c c } \hline
$\gamma$ \textdownarrow $\beta$ $\rightarrow$ 
% & 0 & 0.1 & 0.2 & 0.3 & 0.4 & 0.5 & 0.6 & 0.7 & 0.8 & 0.9 & 1 \\ 
& 0 & 0.2  & 0.4  &  0.6 & 0.8 & 1 \\ 
\hline
% gamma/beta & 0 & 0.2 & 0.4 & 0.6 & 0.8 & 1 \\ 
$-1$ & -- & 0.518 & 0.511 & 0.511 & 0.513 & 0.515 \\ 
% $-0.9$ & 0.544 & 0.515 & 0.51 & 0.511 & 0.512 & 0.515 \\ 
% $-0.8$ & 0.527 & 0.512 & 0.509 & 0.51 & 0.512 & 0.515 \\ 
$-0.7$ & 0.519 & 0.51 & 0.509 & 0.51 & 0.512 & 0.515 \\ 
% $-0.6$ & 0.514 & 0.509 & 0.508 & 0.51 & 0.512 & 0.515 \\ 
$-0.5$ & 0.510 & 0.508 & 0.508 & 0.51 & 0.512 & 0.515 \\ 
% $-0.4$ & 0.507 & 0.506 & 0.508 & 0.51 & 0.512 & 0.515 \\ 
$-0.3$ & 0.504 & 0.505 & 0.507 & 0.51 & 0.512 & 0.515 \\ 
% $-0.2$ & 0.502 & 0.504 & 0.507 & 0.509 & 0.512 & 0.515 \\ 
$-0.1$ & 0.499 & 0.503 & 0.506 & 0.509 & 0.512 & 0.515 \\ 
0 & 0.496 & 0.502 & 0.506 & 0.509 & 0.512 & 0.515 \\ 
% 0.1 & 0.493 & 0.501 & 0.506 & 0.509 & 0.512 & 0.515 \\ 
% 0.2 & 0.49 & 0.5 & 0.505 & 0.509 & 0.512 & 0.515 \\ 
% 0.3 & 0.487 & 0.499 & 0.505 & 0.509 & 0.512 & 0.515 \\ 
% 0.4 & 0.483 & 0.497 & 0.504 & 0.508 & 0.512 & 0.515 \\ 
0.5 & 0.48 & 0.496 & 0.504 & 0.508 & 0.512 & 0.515 \\ 
% 0.6 & 0.476 & 0.494 & 0.503 & 0.508 & 0.512 & 0.515 \\ 
% 0.7 & 0.472 & 0.492 & 0.503 & 0.508 & 0.512 & 0.515 \\ 
% 0.8 & 0.469 & 0.491 & 0.502 & 0.508 & 0.512 & 0.515 \\ 
% 0.9 & 0.465 & 0.489 & 0.502 & 0.508 & 0.512 & 0.515 \\ 
1 & 0.461 & 0.486 & 0.501 & 0.507 & 0.512 & 0.515 \\ 
% 1.1 & 0.457 & 0.484 & 0.5 & 0.507 & 0.512 & 0.515 \\ 
% 1.2 & 0.454 & 0.482 & 0.499 & 0.507 & 0.511 & 0.515 \\ 
1.3 & 0.45 & 0.479 & 0.499 & 0.507 & 0.511 & 0.515 \\ 
% 1.4 & 0.447 & 0.477 & 0.498 & 0.507 & 0.511 & 0.515 \\ 
% 1.5 & 0.444 & 0.474 & 0.497 & 0.3 & 0.511 & 0.515 \\ 
% 1.6 & 0.441 & 0.471 & 0.496 & 0.506 & 0.511 & 0.515 \\ 
1.7 & 0.438 & 0.469 & 0.495 & 0.506 & 0.511 & 0.515 \\ 
% 1.8 & 0.435 & 0.466 & 0.493 & 0.506 & 0.511 & 0.515 \\ 
% 1.9 & 0.433 & 0.463 & 0.492 & 0.506 & 0.511 & 0.515 \\ 
2 & 0.43 & 0.461 & 0.491 & 0.505 & 0.511 & 0.515 \\ 
\hline
		\end{tabular}
\label{tab:kidney_w_out}
	 %{tab:Dwo}
\end{table}
 \begin{table}

	\centering 
	\caption{The estimates of the parameter 
of the geometric model for 
% Geometric parameter values for 
different values of $\gamma$ and $\beta$ for the 
Peritonitis Incidence Data without the two outliers 
}
		\begin{tabular}{c c c c c c c } \hline
$\gamma$ \textdownarrow $\beta$ $\rightarrow$ 
% & 0 & 0.1 & 0.2 & 0.3 & 0.4 & 0.5 & 0.6 & 0.7 & 0.8 & 0.9 & 1 \\ 
& 0 & 0.2  & 0.4  &  0.6 & 0.8 & 1 \\ 
\hline
% gamma/beta & 0 & 0.2 & 0.4 & 0.6 & 0.8 & 1 \\ 
$-1$ & -- & 0.521 & 0.512 & 0.511 & 0.513 & 0.515 \\ 
% $-0.9$ & 0.551 & 0.517 & 0.511 & 0.511 & 0.513 & 0.515 \\ 
% $-0.8$ & 0.534 & 0.515 & 0.51 & 0.511 & 0.513 & 0.515 \\ 
$-0.7$ & 0.526 & 0.513 & 0.51 & 0.511 & 0.513 & 0.515 \\ 
% $-0.6$ & 0.521 & 0.512 & 0.51 & 0.51 & 0.512 & 0.515 \\ 
$-0.5$ & 0.518 & 0.511 & 0.509 & 0.51 & 0.512 & 0.515 \\ 
% $-0.4$ & 0.515 & 0.51 & 0.509 & 0.51 & 0.512 & 0.515 \\ 
$-0.3$ & 0.513 & 0.509 & 0.508 & 0.51 & 0.512 & 0.515 \\ 
% $-0.2$ & 0.512 & 0.508 & 0.508 & 0.51 & 0.512 & 0.515 \\ 
$-0.1$ & 0.510 & 0.508 & 0.508 & 0.51 & 0.512 & 0.515 \\ 
0 & 0.509 & 0.507 & 0.508 & 0.51 & 0.512 & 0.515 \\ 
% 0.1 & 0.508 & 0.506 & 0.507 & 0.509 & 0.512 & 0.515 \\ 
% 0.2 & 0.507 & 0.506 & 0.507 & 0.509 & 0.512 & 0.515 \\ 
% 0.3 & 0.506 & 0.505 & 0.507 & 0.509 & 0.512 & 0.515 \\ 
% 0.4 & 0.506 & 0.505 & 0.507 & 0.509 & 0.512 & 0.515 \\ 
0.5 & 0.505 & 0.505 & 0.506 & 0.509 & 0.512 & 0.515 \\ 
% 0.6 & 0.504 & 0.504 & 0.506 & 0.509 & 0.512 & 0.515 \\ 
% 0.7 & 0.503 & 0.504 & 0.506 & 0.509 & 0.512 & 0.515 \\ 
% 0.8 & 0.503 & 0.503 & 0.506 & 0.509 & 0.512 & 0.515 \\ 
% 0.9 & 0.502 & 0.503 & 0.505 & 0.508 & 0.512 & 0.515 \\ 
1 & 0.501 & 0.503 & 0.505 & 0.508 & 0.512 & 0.515 \\ 
% 1.1 & 0.501 & 0.502 & 0.505 & 0.508 & 0.512 & 0.515 \\ 
% 1.2 & 0.5 & 0.502 & 0.505 & 0.508 & 0.512 & 0.515 \\ 
1.3 & 0.5 & 0.501 & 0.504 & 0.508 & 0.512 & 0.515 \\ 
% 1.4 & 0.499 & 0.501 & 0.504 & 0.508 & 0.512 & 0.515 \\ 
% 1.5 & 0.499 & 0.501 & 0.504 & 0.3 & 0.512 & 0.515 \\ 
% 1.6 & 0.498 & 0.5 & 0.504 & 0.508 & 0.511 & 0.515 \\ 
1.7 & 0.498 & 0.5 & 0.504 & 0.508 & 0.511 & 0.515 \\ 
% 1.8 & 0.497 & 0.5 & 0.503 & 0.507 & 0.511 & 0.515 \\ 
% 1.9 & 0.497 & 0.499 & 0.503 & 0.507 & 0.511 & 0.515 \\ 
2 & 0.496 & 0.499 & 0.503 & 0.507 & 0.511 & 0.515 \\ 
\hline
		\end{tabular}
\label{tab:kidney_wo_out}
	 %{tab:Dwo}
\end{table}
\subsection{An Example with a Poisson Model}
% {Drosophila Data}
% \paragraph{}
This example 
% (Table 
% [\ref{droso_table1}]) 
gives us the observed frequencies and 
corresponding estimated frequencies (Table 
\ref{droso_table1}) for several 
minimum LSD 
estimators 
% of LSD family 
under the Poisson model
for a sex linked recessive lethal test in drosophila (fruit flies)
exposed to a certain chemical. For each of several male flies
one samples about $100$ daughter flies, and 
then determines the frequency of the number of 
daughter flies having a recessive lethal mutation 
in its $X$-chromosome. The data represent a frequency 
of frequencies; refer to 
% ; 
% refer 
Woodruff et al.~(1984) for details. There is a possible case
of outliers corresponding to the observations at $x=3,4$. 
\paragraph{}
Table \ref{droso_table1} provides the estimators and the predicted 
frequencies for a small number of $\gamma,\beta$ combinations,
together with the fits of the maximum likelihood estimator
(denoted by ML) and the outlier deleted maximum likelihood
estimator obtained by removing the two outliers (denoted
by ML$+$D). Clearly the estimators (and the estimated frequencies)
are substantially different for the ML 
and ML$+$D cases, demonstrating that the maximum 
likelihood estimator is significantly affected by the presence 
of these outliers. Also apart from the ML, the ($\gamma=1,\beta=0.1$)
combination leads to highly unstable estimators.  
% The ML method and ML$+$D 
% method fits are shown in the rows corresponding to ML and ML$+$D.
% Note that the ML method corresponds to the minimization of LSD 
% with parameter $\alpha=\lambda=0$. 
% While the estimates corresponding to only a few 
% ($\lambda,\alpha$) combinations are presented in the table,
% the general trends observed by us are as follows:
While Table \ref{droso_table1} provides a small number of ($\gamma,\beta$) 
combinations, a large selection is presented in Table \ref{droso_estimate}, 
where the salient features may be described as follows: 
\begin{enumerate}[(a)]
 \item All the estimators corresponding to large negative values of 
$\gamma$ and/or values of $\beta$ close to $1$ generate outlier 
resistant methods,
\item Estimators corresponding to large positive values of $\gamma$
are relatively poor in terms of robustness, especially for small $\beta$
\item All estimators for $\beta=1$ are identical,
irrespective of the value of $\gamma$. 
\end{enumerate}
Once again this example shows that large positive values of $\gamma$ 
with $\beta$ close to zero are the more unstable distances 
within the LSD class. 
% show values by retaining and deleting
% outliers respectively. LSD estimator with $\lambda=1$ and $\alpha=0.1$ gives estimate as
% $0.6311$ and expected frequencies very far from the observed frequencies 
%  whereas LSD estimator with $\lambda=-1$ and $\alpha=0.1$ overestimates as parameter
% estiamte comes out to be $0.0762$. LSD estimator with $\alpha=1$, irrespective of $\lambda$ 
% gives (see Table [\ref{droso_estimate}]) gives estimate $\theta$ as $0.1297$
% which is close to estimate for ML$+$D case and estiamted frequencies are close to
% observed freuencies.  
\begin{table}
\caption{ 
% drosophila Data and Estimation Using Maximum Likelihhod Estimate
Fits of the Poisson model to the Drosophila Data using several estimation methods
}
\vspace{2 mm}
\centering % used for centering table
  \begin{tabular}{l c c c c c c c} % centered columns (4 columns)
\hline %inserts double horizontal lines
& \multicolumn{7}{c}{Recessive lethal count}\\\cline{2-8}
& 0 & 1 & 2 & 3 & 4 & $\geq 5$ 
& $\hat{\theta}$
\\
Observed & 23 & 3 & 0 & 1& 1& 0\\
ML & 19.59 & 7.00 & 1.25 & 0.15 & 0.01 & -- & 0.3571\\
ML+D & 24.95 & 2.88 & 0.17 & 0.01 & -- & -- & 0.1154\\
{\rm LSD}$_{\gamma = 1~~\&~~\beta = 0.1}$ & 14.90 & 9.40 & 2.97 & 0.62 & 0.10 & 0.01 & 0.6311\\
{\rm LSD}$_{\gamma = -1~~\&~~\beta = 0.1}$ & 25.95 & 1.98 & 0.07 & -- & -- & -- & 0.0762\\
{\rm LSD}$_{\gamma = 1~~\&~~\beta = 1}$ & 24.59 & 3.19 & 0.21 & 0.01 & -- & -- & 0.1297\\
{\rm LSD}$_{\gamma =-1~~\&~~\beta = 1}$ & 24.59 & 3.19 & 0.21 & 0.01 & -- & -- & 0.1297\\
\hline
% \hline 
\end{tabular}
% \hline\hline 
\label{droso_table1}
 \end{table}
 \begin{table}
	\centering 
	\caption{Estimates of the parameter for drosophila data in Table {\ref{droso_table1}}}
		\begin{tabular}{c c c c c c c c c c c c} \hline
$\gamma$ \textdownarrow $\beta$ $\rightarrow$ 
% & 0 & 0.1 & 0.2 & 0.3 & 0.4 & 0.5 & 0.6 & 0.7 & 0.8 & 0.9 & 1 \\ 
& 0 & 0.2  & 0.4  &  0.6 & 0.8 & 0.9 & 1 \\ 
% 0.1 & 0.3147 & 0.2283 & 0.1742 & 0.1483 & 0.1373 & 0.1327 & 0.1307 & 0.13 & 0.1297 & 0.1297 \\ 
% 0.2 & 0.3666 & 0.2685 & 0.1958 & 0.157 & 0.1405 & 0.1338 & 0.1311 & 0.1301 & 0.1298 & 0.1297 \\ 
% 0.3 & 0.4144 & 0.3112 & 0.223 & 0.1688 & 0.1447 & 0.1353 & 0.1317 & 0.1303 & 0.1298 & 0.1297 \\ 
% 0.4 & 0.4572 & 0.3533 & 0.2546 & 0.1842 & 0.1501 & 0.1371 & 0.1322 & 0.1305 & 0.1298 & 0.1297 \\ 
% 0.5 & 0.4952 & 0.3929 & 0.2885 & 0.2035 & 0.1571 & 0.1392 & 0.1329 & 0.1307 & 0.1299 & 0.1297 \\ 
% 0.6 & 0.529 & 0.4296 & 0.3227 & 0.2262 & 0.1662 & 0.1419 & 0.1336 & 0.1308 & 0.13 & 0.1297 \\ 
% 0.7 & 0.5589 & 0.4631 & 0.3561 & 0.2515 & 0.1774 & 0.1451 & 0.1345 & 0.1311 & 0.13 & 0.1297 \\ 
% 0.8 & 0.5856 & 0.4936 & 0.388 & 0.2783 & 0.1912 & 0.1491 & 0.1355 & 0.1313 & 0.1301 & 0.1297 \\ 
% 0.9 & 0.6096 & 0.5213 & 0.4179 & 0.3056 & 0.2072 & 0.1541 & 0.1366 & 0.1316 & 0.1301 & 0.1297 \\ 
% 1 & 0.6311 & 0.5464 & 0.4458 & 0.3326 & -- & 0.1602 & 0.138 & 0.1319 & 0.1302 & 0.1297 \\ 
% \hline
% $-0.1$ & 0.2136 & 0.1687 & 0.1471 & 0.1372 & 0.1328 & 0.1308 & 0.13 & 0.1297 & 0.1297 & 0.1297 \\ 
% $-0.2$ & 0.1764 & 0.1511 & 0.1393 & 0.1337 & 0.1312 & 0.1301 & 0.1297 & 0.1296 & 0.1296 & 0.1297 \\ 
% $-0.3$ & 0.1514 & 0.1394 & 0.1337 & 0.1311 & 0.1299 & 0.1294 & 0.1294 & 0.1294 & 0.1296 & 0.1297 \\ 
% $-0.4$ & 0.1356 & 0.1315 & 0.1296 & 0.1289 & 0.1288 & 0.1288 & 0.1291 & 0.1293 & 0.1296 & 0.1297 \\ 
% $-0.5$ & 0.1252 & 0.1257 & 0.1264 & 0.1271 & 0.1278 & 0.1284 & 0.1288 & 0.1292 & 0.1295 & 0.1297 \\ 
% $-0.6$ & 0.1174 & 0.1211 & 0.1237 & 0.1256 & 0.1269 & 0.1279 & 0.1286 & 0.1291 & 0.1294 & 0.1297 \\ 
% $-0.7$ & 0.1105 & 0.117 & 0.1212 & 0.1242 & 0.1261 & 0.1274 & 0.1284 & 0.129 & 0.1294 & 0.1297 \\ 
% $-0.8$ & 0.103 & 0.1126 & 0.1188 & 0.1228 & 0.1253 & 0.127 & 0.1281 & 0.1289 & 0.1294 & 0.1297 \\ 
% $-0.9$ & 0.0931 & 0.1075 & 0.116 & 0.1212 & 0.1245 & 0.1266 & 0.1279 & 0.1288 & 0.1293 & 0.1297 \\ 
% $-1$ & 0.0762 & 0.1003 & 0.1127 & 0.1196 & 0.1237 & 0.1262 & 0.1277 & 0.1287 & 0.1293 & 0.1297 \\ 
\hline
% $-1$ & 0 & 0.076 & 0.1 & 0.113 & 0.12 & 0.124 & 0.126 & 0.128 & 0.129 & 0.129 & 0.13 \\ 
% $-0.9$ & 0.068 & 0.093 & 0.108 & 0.116 & 0.121 & 0.125 & 0.127 & 0.128 & 0.129 & 0.129 & 0.13 \\ 
% $-0.8$ & 0.088 & 0.103 & 0.113 & 0.119 & 0.123 & 0.125 & 0.127 & 0.128 & 0.129 & 0.129 & 0.13 \\ 
% $-0.7$ & 0.101 & 0.111 & 0.117 & 0.121 & 0.124 & 0.126 & 0.127 & 0.128 & 0.129 & 0.129 & 0.13 \\ 
% $-0.6$ & 0.112 & 0.117 & 0.121 & 0.124 & 0.126 & 0.127 & 0.128 & 0.129 & 0.129 & 0.129 & 0.13 \\ 
% $-0.5$ & 0.125 & 0.125 & 0.126 & 0.126 & 0.127 & 0.128 & 0.128 & 0.129 & 0.129 & 0.13 & 0.13 \\ 
% $-0.4$ & 0.145 & 0.136 & 0.132 & 0.13 & 0.129 & 0.129 & 0.129 & 0.129 & 0.129 & 0.13 & 0.13 \\ 
% $-0.3$ & 0.177 & 0.151 & 0.139 & 0.134 & 0.131 & 0.13 & 0.129 & 0.129 & 0.129 & 0.13 & 0.13 \\ 
% $-0.2$ & 0.227 & 0.176 & 0.151 & 0.139 & 0.134 & 0.131 & 0.13 & 0.13 & 0.13 & 0.13 & 0.13 \\ 
% $-0.1$ & 0.291 & 0.214 & 0.169 & 0.147 & 0.137 & 0.133 & 0.131 & 0.13 & 0.13 & 0.13 & 0.13 \\ 
% 0 & 0 & 0.262 & 0.194 & 0.158 & 0.142 & 0.135 & 0.132 & 0.13 & 0.13 & 0.13 & 0.13 \\ 
% 0.1 & 0.417 & 0.315 & 0.228 & 0.174 & 0.148 & 0.137 & 0.133 & 0.131 & 0.13 & 0.13 & 0.13 \\ 
% 0.2 & 0.47 & 0.367 & 0.269 & 0.196 & 0.157 & 0.141 & 0.134 & 0.131 & 0.13 & 0.13 & 0.13 \\ 
% 0.3 & 0.514 & 0.414 & 0.311 & 0.223 & 0.169 & 0.145 & 0.135 & 0.132 & 0.13 & 0.13 & 0.13 \\ 
% 0.4 & 0.553 & 0.457 & 0.353 & 0.255 & 0.184 & 0.15 & 0.137 & 0.132 & 0.131 & 0.13 & 0.13 \\ 
% 0.5 & 0.586 & 0.495 & 0.393 & 0.289 & 0.204 & 0.157 & 0.139 & 0.133 & 0.131 & 0.13 & 0.13 \\ 
% 0.6 & 0.615 & 0.529 & 0.43 & 0.323 & 0.226 & 0.166 & 0.142 & 0.134 & 0.131 & 0.13 & 0.13 \\ 
% 0.7 & 0.641 & 0.559 & 0.463 & 0.356 & 0.252 & 0.177 & 0.145 & 0.135 & 0.131 & 0.13 & 0.13 \\ 
% 0.8 & 0.663 & 0.586 & 0.494 & 0.388 & 0.278 & 0.191 & 0.149 & 0.136 & 0.131 & 0.13 & 0.13 \\ 
% 0.9 & 0.683 & 0.61 & 0.521 & 0.418 & 0.306 & 0.207 & 0.154 & 0.137 & 0.132 & 0.13 & 0.13 \\ 
% 1 & 0.701 & 0.631 & 0.546 & 0.446 & 0.333 & 0 & 0.16 & 0.138 & 0.132 & 0.13 & 0.13 \\ 
% $-1$ & -- & 0.1 & 0.12 & 0.126 & 0.129 & 0.129 & 0.13 \\ 
% $-0.9$ & 0.068 & 0.108 & 0.121 & 0.127 & 0.129 & 0.129 & 0.13 \\ 
$-0.8$ & 0.088 & 0.113 & 0.123 & 0.127 & 0.129 & 0.129 & 0.13 \\ 
$-0.7$ & 0.101 & 0.117 & 0.124 & 0.127 & 0.129 & 0.129 & 0.13 \\ 
$-0.6$ & 0.112 & 0.121 & 0.126 & 0.128 & 0.129 & 0.129 & 0.13 \\ 
$-0.5$ & 0.125 & 0.126 & 0.127 & 0.128 & 0.129 & 0.13 & 0.13 \\ 
$-0.4$ & 0.145 & 0.132 & 0.129 & 0.129 & 0.129 & 0.13 & 0.13 \\ 
$-0.3$ & 0.177 & 0.139 & 0.131 & 0.129 & 0.129 & 0.13 & 0.13 \\ 
$-0.2$ & 0.227 & 0.151 & 0.134 & 0.13 & 0.13 & 0.13 & 0.13 \\ 
$-0.1$ & 0.291 & 0.169 & 0.137 & 0.131 & 0.13 & 0.13 & 0.13 \\ 
0 & 0.357 & 0.194 & 0.142 & 0.132 & 0.13 & 0.13 & 0.13 \\ 
0.1 & 0.417 & 0.228 & 0.148 & 0.133 & 0.13 & 0.13 & 0.13 \\ 
0.2 & 0.47 & 0.269 & 0.157 & 0.134 & 0.13 & 0.13 & 0.13 \\ 
0.3 & 0.514 & 0.311 & 0.169 & 0.135 & 0.13 & 0.13 & 0.13 \\ 
0.4 & 0.553 & 0.353 & 0.184 & 0.137 & 0.131 & 0.13 & 0.13 \\ 
0.5 & 0.586 & 0.393 & 0.204 & 0.139 & 0.131 & 0.13 & 0.13 \\ 
0.6 & 0.615 & 0.43 & 0.226 & 0.142 & 0.131 & 0.13 & 0.13 \\ 
0.7 & 0.641 & 0.463 & 0.252 & 0.145 & 0.131 & 0.13 & 0.13 \\ 
0.8 & 0.663 & 0.494 & 0.278 & 0.149 & 0.131 & 0.13 & 0.13 \\ 
% 0.9 & 0.683 & 0.521 & 0.306 & 0.154 & 0.132 & 0.13 & 0.13 \\ 
% 1 & 0.701 & 0.546 & 0.333 & 0.16 & 0.132 & 0.13 & 0.13 \\ 
% $-1$ & -- & 0.0762 & 0.1003 & 0.1127 & 0.1196 & 0.1237 & 0.1262 & 0.1277 & 0.1287 & 0.1293 & 0.1297 \\ 
% $-0.9$ & 0.0677 & 0.0931 & 0.1075 & 0.116 & 0.1212 & 0.1245 & 0.1266 & 0.1279 & 0.1288 & 0.1293 & 0.1297 \\ 
% $-0.8$ & 0.0879 & 0.103 & 0.1126 & 0.1188 & 0.1228 & 0.1253 & 0.127 & 0.1281 & 0.1289 & 0.1294 & 0.1297 \\ 
% $-0.7$ & 0.1008 & 0.1105 & 0.117 & 0.1212 & 0.1242 & 0.1261 & 0.1274 & 0.1284 & 0.129 & 0.1294 & 0.1297 \\ 
% $-0.6$ & 0.1121 & 0.1174 & 0.1211 & 0.1237 & 0.1256 & 0.1269 & 0.1279 & 0.1286 & 0.1291 & 0.1294 & 0.1297 \\ 
% $-0.5$ & 0.1252 & 0.1252 & 0.1257 & 0.1264 & 0.1271 & 0.1278 & 0.1284 & 0.1288 & 0.1292 & 0.1295 & 0.1297 \\ 
% $-0.4$ & 0.1446 & 0.1356 & 0.1315 & 0.1296 & 0.1289 & 0.1288 & 0.1288 & 0.1291 & 0.1293 & 0.1296 & 0.1297 \\ 
% $-0.3$ & 0.1768 & 0.1514 & 0.1394 & 0.1337 & 0.1311 & 0.1299 & 0.1294 & 0.1294 & 0.1294 & 0.1296 & 0.1297 \\ 
% $-0.2$ & 0.2272 & 0.1764 & 0.1511 & 0.1393 & 0.1337 & 0.1312 & 0.1301 & 0.1297 & 0.1296 & 0.1296 & 0.1297 \\ 
% $-0.1$ & 0.2913 & 0.2136 & 0.1687 & 0.1471 & 0.1372 & 0.1328 & 0.1308 & 0.13 & 0.1297 & 0.1297 & 0.1297 \\ 
% 0 & 0.3571 & 0.2618 & 0.1943 & 0.1583 & 0.142 & 0.1348 & 0.1317 & 0.1303 & 0.1298 & 0.1297 & 0.1297 \\ 
% 0.1 & 0.4172 & 0.3147 & 0.2283 & 0.1742 & 0.1483 & 0.1373 & 0.1327 & 0.1307 & 0.13 & 0.1297 & 0.1297 \\ 
% 0.2 & 0.4695 & 0.3666 & 0.2685 & 0.1958 & 0.157 & 0.1405 & 0.1338 & 0.1311 & 0.1301 & 0.1298 & 0.1297 \\ 
% 0.3 & 0.5143 & 0.4144 & 0.3112 & 0.223 & 0.1688 & 0.1447 & 0.1353 & 0.1317 & 0.1303 & 0.1298 & 0.1297 \\ 
% 0.4 & 0.5528 & 0.4572 & 0.3533 & 0.2546 & 0.1842 & 0.1501 & 0.1371 & 0.1322 & 0.1305 & 0.1298 & 0.1297 \\ 
% 0.5 & 0.5862 & 0.4952 & 0.3929 & 0.2885 & 0.2035 & 0.1571 & 0.1392 & 0.1329 & 0.1307 & 0.1299 & 0.1297 \\ 
% 0.6 & 0.6152 & 0.529 & 0.4296 & 0.3227 & 0.2262 & 0.1662 & 0.1419 & 0.1336 & 0.1308 & 0.13 & 0.1297 \\ 
% 0.7 & 0.6407 & 0.5589 & 0.4631 & 0.3561 & 0.2515 & 0.1774 & 0.1451 & 0.1345 & 0.1311 & 0.13 & 0.1297 \\ 
% 0.8 & 0.6633 & 0.5856 & 0.4936 & 0.388 & 0.2783 & 0.1912 & 0.1491 & 0.1355 & 0.1313 & 0.1301 & 0.1297 \\ 
% 0.9 & 0.6834 & 0.6096 & 0.5213 & 0.4179 & 0.3056 & 0.2072 & 0.1541 & 0.1366 & 0.1316 & 0.1301 & 0.1297 \\ 
% 1 & 0.7014 & 0.6311 & 0.5464 & 0.4458 & 0.3326 & -- & 0.1602 & 0.138 & 0.1319 & 0.1302 & 0.1297 \\ 
\hline
		\end{tabular}
\label{droso_estimate}	 %{tab:Dwo}
\end{table}

\section{Testing Parametric Hypothesis using the LSD Measures}
\label{SEC:testing}

Now we focus our attention on hypothesis testing, 
the other important \\ paradigm of statistical inference. 
\subsection{One Sample problem}
	We consider a parametric family of densities 
$\mathcal{F} = \{f_\theta : \theta \in \Theta \subseteq \mathbb{R}^p\}$ 
as above. Suppose we are given a random sample $X_1, \ldots, X_n$ of 
size $n$ from the population. Based on this sample, we want to test the hypothesis 
$$
 H_0 : \theta = \theta_0 ~~~ \mbox{against} ~~~~ H_1 : \theta \ne \theta_0.
$$
% There is no uniformly most powerful test for the problem under 
% consideration for most parametric models. 
When the model is correctly 
specified and the null hypothesis is correct, $f_{\theta_0}$ is the data 
generating density. 
% We can test for this hypothesis by using the LSD 
% measure between $f_{\theta_0}$ and $f_{\hat{\theta}}$ for any 
% estimator $\hat{\theta}$ of $\theta$ based on the given sample. Since 
% the Density Power Divergence is the only divergence in the family of 
% LSD that does not require the consideration of the kernel density 
% estimation while obtaining the minimum divergence estimator, we will 
% consider the estimator $\hat{\theta}_\beta$ of $\theta$ obtained by 
% minimizing the DPD having parameter $\beta$. And then 
We consider the 
test statistics based on the LSD with parameter 
% $A$ and $B$
$\beta$ and $\gamma$ 
as follows:
\begin{equation}
 W_{\beta,\gamma}(\hat{\theta}_{\beta,\gamma},\theta_0)
=2n~{\rm LSD}_{\beta,\gamma}(f_{\hat{\theta}_{\beta,\gamma}},f_{\theta_0}),
%  T_{\beta,\gamma}(\hat{\theta}_{\beta},\theta_0)
% =2n~{\rm LSD}_{\alpha,\gamma}(\hat{\theta}_{\beta},\theta_0),
\end{equation}
where ${\rm LSD}_{\beta,\gamma}(f_{\hat{\theta}_{\beta,\gamma}},f_{\theta_0})$ has the form 
given in (\ref{lsd_form}). 
\begin{theorem}
% 	Suppose the model density satisfies the Lehmann and 
% GPD conditions.
 The 
asymptotic distribution of the test statistic 	$W_{\beta,\gamma}
(f_{{\hat{\theta}}_{\beta,\gamma}},f_{{\theta_0}})$, under the null hypothesis 
$H_0 : \theta = \theta_0$, coincides with the distribution of 
$$
\sum_{i=1}^r ~  \zeta_i^{\beta}(\theta_0)Z_i^2
$$
where $Z_1, \ldots,Z_r$ are independent standard normal variables, 
$\zeta_1^{\beta}(\theta_0), \ldots, \\ \zeta_r^{\beta}
(\theta_0)$  are the nonzero eigenvalues of $A_{\beta}(\theta_0) 
J_{\beta}^{-1}(\theta_0) K_{\beta}(\theta_0) J_{\beta}^{-1}(\theta_0)$, 
with $J_{\beta}(\cdot)$ and $K_{\beta}(\cdot)$ as defined in Theorem
\ref{th:est_th}
% (\ref{k_j_exp})
% (\ref{j_g_v_g})
and the matrix $A_{\beta}(\theta_0)$ is defined as
$$
A_{\beta}(\theta_0) = \nabla [  \nabla {\rm LSD}_{\beta,\gamma}(f_\theta, f_{\theta_0}) ]
|_{\theta = \theta_0} 
% = \left( (1+\alpha) \int f_{\theta_0}^{\alpha -1} 
% \frac{\delta f_{\theta_0}}{\delta \theta_i} \frac{\delta f_{\theta_0}}
% {\delta \theta_j} \right)_{i,j=1,\ldots,p}
$$
 and
$$
r = rank\left( J_{\beta}^{-1}(\theta_0) 
K_{\beta}(\theta_0) J_{\beta}^{-1}(\theta_0)
A_{\beta}(\theta_0) J_{\beta}^{-1}(\theta_0) 
K_{\beta}(\theta_0) J_{\beta}^{-1}(\theta_0)\right).
$$
% Here $\nabla$ represents the gradient with respect to $\theta$. 
\begin{proof}
  We consider the second order Taylor series expansion 
of ${\rm LSD}_{\beta,\gamma}(f_\theta, f_{\theta_0})$ around 
$\theta = \theta_0$ at $\theta = \hat{\theta}_{\beta}$ as,
\begin{eqnarray}
\label{EQ:lsd_expansion}
    {\rm LSD}_{\beta,\gamma}\left(f_{\hat{\theta}_{\beta}},f_{\theta_0}\right) 
&=& {\rm LSD}_{\beta,\gamma}(f_{\theta_0}, f_{\theta_0}) + \sum_{i=1}^p
 \nabla_i {\rm LSD}_{\beta,\gamma}(f_\theta, 
f_{\theta_0})|_{\theta = \theta_0} (\hat{\theta}_{\beta}^i - \theta_0^i) \nonumber 
\\
& & + \frac{1}{2} \sum_{i,j} \nabla_{ij} {\rm LSD}_{\beta,\gamma}
(f_\theta, f_{\theta_0})|_{\theta = \theta_0} 
(\hat{\theta}_{\beta}^i - \theta_0^i) (\hat{\theta}_{\beta}^j - 
\theta_0^j) \nonumber \\ 
& & + o(||\hat{\theta}_{\beta} - \theta_0||^2), 
\end{eqnarray}
where 
$\nabla_i$ and $\nabla_{ij}$ represent the indicated partial 
derivatives with respect to the components of $\theta$. 
% superscripts denotes the corresponding components. 
Now 
we have 
$$ {\rm LSD}_{\beta,\gamma}(f_{\theta_0}, f_{\theta_0}) = 0 $$ 
and $$\nabla_i {\rm LSD}_{\beta,\gamma}(f_\theta, 
f_{\theta_0})|_{\theta = \theta_0} =0. $$ 
% Also, we have
% $$
%  \nabla_{ij}^2 C(f_\theta, 
% f_{\theta_0})|_{\theta = \theta_0} =  (1+\gamma) 
% \int f_{\theta_0}^{\gamma -1} \frac{\delta f_{\theta_0}}
% {\delta \theta_i} \frac{\delta f_{\theta_0}}{\delta \theta_j} 
% $$
Note that the above second order partial derivative of 
${\rm LSD}_{\beta,\gamma}(f_\theta, f_{\theta_0})$ at 
$\theta = \theta_0 $ is independent of $\gamma$ and 
so we will denote that as function of $\beta$ only.
We will denote the second order partial derivatives of
${\rm LSD}_{\beta,\gamma}(f_\theta, f_{\theta_0})$  in
(\ref{EQ:lsd_expansion})
by $a_{ij}^{\beta} (\theta_0)$. 
% Then 
% note that 
Also denote
$A_{\beta}(\theta_0) = \left( a_{ij}^{\beta} (\theta_0) 
\right)_{i,j=1,\ldots,p}$.
Now from the above Taylor series expansion it is clear 
that the random variables 
\begin{eqnarray}
     W_{\beta,\gamma}({\hat{\theta}_{\beta,\gamma}},{\theta_0}) = 
2 n {\rm LSD}_{\beta,\gamma}(f_{\hat{\theta}_{\beta,\gamma}},f_{\theta_0}) \nonumber \\
~~~~ \mbox{and} ~~~~ \sqrt{n} (\hat{\theta}_{\beta,\gamma} - \theta_0)^T 
A_{\beta} (\theta_0) \sqrt{n} (\hat{\theta}_{\beta,\gamma} - \theta_0) \nonumber
\end{eqnarray}
% 
% 
% $$ 
% 
% $$ 
have the same asymptotic distribution. Now we know from 
the previous section
% Basu et al. (1998) 
that the 
asymptotic distribution of $\sqrt{n} (\hat{\theta}_{\beta,\gamma}-\theta_0)$ 
is normal with mean zero and variance $J_{\beta}^{-1}(\theta_0) 
K_{\beta}(\theta_0) J_{\beta}^{-1}(\theta_0)$.
   Further we know that for $X \sim N_q(0,\Sigma)$, and a $q-$dimensional 
real symmetric matrix $A$, the distribution of the quadratic form 
$X^T A X$ is the same as that of
$\sum \limits_{i=1}^r \zeta_i^{\beta} Z_i^2$, where $Z_1, \ldots ,Z_r$ are independent 
standard normal variables, $r = rank(\Sigma A \Sigma)$, $r \ge 1$ and 
$\zeta_1^{\beta}, \ldots , \zeta_r^{\beta}$ are the nonzero eigenvalues of $A \Sigma$ 
(Dik and Gunst, 1985, Corollary 2.1). Applying this result with 
$ X =  \sqrt{n} (\hat{\theta}_{\beta,\gamma} - \theta_0)$ 
% we get 
the theorem is established. It
% Thus it 
is evident that the asymptotic distribution of the statistic depends on $\beta$ only
and is independent of $\gamma$.
\end{proof}
\end{theorem}

\begin{theorem}
% 	Suppose the model density satisfies the Lehmann and 
% Ghosh-Basu conditions. 
An approximation to the power 
function of the test statistic $W_{\beta,\gamma}( {\hat{\theta}_{\beta,\gamma}} 
, {\theta_0})$ for testing $H_0 : \theta =\theta_0$ against 
$H_1: \theta \ne \theta_0$ at the significance level $\alpha$ is given by
\begin{eqnarray}
    \pi_{n,\alpha}^{\beta,\gamma} (\theta^*) = 
1 - \Phi \left( \frac{\sqrt{n}}{\sigma_{\beta,\gamma 
}(\theta^*)} \left( \frac{t_\alpha^{\beta,\gamma}}{2 n} 
- {\rm LSD}_{\beta,\gamma}(f_{\theta^*}, f_{\theta_0}) \right) \right), ~~~ \theta^* \ne \theta_0
\end{eqnarray}
where $t_\alpha^{\beta,\gamma}$ is the $(1-\alpha)^{th}$ 
quantile of the asymptotic distribution of \\ $W_{\beta,\gamma}
( {\hat{\theta}_{\beta,\gamma}} , {\theta_0})$, and $\sigma_{\beta,\gamma}(\theta^*)$ is defined as
\begin{eqnarray}
\label{EQ:sigma_defn}
    \sigma_{\beta,\gamma}^2(\theta) = 
M_{\beta,\gamma}(\theta)^T J_\beta^{-1}(\theta) 
K_\beta(\theta) J_\beta^{-1}(\theta) M_{\beta,\gamma}(\theta)
\end{eqnarray}
with 
$$
M_{\beta,\gamma}(\theta) = \nabla {\rm LSD}_{\beta,\gamma}
( f_\theta , f_{\theta_0}). 
% = \frac{1+\alpha}{B} 
% \left[ \int f_\theta^{1+\alpha} u_\theta - 
% \int f_{\theta_0}^B f_\theta^A u_\theta \right]
$$
\begin{proof}
%  
% \noindent 
% \textbf{Proof :} 
Fix some $\theta^* \ne \theta_0$. Consider 
the first order Taylor series expansion of 
${\rm LSD}_{\beta,\gamma}( f_{\hat{\theta}_{\beta,\gamma}} , 
f_{\theta_0})$ under $f_{\theta^*}$ as
$$
{\rm LSD}_{\beta,\gamma}( f_{\hat{\theta}_{\beta,\gamma}} , 
f_{\theta_0}) = {\rm LSD}_{\beta,\gamma}( f_{\theta^*} , 
f_{\theta_0}) + M_{\beta,\gamma}(\theta^*)^T 
(\hat{\theta}_{\beta,\gamma} - \theta^*) + o(||\hat{\theta}_{\beta,\gamma} - \theta^*||)
$$
where $M_{\beta,\gamma}$ is as defined in the theorem. Now 
we know that, under $\theta^*$, 
$$
\sqrt{n} (\hat{\theta}_{\beta,\gamma} - \theta^*) \rightarrow 
N(0, J_\beta^{-1}(\theta^*) K_\beta(\theta^*) J_\beta^{-1}(\theta^*) )
 ~~~as ~ n \rightarrow \infty
$$
and $ \sqrt{n} o(||\hat{\theta}_{\beta,\gamma} - \theta^*||) = o_p(1)$. Thus 
we get that the random variables 
$$
\sqrt{n} \left[ {\rm LSD}_{\beta,\gamma}( f_{\hat{\theta}_{\beta,\gamma}} , 
f_{\theta_0}) - {\rm LSD}_{\beta,\gamma}( f_{\theta^*} , f_{\theta_0}) 
\right] ~~~~~\mbox{ and } ~~~~~ M_{\beta,\gamma}(\theta^*)^T 
\sqrt{n}(\hat{\theta}_{\beta,\gamma} - \theta^*) 
$$
have the same asymtotic distribution . Therefore, we have 
$$
\sqrt{n} \left[ {\rm LSD}_{\beta,\gamma}( f_{\hat{\theta}_{\beta,\gamma}} , 
f_{\theta_0}) - {\rm LSD}_{\beta,\gamma}( f_{\theta^*} , f_{\theta_0}) 
\right] \rightarrow N( 0, \sigma_{\beta,\gamma}(\theta^*) )
$$
where $\sigma_{\beta,\gamma}(\theta^*)$ is as given in 
(\ref{EQ:sigma_defn}) above. Hence the desired approximation to the power function 
follows from the above asymptotic distribution. 
\end{proof}

% \\
% \begin{flushright}$\square$\end{flushright} 
\end{theorem}
% \begin{theorem}
% % 	Suppose the model density satisfies the Lehmann and 
% % GPD conditions.
%  The 
% asymptotic distribution of the test statistic 	$W_{\beta,\gamma}
% (f_{{\hat{\theta}}_{\beta,\gamma}},f_{{\theta_0}})$, under the null hypothesis 
% $H_0 : \theta = \theta_0$, coincides with the distribution of 
% $$
% \sum_{i=1}^r ~  \zeta_i^{\beta,\gamma}(\theta_0)Z_i^2
% $$
% where $Z_1, \ldots,Z_r$ are independent standard normal variables, 
% $\zeta_1^{\beta,\gamma}(\theta_0), \ldots, \zeta_r^{\beta,\gamma}
% (\theta_0)$ are the nonzero eigenvalues of $A_{\beta,\gamma}(\theta_0) 
% J_{\beta,\gamma}^{-1}(\theta_0) K_{\beta,\gamma}(\theta_0) J_{\beta,\gamma}^{-1}(\theta_0)$, 
% with $J_{\beta,\gamma}(\cdot)$ and $K_{\beta,\gamma}(\cdot)$ as defined in 
% (\ref{k_j_exp})
% % (\ref{j_g_v_g})
% and the matrix $A_{\beta,\gamma}(\theta_0)$ is defined as
% $$
% A_{\beta,\gamma}(\theta_0) = \nabla [  \nabla {\rm LSD}_{\beta,\gamma}(f_\theta, f_{\theta_0}) ]
% |_{\theta = \theta_0} 
% % = \left( (1+\alpha) \int f_{\theta_0}^{\alpha -1} 
% % \frac{\delta f_{\theta_0}}{\delta \theta_i} \frac{\delta f_{\theta_0}}
% % {\delta \theta_j} \right)_{i,j=1,\ldots,p}
% $$
%  and
% $$
% r = rank\left( J_{\beta,\gamma}^{-1}(\theta_0) 
% K_{\beta,\gamma}(\theta_0) J_{\beta,\gamma}^{-1}(\theta_0)
% A_{\beta,\gamma}(\theta_0) J_{\beta,\gamma}^{-1}(\theta_0) 
% K_{\beta,\gamma}(\theta_0) J_{\beta,\gamma}^{-1}(\theta_0)\right).
% $$
% Here $\nabla$ represents the gradient with respect to $\theta$. 
% \begin{proof}
%   We consider the second order Taylor series expansion 
% of ${\rm LSD}_{\beta,\gamma}(f_\theta, f_{\theta_0})$ around 
% $\theta = \theta_0$ at $\theta = \hat{\theta}_{\beta,\gamma}$ as,
% \begin{eqnarray}
% \label{EQ:lsd_expansion}
%     {\rm LSD}_{\beta,\gamma}\left(f_{\hat{\theta}_{\beta,\gamma}},f_{\theta_0}\right) 
% &=& {\rm LSD}_{\beta,\gamma}(f_{\theta_0}, f_{\theta_0}) + \sum_{i=1}^p
%  \nabla_i {\rm LSD}_{\beta,\gamma}(f_\theta, 
% f_{\theta_0})|_{\theta = \theta_0} (\hat{\theta}_{\beta,\gamma}^i - \theta_0^i) \nonumber 
% \\
% & & + \frac{1}{2} \sum_{i,j} \nabla_{ij} {\rm LSD}_{\beta,\gamma}
% (f_\theta, f_{\theta_0})|_{\theta = \theta_0} 
% (\hat{\theta}_{\beta,\gamma}^i - \theta_0^i) (\hat{\theta}_{\beta,\gamma}^j - 
% \theta_0^j) + o(||\hat{\theta}_{\beta,\gamma} - \theta_0||^2), 
% \end{eqnarray}
% where 
% $\nabla_i$ and $\nabla_{ij}$ represent the indicated partial 
% derivatives with respect to the components of $\theta$. 
% % superscripts denotes the corresponding components. 
% Now 
% we have 
% $$ {\rm LSD}_{\beta,\gamma}(f_{\theta_0}, f_{\theta_0}) = 0 $$ 
% and $$\nabla_i {\rm LSD}_{\beta,\gamma}(f_\theta, 
% f_{\theta_0})|_{\theta = \theta_0} =0. $$ 
% % Also, we have
% % $$
% %  \nabla_{ij}^2 C(f_\theta, 
% % f_{\theta_0})|_{\theta = \theta_0} =  (1+\gamma) 
% % \int f_{\theta_0}^{\gamma -1} \frac{\delta f_{\theta_0}}
% % {\delta \theta_i} \frac{\delta f_{\theta_0}}{\delta \theta_j} 
% % $$
% % Note that the above second order partial derivative of 
% % ${\rm LSD}_{\alpha,\lambda}(f_\theta, f_{\theta_0})$ at 
% % $\theta = \theta_0 $ is independent of $\lambda$ and 
% % so 
% We will denote the second order partial derivatives of
% ${\rm LSD}_{\beta,\gamma}(f_\theta, f_{\theta_0})$  in
% (\ref{EQ:lsd_expansion})
% by $a_{ij}^{\beta,\gamma} (\theta_0)$. 
% % Then 
% % note that 
% Also denote
% $A_{\beta,\gamma}(\theta_0) = \left( a_{ij}^{\beta,\gamma} (\theta_0) 
% \right)_{i,j=1,\ldots,p}$.
% Now from the above Taylor series expansion it is clear 
% that the random variables 
% $$ 
%     W_{\beta,\gamma}({\hat{\theta}_{\beta,\gamma}},{\theta_0}) = 
% 2 n {\rm LSD}_{\beta,\gamma}(f_{\hat{\theta}_{\beta,\gamma}},f_{\theta_0}) 
% ~~~~ \mbox{and} ~~~~ \sqrt{n} (\hat{\theta}_{\beta,\gamma} - \theta_0)^T 
% A_{\beta,\gamma} (\theta_0) \sqrt{n} (\hat{\theta}_{\beta,\gamma} - \theta_0)
% $$ 
% have the same asymptotic distribution. Now we know from 
% the previous section
% % Basu et al. (1998) 
% that the 
% asymptotic distribution of $\sqrt{n} (\hat{\theta}_{\beta,\gamma}-\theta_0)$ 
% is normal with mean zero and variance $J_{\beta,\gamma}^{-1}(\theta_0) 
% K_{\beta,\gamma}(\theta_0) J_{\beta,\gamma}^{-1}(\theta_0)$.
%    Further we know that for $X \sim N_q(0,\Sigma)$, and a $q-$dimensional 
% real symmetric matrix $A$, the distribution of the quadratic form 
% $X^T A X$ is the same as that of
% $\sum \limits_{i=1}^r \zeta_i^{\beta,\gamma} Z_i^2$, where $Z_1, \ldots ,Z_r$ are independent 
% standard normal variables, $r = rank(\Sigma A \Sigma)$, $r \ge 1$ and 
% $\zeta_1^{\beta,\gamma}, \ldots , \zeta_r^{\beta,\gamma}$ are the nonzero eigenvalues of $A \Sigma$ 
% (Dik and Gunst, 1985, Corollary 2.1). Applying this result with 
% $ X =  \sqrt{n} (\hat{\theta}_{\beta,\gamma} - \theta_0)$ we get the theorem.
% \end{proof}
% \end{theorem}
% \begin{theorem}
% \label{Theorem3} The asymptotic distribution of $T_{\boldsymbol{\beta,\gamma}}(%
% \widehat{\boldsymbol{\theta }}_{\beta,\gamma},\widetilde{\boldsymbol{\theta }}%
% _{\beta,\gamma})$ defined 
% % in (\ref{TRES.1}) coincides, under the null hypothesis $%
% % H_{0}$ given in (\ref{1}), 
% with the distribution of the random variable
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{equation*}
% {\textstyle\sum\limits_{i=1}^{k}}\lambda _{i}^{\beta ,\gamma }(\boldsymbol{%
% \theta }_{0})Z_{i}^{2}
% \end{equation*}%
% %
% %
% %
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% where $Z_{1},\ldots ,Z_{k}$ are independent standard normal variables, $%
% \lambda _{1}^{\beta ,\gamma },\ldots ,\lambda _{k}^{\beta ,\gamma }$ are the
% nonzero eigenvalues of \\
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% $\boldsymbol{A}_{\gamma }\left( \boldsymbol{\theta }_{0}\right) \boldsymbol{B%
% }_{\beta,\gamma}\left( \boldsymbol{\theta }_{0}\right) \boldsymbol{K}_{\beta,\gamma}(%
% \boldsymbol{\theta }_{0})\boldsymbol{B}_{\beta,\gamma}\left( \boldsymbol{\theta }%
% _{0}\right) $
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% and %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{equation}
% %k=\min \{r,rank\left( \boldsymbol{A}_{\gamma }\left( \boldsymbol{\theta }%
% %_{0}\right) \right) \}.  \label{k}
% k = rank\left(\boldsymbol{B}_{\beta,\gamma}\left(\boldsymbol{\theta}_{0}\right)
% \boldsymbol{K}_{\beta,\gamma}  (\boldsymbol{\theta }_{0})
% \boldsymbol{B}_{\beta,\gamma}\left(\boldsymbol{\theta}_{0}\right)  
% \boldsymbol{A}_{\gamma}\left(\boldsymbol{\theta}_{0}\right)
% \boldsymbol{B}_{\beta,\gamma}\left(\boldsymbol{\theta}_{0}\right)
% \boldsymbol{K}_{\beta,\gamma} (\boldsymbol{\theta }_{0}) \boldsymbol{B}_{\beta,\gamma}\left(\boldsymbol{\theta}_{0}\right)
% \right)  . \label{k}
% \end{equation}%
% %
% %
% %
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% The matrices $\boldsymbol{A}_{\gamma }\left( \boldsymbol{\theta }_{0}\right)
% $ and $\boldsymbol{B}_{\beta,\gamma}\left( \boldsymbol{\theta }_{0}\right) $ are
% defined by%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{equation}
% \boldsymbol{A}_{\gamma }\left( \boldsymbol{\theta }_{0}\right) =\left(
% a_{ij}^{\gamma }\left( \boldsymbol{\theta }_{0}\right) \right)
% _{i,j=1,...,p}=\left( 1+\gamma \right) \left( \int\nolimits_{\mathcal{X}}f_{%
% \boldsymbol{\theta }_{0}}^{\gamma -1}\left( x\right) \frac{\partial f_{%
% \boldsymbol{\theta }_{0}}\left( x\right) }{\partial \theta _{j}}\frac{%
% \partial f_{\boldsymbol{\theta }_{0}}\left( x\right) }{\partial \theta _{i}}%
% dx\right) _{i,j=1,...,p}  \label{AA}
% \end{equation}%
% %
% %
% %
% %
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% and%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{equation}
% \boldsymbol{B}_{\beta,\gamma}\left( \boldsymbol{\theta }_{0}\right) =\boldsymbol{J}%
% _{\beta,\gamma}^{-1}(\boldsymbol{\theta }_{0})\boldsymbol{G}(\boldsymbol{\theta }%
% _{0})\left[ \boldsymbol{G}^{T}(\boldsymbol{\theta }_{0})\boldsymbol{J}%
% _{\beta,\gamma}^{-1}(\boldsymbol{\theta }_{0})\boldsymbol{G}(\boldsymbol{\theta }%
% _{0})\right] ^{-1}\boldsymbol{G}^{T}(\boldsymbol{\theta }_{0}) \boldsymbol{J}%
% _{\beta,\gamma}^{-1}(\boldsymbol{\theta }_{0}).  \label{B}
% \end{equation}%
% %
% %
% %
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \end{theorem}
% 
% \begin{proof}
% % See Appendix.
% % 
% % \hspace*{\fill}\bigskip
% % \noindent \textbf{Proof of Theorem \ref{Theorem3}:} 
% Consider the expression ${\rm LSD}_{\beta,\gamma }(f_{\boldsymbol{\theta }},f_{\widetilde{%
% \boldsymbol{\theta }}_{\beta,\gamma }})$. A Taylor expansion for an arbitrary $%
% \boldsymbol{\theta }\in \Theta $, around $\widetilde{\boldsymbol{\theta }}%
% _{\beta,\gamma }$ leads to the relation
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{align*}
% {\rm LSD}_{\beta,\gamma }(f_{\boldsymbol{\theta }},f_{\widetilde{\boldsymbol{\theta }}%
% _{\beta,\gamma }})& ={\rm LSD}_{\beta,\gamma}(f_{\widetilde{\boldsymbol{\theta }}_{\beta,\gamma}},f_{%
% \widetilde{\boldsymbol{\theta }}_{\beta,\gamma}})+{\textstyle\sum\limits_{i=1}^{p}}%
% \left( \frac{\partial {\rm LSD}_{\beta,\gamma}(f_{\boldsymbol{\theta }},f_{\widetilde{%
% \boldsymbol{\theta }}_{\beta,\gamma}})}{\partial \theta _{i}}\right) _{\boldsymbol{%
% \theta =}\widetilde{\boldsymbol{\theta }}_{\beta,\gamma}}\left( \theta _{i}-%
% \widetilde{\theta }_{i,{\beta,\gamma}}\right) \\
% & +\frac{1}{2}{\textstyle\sum\limits_{i=1}^{p}}{\textstyle%
% \sum\limits_{j=1}^{p}}\left( \frac{\partial ^{2}{\rm LSD}_{\beta,\gamma}(f_{\boldsymbol{%
% \theta }},f_{\widetilde{\boldsymbol{\theta }}_{\beta,\gamma}})}{\partial \theta
% _{i}\partial \theta _{j}}\right) _{\boldsymbol{\theta =}\widetilde{%
% \boldsymbol{\theta }}_{\beta,\gamma}}\left( \theta _{i}-\widetilde{\theta }_{i,{%
% \beta }}\right) \left( \theta _{j}-\widetilde{\theta }_{j,{\beta,\gamma}}\right)
% +o\left( \left\Vert \boldsymbol{\theta }-\widetilde{\boldsymbol{\theta }}%
% _{\beta,\gamma}\right\Vert ^{2}\right) .
% \end{align*}%
% %
% %
% %
% %
% %
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% It is clear that ${\rm LSD}_{\beta,\gamma}(f_{\widetilde{\boldsymbol{\theta }}_{\beta
% }},f_{\widetilde{\boldsymbol{\theta }}_{\beta,\gamma}})=0$, $\left( \frac{\partial
% {\rm LSD}_{\beta,\gamma}(f_{\boldsymbol{\theta }},f_{\widetilde{\boldsymbol{\theta }}%
% _{\beta,\gamma}})}{\partial \boldsymbol{\theta }_{i}}\right) _{\boldsymbol{\theta =%
% }\widetilde{\boldsymbol{\theta }}_{\beta,\gamma}}=0$ for each $i$, and
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{equation*}
% a_{ij}^{\gamma }\left( \widetilde{\boldsymbol{\theta }}_{\beta,\gamma}\right)
% =\left( \frac{\partial ^{2}{\rm LSD}_{\beta,\gamma}(f_{\boldsymbol{\theta }},f_{%
% \widetilde{\boldsymbol{\theta }}_{\beta,\gamma}})}{\partial \theta _{i}\partial
% \theta _{j}}\right) _{\boldsymbol{\theta =}\widetilde{\boldsymbol{\theta }}%
% _{\beta,\gamma}}=\left( 1+\gamma \right) \int\nolimits_{\mathcal{X}}f_{\widetilde{%
% \boldsymbol{\theta }}_{\beta,\gamma}}^{\gamma -1}\left( x\right) \frac{\partial f_{%
% \widetilde{\boldsymbol{\theta }}_{\beta,\gamma}}\left( x\right) }{\partial \theta
% _{i}}\frac{\partial f_{\widetilde{\boldsymbol{\theta }}_{\beta,\gamma}}\left(
% x\right) }{\partial \theta _{j}}dx.
% \end{equation*}%
% %
% %
% %
% %
% %
% %
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Therefore,
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{equation*}
% T_{\boldsymbol{\gamma }}(\widehat{\boldsymbol{\theta }}_{\beta,\gamma},\widetilde{%
% \boldsymbol{\theta }}_{\beta,\gamma})=2n{\rm LSD}_{\beta,\gamma}(f_{\widehat{\boldsymbol{\theta
% }}},f_{\widetilde{\boldsymbol{\theta }}_{\beta,\gamma}})=n^{1/2}(\widehat{%
% \boldsymbol{\theta }}_{\beta,\gamma}-\widetilde{\boldsymbol{\theta }}_{\beta,\gamma})^{T}%
% \boldsymbol{A}_{\gamma }(\widetilde{\boldsymbol{\theta }}_{\beta,\gamma})n^{1/2}(%
% \widehat{\boldsymbol{\theta }}_{\beta,\gamma}-\widetilde{\boldsymbol{\theta }}%
% _{\beta,\gamma})+n\times o\left( \left\Vert \widehat{\boldsymbol{\theta }}_{\beta
% }-\widetilde{\boldsymbol{\theta }}_{\beta,\gamma})\right\Vert ^{2}\right) .
% \end{equation*}%
% %
% %
% %
% %
% %
% %
% %
% %
% %
% %
% %
% %
% %
% %
% %
% %
% %
% %
% %
% %
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Under $\boldsymbol{\theta }_{0}\in \Theta _{0}$,
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{equation*}
% \boldsymbol{A}_{\gamma }(\widetilde{\boldsymbol{\theta }}_{\beta,\gamma})\underset{%
% n\longrightarrow \infty }{\overset{\mathcal{P}}{\longrightarrow }}%
% \boldsymbol{A}_{\gamma }\left( \boldsymbol{\theta }_{0}\right) .
% \end{equation*}%
% %
% %
% %
% %
% %
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Using 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % (\ref{[E]}) and
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{equation*}
% n^{1/2}\tfrac{\partial }{\partial \boldsymbol{\theta }}\left. h_{n}(%
% \boldsymbol{\theta })\right\vert _{\boldsymbol{\theta =\theta }_{0}}=-n^{1/2}%
% \boldsymbol{J}\left( \boldsymbol{\theta }_{0}\right) (\widehat{\boldsymbol{%
% \theta }}_{\beta,\gamma}-\boldsymbol{\theta }_{0})+o_{p}(1),
% \end{equation*}%
% %
% %
% %
% %
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% we get
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{align*}
% n^{1/2}(\widetilde{\boldsymbol{\theta }}_{\beta,\gamma}-\boldsymbol{\theta }_{0})&
% =\boldsymbol{P}(\boldsymbol{\theta }_{0})n^{1/2}\boldsymbol{J}\left(
% \boldsymbol{\theta }_{0}\right) (\widehat{\boldsymbol{\theta }}_{\beta,\gamma}-%
% \boldsymbol{\theta }_{0})+o_{p}(1) \\
% & =\boldsymbol{J}^{-1}\left( \boldsymbol{\theta }_{0}\right) n^{1/2}%
% \boldsymbol{J}\left( \boldsymbol{\theta }_{0}\right) (\widehat{\boldsymbol{%
% \theta }}_{\beta,\gamma}-\boldsymbol{\theta }_{0})-\boldsymbol{Q}\left(
% \boldsymbol{\theta }_{0}\right) \boldsymbol{G}^{T}(\boldsymbol{\theta }_{0}) n^{1/2}(\widehat{\boldsymbol{\theta }}%
% _{\beta,\gamma}-\boldsymbol{\theta }_{0})+o_{p}(1) \\
% & =n^{1/2}(\widehat{\boldsymbol{\theta }}_{\beta,\gamma}-\boldsymbol{\theta }_{0})-%
% \boldsymbol{Q}\left( \boldsymbol{\theta }_{0}\right) \boldsymbol{G}^{T}(\boldsymbol{\theta }_{0}) n^{1/2}(\widehat{%
% \boldsymbol{\theta }}_{\beta,\gamma}-\boldsymbol{\theta }_{0})+o_{p}(1).
% \end{align*}%
% %
% %
% %
% %
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Therefore
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{equation}
% n^{1/2}(\widehat{\boldsymbol{\theta }}_{\beta,\gamma}-\widetilde{\boldsymbol{%
% \theta }}_{\beta,\gamma})=\boldsymbol{Q}(\boldsymbol{\theta }_{0}) \boldsymbol{G}^{T}(\boldsymbol{\theta }_{0}) n^{1/2}(\widehat{%
% \boldsymbol{\theta }}_{\beta,\gamma}-\boldsymbol{\theta }_{0})+o_{p}(1).  
% \label{curl_theta}
% \end{equation}%
% %
% %
% %
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% On the other hand, $n^{1/2}(\widehat{\boldsymbol{\theta }}_{\beta,\gamma}-%
% \boldsymbol{\theta }_{0})\underset{n\longrightarrow \infty }{\overset{%
% \mathcal{L}}{\longrightarrow }}\mathcal{N}(0,\boldsymbol{J}_{\beta,\gamma}^{-1}(%
% \boldsymbol{\theta }_{0})\boldsymbol{K}(\boldsymbol{\theta }_{0})\boldsymbol{%
% J}_{\beta,\gamma}^{-1}(\boldsymbol{\theta }_{0}))$. From equations 
% % (\ref{Q}) and (\ref{B}) 
% we have $\boldsymbol{B}_{\beta,\gamma} 
% \left( \boldsymbol{\theta }_{0}\right) = \boldsymbol{Q}_{\beta,\gamma} \left( \boldsymbol{%
% \theta }_{0}\right) \boldsymbol{G}^{T}(\boldsymbol{\theta }_{0}) \boldsymbol{J}_{\beta,\gamma}^{-1}(%
% \boldsymbol{\theta }_{0})$. Therefore it follows that
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{equation*}
% n^{1/2}(\widehat{\boldsymbol{\theta }}_{\beta,\gamma}-\widetilde{\boldsymbol{%
% \theta }}_{\beta,\gamma})\underset{n\longrightarrow \infty }{\overset{\mathcal{L}}{%
% \longrightarrow }}\mathcal{N}(0,\boldsymbol{B}_{\beta,\gamma}\left( \boldsymbol{%
% \theta }_{0}\right) \boldsymbol{K}_{\beta,\gamma}(\boldsymbol{\theta }_{0})%
% \boldsymbol{B}_{\beta,\gamma}\left( \boldsymbol{\theta }_{0}\right) ).
% \end{equation*}%
% Now the asymptotic distribution of the random variables
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% $T_{\boldsymbol{\gamma }}(\widehat{\boldsymbol{\theta }}_{\beta,\gamma},\widetilde{%
% \boldsymbol{\theta }}_{\beta,\gamma})=2n{\rm LSD}_{\beta,\gamma}(f_{\widehat{\boldsymbol{\theta
% }}_{\beta,\gamma}},f_{\widetilde{\boldsymbol{\theta }}_{\beta,\gamma}})$
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% and
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{equation*}
% n^{1/2}(\widehat{\boldsymbol{\theta }}_{\beta,\gamma}-\widetilde{\boldsymbol{%
% \theta }}_{\beta,\gamma})^{T}\boldsymbol{A}_{\gamma }\left( \boldsymbol{\theta }%
% _{0}\right) n^{1/2}(\widehat{\boldsymbol{\theta }}_{\beta,\gamma}-\widetilde{%
% \boldsymbol{\theta }}_{\beta,\gamma})
% \end{equation*}%
% %
% %
% %
% %
% %
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% are the same because
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{equation*}
% n\times o\left( \left\Vert \widehat{\boldsymbol{\theta }}_{\beta,\gamma}-%
% \widetilde{\boldsymbol{\theta }}_{\beta,\gamma}\right\Vert ^{2}\right)
% =o_{p}\left( 1\right) .
% \end{equation*}%
% %
% %
% %
% %
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Now we apply Corollary 2.1 in Dik and Gunst (1985), which states:
% \textquotedblleft Let $\boldsymbol{X}$ be a $q$-variate normal random
% variable with mean vector $\boldsymbol{0}$ and variance-covariance matrix $%
% \boldsymbol{\Sigma }$. Let $\boldsymbol{M}$ be a real symmetric matrix of
% order $q$. Let 
% % $k=\rm{rank}(\boldsymbol{\Sigma M\Sigma )}$, $k\geq 1$ and let 
% $%
% \lambda _{1},\ldots ,\lambda _{k},$ be the nonzero eigenvalues of $%
% \boldsymbol{M\Sigma .}$ Then the distribution of the quadratic form $%
% \boldsymbol{X}^{T}\boldsymbol{MX}$ coincides with the distribution of the
% random variable ${\textstyle\sum\limits_{i=1}^{k}}\lambda _{i}Z_{i}^{2},$
% where $Z_{1},\ldots ,Z_{k}$ are independent, each having a standard normal
% variable \textquotedblright . In our case the asymptotic distribution of
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% $T_{\gamma }(\widehat{\boldsymbol{\theta }}_{\beta,\gamma},\widetilde{\boldsymbol{%
% \theta }}_{\beta,\gamma})$
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% coincides with the distribution of the random variable
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% ${\textstyle\sum\limits_{i=1}^{k}}\lambda _{i}^{\beta ,\gamma }Z_{i}^{2}$
% where $\lambda _{1}^{\beta ,\gamma },\ldots ,\lambda _{k}^{\beta ,\gamma }$,
% are the nonzero eigenvalues of $\boldsymbol{A}_{\gamma }\left( \boldsymbol{%
% \theta }_{0}\right) \boldsymbol{B}_{\beta,\gamma}\left( \boldsymbol{\theta }%
% _{0}\right) \boldsymbol{K}_{\beta,\gamma}(\boldsymbol{\theta }_{0})\boldsymbol{B}%
% _{\beta,\gamma}\left( \boldsymbol{\theta }_{0}\right) $,
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % and
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % $k$ is as given in (\ref{k}).%
% where
% \begin{equation*}
% k={\rm rank}\left( \boldsymbol{B}_{\beta,\gamma}\left( \boldsymbol{\theta }_{0}\right)
% \boldsymbol{K}_{\beta,\gamma}(\boldsymbol{\theta }_{0})\boldsymbol{B}_{\beta
% }\left( \boldsymbol{\theta }_{0}\right) \boldsymbol{A}_{\gamma }\left(
% \boldsymbol{\theta }_{0}\right) \boldsymbol{B}_{\beta,\gamma}\left( \boldsymbol{%
% \theta }_{0}\right) \boldsymbol{K}_{\beta,\gamma}(\boldsymbol{\theta }_{0})%
% \boldsymbol{B}_{\beta,\gamma}\left( \boldsymbol{\theta }_{0}\right) \right) .%
% \end{equation*}%
% %
% %
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \end{proof}


\subsection{Two-Sample Problem}
	Again consider a parametric family of densities 
$\{f_\theta : \theta \in \Theta \subseteq \mathbb{R}^p\}$ as above in 
one sample problem, but here we are given two random 
samples $X_1, \ldots, X_n$ of size $n$ and $Y_1, \ldots, Y_m$ of 
size $m$ from two 
% populations 
distributions within the model
having parameters $\theta_1$ and 
$\theta_2$ respectively and based on these two samples, we want to 
test for the homogeneity of the two samples, i.e. to test the hypothesis 
$$
 H_0 : \theta_1 = \theta_2 ~~~ \mbox{against} ~~~~ H_1 : \theta_1 \ne \theta_2.
$$
 We will consider the estimator $^{(1)}\hat{\theta}_{\beta,\gamma}$  and 
$^{(2)}\hat{\theta}_{\beta,\gamma}$ of $\theta_1$ and $\theta_2$ respectively, 
obtained by minimizing the LSD having parameters ${\beta,\gamma}$ and,
%  then 
as before, 
% we 
will 
consider the test statistic based on the LSD with parameter $\beta$ 
and $\gamma$ as 
given by 
% follows
\begin{eqnarray}
    S_{\beta,\gamma}\left( ^{(1)}\hat{\theta}_{\beta,\gamma} , 
^{(2)}\hat{\theta}_{\beta,\gamma}\right) 
= \frac{2nm}{n+m} ~  {\rm LSD}_{\beta,\gamma}\left( f_{^{(1)}\hat{\theta}_{\beta,\gamma}} , 
f_{^{(2)}\hat{\theta}_{\beta,\gamma}}\right).
\end{eqnarray}
%  where $GPD( . , .)$ has the form given by (22), (23) or (24) 
% based on the values of the parameter $\alpha$ and $\lambda$. 
Now, first let us 
consider the asymptotic distribution of the test statistic $S_{\beta,\gamma}
\left( ^{(1)}\hat{\theta}_{\beta,\gamma}, ^{(2)}\hat{\theta}_{\beta,\gamma}\right)$ under $H_0$ 
in the following theorem. 
Assume that $\frac{m}{m+n}\rightarrow \omega, (0<\omega<1), \mbox{ as } m\rightarrow \infty
\mbox{ and } n\rightarrow\infty$. 
\begin{theorem}
\label{th_two}
% 	Suppose the model density satisfies the Lehmann and GPD 
% conditions. 
The asymptotic distribution of the test 
statistic \\	$S_{\beta,\gamma}\left( ^{(1)}\hat{\theta}_{\beta,\gamma} , 
^{(2)}\hat{\theta}_{\beta,\gamma}\right)$, under the null hypothesis $H_0 : \theta_1 = \theta_2$, 
coincides with the distribution of 
$$
\sum_{i=1}^r ~  \zeta_i^{\beta}(\theta_1)Z_i^2
$$
where $Z_1, \ldots,Z_r$ are independent standard normal variables, 
$\zeta_1^{\beta}(\theta_1), \ldots, \\ \zeta_r^{\beta}(\theta_1)$ 
are the nonzero eigenvalues of $A_{\beta}(\theta_1) J_{\beta}^{-1}(\theta_1) 
K_{\beta}(\theta_1) J_{\beta}^{-1}(\theta_1)$, with \\
$J_{\beta}(\cdot)$, $K_{\beta}(\cdot)$ 
and $A_{\beta}(\cdot)$ as defined in the previous section and
$$
r = rank\left( J_{\beta}^{-1}(\theta_1) 
K_{\beta}(\theta_1) J_{\beta}^{-1}(\theta_1)
A_{\beta}(\theta_1) J_{\beta}^{-1}(\theta_1) 
K_{\beta}(\theta_1) J_{\beta}^{-1}(\theta_1)\right).
$$
\begin{proof} 
 We have 
$$
\sqrt{n} \left( ^{(1)}\hat{\theta}_{\beta,\gamma} - 
\theta_1 \right) \rightarrow N( 0, J_{\beta}^{-1}
(\theta_1) K_{\beta}(\theta_1) J_{\beta}^{-1}(\theta_1) )
$$
and
$$
\sqrt{m} \left( ^{(2)}\hat{\theta}_{\beta,\gamma} - \theta_2 \right) 
\rightarrow N( 0, J_{\beta}^{-1}(\theta_2) K_{\beta}(\theta_2) 
J_{\beta}^{-1}(\theta_2) ).
$$
Let $\frac{m}{m+n} \rightarrow \omega \in (0,1)$ as $m,n \rightarrow 
\infty $. Then we have
$$
\sqrt{\frac{mn}{m+n}} \left( ^{(1)}\hat{\theta}_{\beta,\gamma} - \theta_1 
\right) \rightarrow N( 0, \omega J_{\beta}^{-1}(\theta_1) 
K_{\beta}(\theta_1) J_{\beta}^{-1}(\theta_1) )
$$
and
$$
\sqrt{\frac{mn}{m+n}} \left( ^{(2)}\hat{\theta}_{\beta,\gamma} - 
\theta_2 \right) \rightarrow N( 0, (1-\omega) J_{\beta}^{-1}
(\theta_2) K_{\beta}(\theta_2) J_{\beta}^{-1}(\theta_2) ).
$$
Now, under $H_0 : \theta_1 = \theta_2$, we get that
$$
\sqrt{\frac{mn}{m+n}} \left( ^{(1)}\hat{\theta}_{\beta,\gamma} - 
^{(2)}\hat{\theta}_{\beta,\gamma} \right) \rightarrow N( 0,  
J_{\beta}^{-1}(\theta_1) K_{\beta}(\theta_1) J_{\beta}^{-1}(\theta_1) ).
$$
Next consider the second order Taylor series expansion of 
${\rm LSD}_{\beta,\gamma}\left( f_{\theta_1} , f_{\theta_2}\right)$ around 
$\theta_1 = \theta_2$ at $\left( ^{(1)}\hat{\theta}_{\beta,\gamma} , 
^{(2)}\hat{\theta}_{\beta,\gamma} \right)$ as follows
\begin{eqnarray}
    {\rm LSD}_{\beta,\gamma}\left( f_{^{(1)}\hat{\theta}_{\beta,\gamma}} , 
f_{^{(2)}\hat{\theta}_{\beta,\gamma}}\right) & = & \frac{1}{2} \sum_{i,j=1}^p 
\left( \frac{\partial^2  {\rm LSD}_{\beta,\gamma}(f_{\theta_1}, 
f_{\theta_2})}{\partial \theta_{1i} \partial \theta_{1j}}\right)_{\theta_1 
= \theta_2} \nonumber \\
& & \left(\hat{\theta}_{\beta,\gamma}^{1i} - \theta_{1i}\right) 
\left(\hat{\theta}_{\beta,\gamma}^{1j} - \theta_{1j}\right) \nonumber
\\
& & +  \sum_{i,j=1}^p \left( \frac{\partial^2  {\rm LSD}_{\beta,\gamma}
(f_{\theta_1}, f_{\theta_2})}{\partial \theta_{1i} \partial 
\theta_{2j}}\right)_{\theta_1 = \theta_2}\nonumber \\
& &  \left(\hat{\theta}_{\beta,\gamma}^{1i} 
- \theta_{1i}\right) \left(\hat{\theta}_{\beta,\gamma}^{2j} - \theta_{2j}\right) \nonumber
\\
& & +  \frac{1}{2} \sum_{i,j=1}^p \left( \frac{\partial^2  
{\rm LSD}_{\beta,\gamma}(f_{\theta_1}, f_{\theta_2})}{\partial 
\theta_{2i} \partial \theta_{2j}}\right)_{\theta_1 = \theta_2} \nonumber \\
& & 
\left(\hat{\theta}_{\beta,\gamma}^{2i} - \theta_{2i}\right) \left(\hat{\theta}_{\beta,\gamma}^{2j} - 
\theta_{2j}\right) \nonumber
\\
& & + o\left( ||^{(1)}\hat{\theta}_{\beta,\gamma} - 
\theta_1 ||^2 \right) + o\left( ||^{(2)}\hat{\theta}_{\beta,\gamma} - 
\theta_2 ||^2 \right). \nonumber 
\end{eqnarray}
But for $ i = 1, \ldots, p$, we have 
$$
\frac{\partial  {\rm LSD}_{\beta,\gamma}(f_{\theta_1}, 
f_{\theta_2})}{\partial \theta_{1i}} = \frac{1+\beta}{B} 
\left[\frac{\sum f_{\theta_{1i}}^{1+\beta}u_{\theta_{1i}}}
{\sum f_{\theta_{1i}}^{1+\beta}}
- \frac{\sum f_{\theta_{1i}}^A f_{\theta_2}^B u_{\theta_{1i}}}
{\sum f_{\theta_{1i}}^A f_{\theta_2}^B }\right].
% \sum (f_{\theta_1}^B - f_{\theta_2}^B) f_{\theta_1}^{A-1} 
% \frac{\partial f_{\theta_1}}{\partial \theta_{1i}} d\mu(x)
$$
where $B = \beta - \gamma(1-\beta)$ and $A = 1+ 
\gamma(1-\beta)$ and hence 
\begin{eqnarray}
   \left( \frac{\partial^2  {\rm LSD}_{\beta,\gamma}(f_{\theta_1}, 
f_{\theta_2})}{\partial \theta_{1i} \partial \theta_{1j}}\right)_{\theta_1 = \theta_2} 
&=& 
% (1+\alpha) \int f_{\theta_1}^{\alpha - 1}  
% \frac{\partial f_{\theta_1}}{\partial \theta_{1i}}  
% \frac{\partial f_{\theta_1}}{\partial \theta_{1j}} 
% d\mu(x)  
% = 
a_{ij}^{\beta}(\theta_1), \nonumber \\
 \left( \frac{\partial^2  {\rm LSD}_{\beta,\gamma}
(f_{\theta_1}, f_{\theta_2})}{\partial \theta_{1i} 
\partial \theta_{2j}}\right)_{\theta_1 = \theta_2} 
&=& 
% -  
% \left( \frac{\partial^2  {\rm LSD}_{\alpha,\lambda}(f_{\theta_1}, 
% f_{\theta_2})}{\partial \theta_{1i} \partial 
% \theta_{1j}}\right)_{\theta_1 = \theta_2}  = 
- a_{ij}^{\beta}(\theta_1), \nonumber \\
 \left( \frac{\partial^2  {\rm LSD}_{\beta,\gamma}(f_{\theta_1}, 
f_{\theta_2})}{\partial \theta_{2i} \partial \theta_{2j}}\right)_{\theta_1 = \theta_2} 
& =&  
% \left( \frac{\partial^2  {\rm LSD}_{\alpha,\lambda}(f_{\theta_1}, f_{\theta_2})}
% {\partial \theta_{1i} \partial \theta_{1j}}\right)_{\theta_1 = 
% \theta_2} = 
a_{ij}^{\beta}(\theta_1). \nonumber
\end{eqnarray}
As in the one sample case, here also the second order partial
derivatives depend on $\beta$ only.
	Therefore, we get 
\begin{eqnarray}
    2~{\rm LSD}_{\beta,\gamma}\left( f_{^{(1)}\hat{\theta}_{\beta,\gamma}} , 
f_{^{(2)}\hat{\theta}_{\beta,\gamma}}\right) 
 &=& (^{(1)}\hat{\theta}_{\beta,\gamma} - 
\theta_1)^T A_{\beta}(\theta_1)(^{(1)}\hat{\theta}_{\beta,\gamma} - \theta_1) \nonumber \\
& & -  
2 (^{(1)}\hat{\theta}_{\beta,\gamma} - \theta_1)^T A_{\beta}(\theta_1)
(^{(2)}\hat{\theta}_{\beta,\gamma} - \theta_1) \nonumber \\
& & + (^{(2)}\hat{\theta}_{\beta,\gamma} - \theta_1)^T A_{\beta}(\theta_1)
(^{(2)}\hat{\theta}_{\beta,\gamma} - \theta_1) \nonumber \\ 
& & + o\left( ||^{(1)}
\hat{\theta}_{\beta,\gamma} - \theta_1 ||^2 \right) 
+ o\left( ||^{(2)}
\hat{\theta}_{\beta,\gamma} - \theta_2 ||^2 \right) \nonumber \\
&=&  (^{(1)}\hat{\theta}_{\beta,\gamma} - ^{(2)}\hat{\theta}_{\beta,\gamma})^T 
A_{\beta}(\theta_1)(^{(1)}\hat{\theta}_{\beta,\gamma} - 
^{(2)}\hat{\theta}_{\beta,\gamma}) \nonumber \\ 
& & + o\left( ||^{(1)}\hat{\theta}_{\beta,\gamma} 
- \theta_1 ||^2 \right) 
+ o\left( ||^{(2)}\hat{\theta}_{\beta,\gamma} \nonumber
- \theta_2 ||^2 \right), \nonumber 
\end{eqnarray}
with
% and
$$
o\left( ||^{(1)}\hat{\theta}_{\beta,\gamma} - \theta_1 ||^2 \right) = 
o_p\left(\frac{1}{n}\right) ~~~~ \mbox{and} ~~~~  o\left( ||^{(2)}\hat{\theta}_{\beta,\gamma} - 
\theta_2 ||^2 \right) = o_p\left(\frac{1}{m}\right).
$$
Thus the asymptotic distribution of 
$$
S_{\beta,\gamma}\left( ^{(1)}\hat{\theta}_{\beta,\gamma} , ^{(2)}\hat{\theta}_{\beta,\gamma}\right) 
= \frac{2nm}{n+m} ~  {\rm LSD}_{\beta,\gamma}\left( f_{^{(1)}\hat{\theta}_{\beta,\gamma}} 
, f_{^{(2)}\hat{\theta}_{\beta,\gamma}}\right)
$$
coincides with the distribution of the random variable 
$\sum \limits_{i=1}^r ~  \zeta_i^{{\beta}}(\theta_1)Z_i^2$. 
Like the one sample case the asymptotic distribution depends 
on $\beta$ only. 
% We consider the second order Taylor 
% series expansion of ${\rm LSD}_{(\alpha, \lambda)}(f_\theta, f_{\theta_0})$ 
% around $\theta = \theta_0$ at $\theta = \hat{\theta}_{\alpha,\lambda}$ as,
% \begin{eqnarray}
%     {\rm LSD}_{(\alpha, \lambda)}(f_{\hat{\theta}_{\alpha,\lambda}}, f_{\theta_0}) 
% &=& {\rm LSD}_{(\alpha, \lambda)}(f_{\theta_0}, f_{\theta_0}) + \sum_{i=1}^p
%  \nabla_i {\rm LSD}_{(\alpha, \lambda)}(f_\theta, f_{\theta_0})|_{\theta = \theta_0} 
% (\hat{\theta}_{\alpha,\lambda}^i - \theta_0^i) \nonumber \\
% & & + \frac{1}{2} \sum_{i,j} \nabla_{ij} {\rm LSD}_{(\alpha, \lambda)}
% (f_\theta, f_{\theta_0})|_{\theta = \theta_0} (\hat{\theta}_{\alpha,\lambda}^i - 
% \theta_0^i) (\hat{\theta}_{\alpha,\lambda}^j - \theta_0^j) + 
% o(||\hat{\theta}_{\alpha,\lambda} - \theta_0||^2) \nonumber
% \end{eqnarray}
% where the superscripts denotes the corresponding components. Now 
% we have $ {\rm LSD}_{(\alpha, \lambda)}(f_{\theta_0}, f_{\theta_0}) = 0 $ 
% and $$\nabla_i {\rm LSD}_{(\alpha, \lambda)}(f_\theta, 
% f_{\theta_0})|_{\theta = \theta_0} =0.$$ 
% % Also, we have
% % $$
% %  \nabla_{ij}^2 {\rm LSD}_{(\alpha, \lambda)}(f_\theta, 
% % f_{\theta_0})|_{\theta = \theta_0} =  (1+\gamma) 
% % \int f_{\theta_0}^{\gamma -1} \frac{\delta f_{\theta_0}}
% % {\delta \theta_i} \frac{\delta f_{\theta_0}}{\delta \theta_j} 
% % $$
% Note that the above second order partial derivative of 
% ${\rm LSD}_{(\alpha, \lambda)}(f_\theta, f_{\theta_0})$ at 
% $\theta = \theta_0 $ 
% % is independent of $\lambda$ and 
% % so we 
% will be denoted by $a_{ij}^{\alpha,\lambda} (\theta_0)$. Then 
% note that $A_{\alpha,\lambda}(\theta_0) = \left( a_{ij}^{\alpha,\lambda}
% (\theta_0) \right)_{i,j=1,\ldots,p}$.
% % \\
% Now from the above Taylor series expansion it is clear 
% that the random variables 
% $$ 
%     T_{\alpha,\lambda}( {\hat{\theta}_{\alpha,\lambda}} , {\theta_0}) = 
% 2 n {\rm LSD}_{\alpha,\lambda}( f_{\hat{\theta}_{\alpha,\lambda}} , f_{\theta_0}) 
% ~~~~ \mbox{and} ~~~~ \sqrt{n} (\hat{\theta}_{\alpha,\lambda} - \theta_0)^T 
% A_{\alpha,\lambda} (\theta_0) \sqrt{n} (\hat{\theta}_{\alpha,\lambda} - \theta_0)
% $$ 
% have the same asymptotic distribution. Now 
% from the previous section we know that 
% the asymptotic distribution of $\sqrt{n} (\hat{\theta}_{\alpha,\lambda}-\theta_0)$ 
% is normal with mean zero and variance $J_{\alpha,\lambda}^{-1}(\theta_0) 
% K_{\alpha,\lambda}(\theta_0) J_{\alpha,\lambda}^{-1}(\theta_0)$.\\
% Further we know that for $X \sim N_q(0,\Sigma)$, and a $q-$dimensional 
% real symmetric matrix $A$, the distribution of the quadratic form 
% $X^T A X$ is the same as that of
% $\sum_{i=1}^r \zeta_i Z_i^2$, where $Z_1, \ldots ,Z_r$ are independent 
% standard normal variables , $r = rank(\Sigma A \Sigma)$, $r \ge 1$ and 
% $\zeta_1, \ldots , \zeta_r$ are the nonzero eigenvalues of $A \Sigma$ 
% (Dik and Gunst, 1985, Corollary 2.1). Applying this result with 
% $ X =  \sqrt{n} (\hat{\theta}_{\alpha,\lambda} - \theta_0)$ we get the theorem.
\end{proof}
\end{theorem}
We have noted that the asymptotic distribution of the LSD based test statistics 
under the simple null hypothesis is independent of the parameter $\gamma$.
Maji, Ghosh and Basu (2014) have also reported a similar observation for the robustness of 
the corresponding test statistics. They have shown that the first order influence function 
of the test statistics is always zero at the simple null and its second order influence function under null, 
being a quadratic form in the first order influence function of the minimum LSD estimator used,
is independent of the parameter $\gamma$. However, the numerical illustrations reported in 
their paper and in the next section of present paper, this independence is not true for samples
with moderate size. Therefore, as in the case of estimation, the robustness of the LSD based 
test of simple null hypothesis can not e indicated in terms of the influence function analysis
even if we even go up to second order. However, Maji, Ghosh and Basu (2014) showed 
that the robustness of the minimum LSD estimators can be measured quite accurately
in terms of the secord order influence function of the estimator. 
Extending the same idea in case of testing, it is a routine exercise to 
see that the third order influence function of the test statistics at the null, 
being a function of the second order influence function of the corresponding estimator,
can serve a better measure of robustness in this case.
% \begin{theorem}
% % 	Suppose the model density satisfies the Lehmann and 
% % Ghosh-Basu conditions. 
% An approximation to the power function 
% of the test statistic $S_{\beta,\gamma}( ^{(1)}\hat{\theta}_{\beta,\gamma} 
% , ^{(2)}\hat{\theta}_{\beta,\gamma})$ for testing $H_0 : \theta_1 =\theta_2$ 
% against $H_1: \theta_1 \ne \theta_2$ at the significance level 
% $\alpha$ is given by
% \begin{eqnarray}
%     \pi_{m,n,\alpha}^{\beta,\gamma} (\theta_1, \theta_2) 
% = 1 - \Phi \left( \frac{\sqrt{\frac{nm}{n+m}}}{\sigma_{\beta
% }(\theta_1, \theta_2)} \left( \frac{s_\alpha^{\beta}}{2 } 
% \frac{n+m}{nm} - {\rm LSD}_{\beta,\gamma}(f_{\theta_1}, f_{\theta_2}) 
% \right) \right),  \theta_1 \ne \theta_2 \nonumber
% \end{eqnarray}
% where $s_\alpha^{\beta}$ is the $(1-\alpha)^{th}$ quantile 
% of the asymptotic distribution of \\ $S_{\beta,\gamma}( ^{(1)}\hat{\theta}_{\beta,\gamma} 
% , ^{(2)}\hat{\theta}_{\beta,\gamma})$, and $\sigma_{\beta}
% (\theta_1, \theta_2)$ is defined as
% \begin{eqnarray}
%     \sigma_{\beta}^2(\theta_1, \theta_2) &=&
%  \omega G_\beta^T J_\beta^{-1}(\theta_1) K_\beta(\theta_1) 
% J_\beta^{-1}(\theta_1) G_\beta \nonumber \\  
% & & + (1-\omega) H_\beta^T 
% J_\beta^{-1}(\theta_2) K_\beta(\theta_2) J_\beta^{-1}(\theta_2) H_\beta \nonumber
% \end{eqnarray}
% with 
% $$
% G_\beta = (g_1^\beta, \ldots, g_p^\beta)^T ~~~ 
% \mbox{and} ~~~ H_\beta = (h_1^\beta, \ldots, h_p^\beta)^T
% $$
% and for $i=1, \cdots, p$,
% $$
% g_i^\beta = \frac{\partial {\rm LSD}_{\beta,\gamma}(f_{\theta_1}, 
% f_{\theta_2})}{\partial \theta_{1i}}, ~~~~~ h_i^\beta = 
% \frac{\partial  {\rm LSD}_{\beta,\gamma}(f_{\theta_1}, 
% f_{\theta_2})}{\partial \theta_{2i}}  
% $$
% and $\omega$ being the limit of $\frac{m}{n+m}$ as $n,m \rightarrow \infty$.
% % \\
% \end{theorem}
% \begin{theorem}
% \label{th_two}
% % 	Suppose the model density satisfies the Lehmann and GPD 
% % conditions. 
% The asymptotic distribution of the test 
% statistic 	$S_{\beta,\gamma}\left( ^{(1)}\hat{\theta}_{\beta,\gamma} , 
% ^{(2)}\hat{\theta}_{\beta,\gamma}\right)$, under the null hypothesis $H_0 : \theta_1 = \theta_2$, 
% coincides with the distribution of 
% $$
% \sum_{i=1}^r ~  \zeta_i^{\beta,\gamma}(\theta_1)Z_i^2
% $$
% where $Z_1, \ldots,Z_r$ are independent standard normal variables, 
% $\zeta_1^{\beta,\gamma}(\theta_1), \ldots, \zeta_r^{\beta,\gamma}(\theta_1)$ 
% are the nonzero eigenvalues of $A_{\beta,\gamma}(\theta_1) J_{\beta,\gamma}^{-1}(\theta_1) 
% K_{\beta,\gamma}(\theta_1) J_{\beta,\gamma}^{-1}(\theta_1)$, with 
% $J_{\beta,\gamma}(\cdot)$, $K_{\beta,\gamma}(\cdot)$ 
% and $A_{\beta,\gamma}(\cdot)$ as defined in previous section and
% $$
% r = rank\left( J_{\beta,\gamma}^{-1}(\theta_1) 
% K_{\beta,\gamma}(\theta_1) J_{\beta,\gamma}^{-1}(\theta_1)
% A_{\beta,\gamma}(\theta_1) J_{\beta,\gamma}^{-1}(\theta_1) 
% K_{\beta,\gamma}(\theta_1) J_{\beta,\gamma}^{-1}(\theta_1)\right).
% $$
% \begin{proof} 
%  We have 
% $$
% \sqrt{n} \left( ^{(1)}\hat{\theta}_{\beta,\gamma} - 
% \theta_1 \right) \rightarrow N( 0, J_{\beta,\gamma}^{-1}
% (\theta_1) K_{\beta,\gamma}(\theta_1) J_{\beta,\gamma}^{-1}(\theta_1) )
% $$
% and
% $$
% \sqrt{m} \left( ^{(2)}\hat{\theta}_{\beta,\gamma} - \theta_2 \right) 
% \rightarrow N( 0, J_{\beta,\gamma}^{-1}(\theta_2) K_{\beta,\gamma}(\theta_2) 
% J_{\beta,\gamma}^{-1}(\theta_2) ).
% $$
% Let $\frac{m}{m+n} \rightarrow \omega \in (0,1)$ as $m,n \rightarrow 
% \infty $. Then we have
% $$
% \sqrt{\frac{mn}{m+n}} \left( ^{(1)}\hat{\theta}_{\beta,\gamma} - \theta_1 
% \right) \rightarrow N( 0, \omega J_{\beta,\gamma}^{-1}(\theta_1) 
% K_{\beta,\gamma}(\theta_1) J_{\beta,\gamma}^{-1}(\theta_1) )
% $$
% and
% $$
% \sqrt{\frac{mn}{m+n}} \left( ^{(2)}\hat{\theta}_{\beta,\gamma} - 
% \theta_2 \right) \rightarrow N( 0, (1-\omega) J_{\beta,\gamma}^{-1}
% (\theta_2) K_{\beta,\gamma}(\theta_2) J_{\beta,\gamma}^{-1}(\theta_2) ).
% $$
% Now, under $H_0 : \theta_1 = \theta_2$, we get that
% $$
% \sqrt{\frac{mn}{m+n}} \left( ^{(1)}\hat{\theta}_{\beta,\gamma} - 
% ^{(2)}\hat{\theta}_{\beta,\gamma} \right) \rightarrow N( 0,  
% J_{\beta,\gamma}^{-1}(\theta_1) K_{\beta,\gamma}(\theta_1) J_{\beta,\gamma}^{-1}(\theta_1) ).
% $$
% Next consider the second order Taylor series expansion of 
% ${\rm LSD}_{\beta,\gamma}\left( f_{\theta_1} , f_{\theta_2}\right)$ around 
% $\theta_1 = \theta_2$ at $\left( ^{(1)}\hat{\theta}_{\beta,\gamma} , 
% ^{(2)}\hat{\theta}_{\beta,\gamma} \right)$ as follows
% \begin{eqnarray}
%     {\rm LSD}_{\beta,\gamma}\left( f_{^{(1)}\hat{\theta}_{\beta,\gamma}} , 
% f_{^{(2)}\hat{\theta}_{\beta,\gamma}}\right) & = & \frac{1}{2} \sum_{i,j=1}^p 
% \left( \frac{\partial^2  {\rm LSD}_{\beta,\gamma}(f_{\theta_1}, 
% f_{\theta_2})}{\partial \theta_{1i} \partial \theta_{1j}}\right)_{\theta_1 
% = \theta_2} (\hat{\theta}_{\beta,\gamma}^{1i} - \theta_{1i}) 
% (\hat{\theta}_{\beta,\gamma}^{1j} - \theta_{1j}) \nonumber
% \\
% & & +  \sum_{i,j=1}^p \left( \frac{\partial^2  {\rm LSD}_{\beta,\gamma}
% (f_{\theta_1}, f_{\theta_2})}{\partial \theta_{1i} \partial 
% \theta_{2j}}\right)_{\theta_1 = \theta_2} (\hat{\theta}_{\beta,\gamma}^{1i} 
% - \theta_{1i}) (\hat{\theta}_{\beta,\gamma}^{2j} - \theta_{2j}) \nonumber
% \\
% & & +  \frac{1}{2} \sum_{i,j=1}^p \left( \frac{\partial^2  
% {\rm LSD}_{\beta,\gamma}(f_{\theta_1}, f_{\theta_2})}{\partial 
% \theta_{2i} \partial \theta_{2j}}\right)_{\theta_1 = \theta_2} 
% (\hat{\theta}_{\beta,\gamma}^{2i} - \theta_{2i}) (\hat{\theta}_{\beta,\gamma}^{2j} - 
% \theta_{2j}) \nonumber
% \\
% & & + o\left( ||^{(1)}\hat{\theta}_{\beta,\gamma} - 
% \theta_1 ||^2 \right) + o\left( ||^{(2)}\hat{\theta}_{\beta,\gamma} - 
% \theta_2 ||^2 \right). \nonumber 
% \end{eqnarray}
% But for $ i = 1, \ldots, p$, we have 
% $$
% \frac{\partial  {\rm LSD}_{\beta,\gamma}(f_{\theta_1}, 
% f_{\theta_2})}{\partial \theta_{1i}} = \frac{1+\alpha}{B} 
% \left[\frac{\int f_{\theta_{1i}}^{1+\alpha}u_{\theta_{1i}}}
% {\int f_{\theta_{1i}}^{1+\alpha}}
% - \frac{\int f_{\theta_{1i}}^A f_{\theta_2}^B u_{\theta_{1i}}}
% {\int f_{\theta_{1i}}^A f_{\theta_2}^B }\right].
% % \int (f_{\theta_1}^B - f_{\theta_2}^B) f_{\theta_1}^{A-1} 
% % \frac{\partial f_{\theta_1}}{\partial \theta_{1i}} d\mu(x)
% $$
% where $B = \alpha - \lambda(1-\alpha)$ and $A = 1+ 
% \lambda(1-\alpha)$ and hence 
% \begin{eqnarray}
%    \left( \frac{\partial^2  {\rm LSD}_{\beta,\gamma}(f_{\theta_1}, 
% f_{\theta_2})}{\partial \theta_{1i} \partial \theta_{1j}}\right)_{\theta_1 = \theta_2} 
% &=& 
% % (1+\alpha) \int f_{\theta_1}^{\alpha - 1}  
% % \frac{\partial f_{\theta_1}}{\partial \theta_{1i}}  
% % \frac{\partial f_{\theta_1}}{\partial \theta_{1j}} 
% % d\mu(x)  
% % = 
% a_{ij}^{\beta,\gamma}(\theta_1), \nonumber \\
%  \left( \frac{\partial^2  {\rm LSD}_{\beta,\gamma}
% (f_{\theta_1}, f_{\theta_2})}{\partial \theta_{1i} 
% \partial \theta_{2j}}\right)_{\theta_1 = \theta_2} 
% &=& 
% % -  
% % \left( \frac{\partial^2  {\rm LSD}_{\alpha,\lambda}(f_{\theta_1}, 
% % f_{\theta_2})}{\partial \theta_{1i} \partial 
% % \theta_{1j}}\right)_{\theta_1 = \theta_2}  = 
% - a_{ij}^{\beta,\gamma}(\theta_1), \nonumber \\
%  \left( \frac{\partial^2  {\rm LSD}_{\beta,\gamma}(f_{\theta_1}, 
% f_{\theta_2})}{\partial \theta_{2i} \partial \theta_{2j}}\right)_{\theta_1 = \theta_2} 
% & =&  
% % \left( \frac{\partial^2  {\rm LSD}_{\alpha,\lambda}(f_{\theta_1}, f_{\theta_2})}
% % {\partial \theta_{1i} \partial \theta_{1j}}\right)_{\theta_1 = 
% % \theta_2} = 
% a_{ij}^{\beta,\gamma}(\theta_1). \nonumber
% \end{eqnarray}
% 
% 	Therefore, we get 
% \begin{eqnarray}
%     2~{\rm LSD}_{\beta,\gamma}\left( f_{^{(1)}\hat{\theta}_{\beta,\gamma}} , 
% f_{^{(2)}\hat{\theta}_{\beta,\gamma}}\right) 
%  &=& (^{(1)}\hat{\theta}_{\beta,\gamma} - 
% \theta_1)^T A_{\beta,\gamma}(\theta_1)(^{(1)}\hat{\theta}_{\beta,\gamma} - \theta_1) - 
% 2 (^{(1)}\hat{\theta}_{\beta,\gamma} - \theta_1)^T A_{\beta,\gamma}(\theta_1)
% (^{(2)}\hat{\theta}_{\beta,\gamma} - \theta_1) \nonumber \\
% & & + (^{(2)}\hat{\theta}_{\beta,\gamma} - \theta_1)^T A_{\beta,\gamma}(\theta_1)
% (^{(2)}\hat{\theta}_{\beta,\gamma} - \theta_1) + o\left( ||^{(1)}
% \hat{\theta}_{\beta,\gamma} - \theta_1 ||^2 \right) \nonumber \\ 
% & & + o\left( ||^{(2)}
% \hat{\theta}_{\beta,\gamma} - \theta_2 ||^2 \right) \nonumber \\
% &=&  (^{(1)}\hat{\theta}_{\beta,\gamma} - ^{(2)}\hat{\theta}_{\beta,\gamma})^T 
% A_{\beta,\gamma}(\theta_1)(^{(1)}\hat{\theta}_{\beta,\gamma} - 
% ^{(2)}\hat{\theta}_{\beta,\gamma}) + o\left( ||^{(1)}\hat{\theta}_{\beta,\gamma} 
% - \theta_1 ||^2 \right) \nonumber \\ 
% & & + o\left( ||^{(2)}\hat{\theta}_{\beta,\gamma} \nonumber
% - \theta_2 ||^2 \right), \nonumber 
% \end{eqnarray}
% with
% % and
% $$
% o\left( ||^{(1)}\hat{\theta}_{\beta,\gamma} - \theta_1 ||^2 \right) = 
% o_p\left(\frac{1}{n}\right) ~~~~ \mbox{and} ~~~~  o\left( ||^{(2)}\hat{\theta}_{\beta,\gamma} - 
% \theta_2 ||^2 \right) = o_p\left(\frac{1}{m}\right).
% $$
% Thus the asymptotic distribution of 
% $$
% S_{\beta,\gamma}\left( ^{(1)}\hat{\theta}_{\beta,\gamma} , ^{(2)}\hat{\theta}_{\beta,\gamma}\right) 
% = \frac{2nm}{n+m} ~  {\rm LSD}_{\beta,\gamma}\left( f_{^{(1)}\hat{\theta}_{\beta,\gamma}} 
% , f_{^{(2)}\hat{\theta}_{\beta,\gamma}}\right)
% $$
% coincides with the distribution of the random variable 
% $\sum \limits_{i=1}^r ~  \zeta_i^{{\beta,\gamma}}(\theta_1)Z_i^2$.
% % We consider the second order Taylor 
% % series expansion of ${\rm LSD}_{(\alpha, \lambda)}(f_\theta, f_{\theta_0})$ 
% % around $\theta = \theta_0$ at $\theta = \hat{\theta}_{\alpha,\lambda}$ as,
% % \begin{eqnarray}
% %     {\rm LSD}_{(\alpha, \lambda)}(f_{\hat{\theta}_{\alpha,\lambda}}, f_{\theta_0}) 
% % &=& {\rm LSD}_{(\alpha, \lambda)}(f_{\theta_0}, f_{\theta_0}) + \sum_{i=1}^p
% %  \nabla_i {\rm LSD}_{(\alpha, \lambda)}(f_\theta, f_{\theta_0})|_{\theta = \theta_0} 
% % (\hat{\theta}_{\alpha,\lambda}^i - \theta_0^i) \nonumber \\
% % & & + \frac{1}{2} \sum_{i,j} \nabla_{ij} {\rm LSD}_{(\alpha, \lambda)}
% % (f_\theta, f_{\theta_0})|_{\theta = \theta_0} (\hat{\theta}_{\alpha,\lambda}^i - 
% % \theta_0^i) (\hat{\theta}_{\alpha,\lambda}^j - \theta_0^j) + 
% % o(||\hat{\theta}_{\alpha,\lambda} - \theta_0||^2) \nonumber
% % \end{eqnarray}
% % where the superscripts denotes the corresponding components. Now 
% % we have $ {\rm LSD}_{(\alpha, \lambda)}(f_{\theta_0}, f_{\theta_0}) = 0 $ 
% % and $$\nabla_i {\rm LSD}_{(\alpha, \lambda)}(f_\theta, 
% % f_{\theta_0})|_{\theta = \theta_0} =0.$$ 
% % % Also, we have
% % % $$
% % %  \nabla_{ij}^2 {\rm LSD}_{(\alpha, \lambda)}(f_\theta, 
% % % f_{\theta_0})|_{\theta = \theta_0} =  (1+\gamma) 
% % % \int f_{\theta_0}^{\gamma -1} \frac{\delta f_{\theta_0}}
% % % {\delta \theta_i} \frac{\delta f_{\theta_0}}{\delta \theta_j} 
% % % $$
% % Note that the above second order partial derivative of 
% % ${\rm LSD}_{(\alpha, \lambda)}(f_\theta, f_{\theta_0})$ at 
% % $\theta = \theta_0 $ 
% % % is independent of $\lambda$ and 
% % % so we 
% % will be denoted by $a_{ij}^{\alpha,\lambda} (\theta_0)$. Then 
% % note that $A_{\alpha,\lambda}(\theta_0) = \left( a_{ij}^{\alpha,\lambda}
% % (\theta_0) \right)_{i,j=1,\ldots,p}$.
% % % \\
% % Now from the above Taylor series expansion it is clear 
% % that the random variables 
% % $$ 
% %     T_{\alpha,\lambda}( {\hat{\theta}_{\alpha,\lambda}} , {\theta_0}) = 
% % 2 n {\rm LSD}_{\alpha,\lambda}( f_{\hat{\theta}_{\alpha,\lambda}} , f_{\theta_0}) 
% % ~~~~ \mbox{and} ~~~~ \sqrt{n} (\hat{\theta}_{\alpha,\lambda} - \theta_0)^T 
% % A_{\alpha,\lambda} (\theta_0) \sqrt{n} (\hat{\theta}_{\alpha,\lambda} - \theta_0)
% % $$ 
% % have the same asymptotic distribution. Now 
% % from the previous section we know that 
% % the asymptotic distribution of $\sqrt{n} (\hat{\theta}_{\alpha,\lambda}-\theta_0)$ 
% % is normal with mean zero and variance $J_{\alpha,\lambda}^{-1}(\theta_0) 
% % K_{\alpha,\lambda}(\theta_0) J_{\alpha,\lambda}^{-1}(\theta_0)$.\\
% % Further we know that for $X \sim N_q(0,\Sigma)$, and a $q-$dimensional 
% % real symmetric matrix $A$, the distribution of the quadratic form 
% % $X^T A X$ is the same as that of
% % $\sum_{i=1}^r \zeta_i Z_i^2$, where $Z_1, \ldots ,Z_r$ are independent 
% % standard normal variables , $r = rank(\Sigma A \Sigma)$, $r \ge 1$ and 
% % $\zeta_1, \ldots , \zeta_r$ are the nonzero eigenvalues of $A \Sigma$ 
% % (Dik and Gunst, 1985, Corollary 2.1). Applying this result with 
% % $ X =  \sqrt{n} (\hat{\theta}_{\alpha,\lambda} - \theta_0)$ we get the theorem.
% \end{proof}
% \end{theorem}
% \section{Numerical Examples and Simulations}
% % \subsection{Discrete Example}
% \subsection{Drosophila Data}
% % \paragraph{}
% This example 
% % (Table 
% % [\ref{droso_table1}]) 
% gives us the observed frequencies and 
% corresponding estimated frequencies (Table 
% \ref{droso_table1}) for several
% estimators of LSD family under Poisson model
% for a sex linked recessive lethal test in drosophila (fruit flies)
% exposed to a certain chemical. For each of several male flies
% one samples about $100$ daughter flies, and 
% then determines the frequency of the number of 
% daughter flies having a recessive lethal mutation 
% in its $X$-chromosome. The data represent a frequency 
% of frequencies; refer to 
% % ; 
% % refer 
% Woodruff et al.~(1984) for details. There is a possible case
% of outliers corresponding to the observations at $x=3,4$. 
% \paragraph{}
% Table \ref{droso_table1} provides the estimators and the predicted 
% frequencies for a small number of $\lambda,\alpha$ combinations,
% together with the fits of the maximum likelihood estimator
% (denoted by ML) and the outlier deleted maximum likelihood
% estimator obtained by removing the two outliers (denoted
% by ML$+$D). Clearly the estimators (and the estimated frequencies)
% are substantially different, demonstrating that the maximum 
% likelihood estimator is significantly affected by the presence 
% of these outliers.  
% % The ML method and ML$+$D 
% % method fits are shown in the rows corresponding to ML and ML$+$D.
% % Note that the ML method corresponds to the minimization of LSD 
% % with parameter $\alpha=\lambda=0$. 
% % While the estimates corresponding to only a few 
% % ($\lambda,\alpha$) combinations are presented in the table,
% % the general trends observed by us are as follows:
% While Table \ref{droso_table1} provides a small number of ($\lambda,\alpha$) 
% combinations, a large selection is presented in Table \ref{droso_estimate}, 
% where the salient features may be described as follows: 
% \begin{enumerate}[(a)]
%  \item All the estimators corresponding to large negative values of 
% $\lambda$ and/or values of $\alpha$ close to $1$ generate outlier 
% resistant methods,
% \item Estimators corresponding to large positive values of $\alpha$
% are relatively poor in terms of robustness, especially for small $\alpha$
% \item All estimators for $\alpha=1$ are identical,
% irrespective of the value of $\lambda$. 
% \end{enumerate}
% % show values by retaining and deleting
% % outliers respectively. LSD estimator with $\lambda=1$ and $\alpha=0.1$ gives estimate as
% % $0.6311$ and expected frequencies very far from the observed frequencies 
% %  whereas LSD estimator with $\lambda=-1$ and $\alpha=0.1$ overestimates as parameter
% % estiamte comes out to be $0.0762$. LSD estimator with $\alpha=1$, irrespective of $\lambda$ 
% % gives (see Table [\ref{droso_estimate}]) gives estimate $\theta$ as $0.1297$
% % which is close to estimate for ML$+$D case and estiamted frequencies are close to
% % observed freuencies.  
% \begin{table}
% \caption{ 
% % drosophila Data and Estimation Using Maximum Likelihhod Estimate
% Fits of the Poisson model to the drosophila data using several estimation methods
% }
% \vspace{2 mm}
% \centering % used for centering table
%   \begin{tabular}{l c c c c c c c} % centered columns (4 columns)
% \hline %inserts double horizontal lines
% & \multicolumn{7}{c}{Recessive lethal count}\\\cline{2-8}
% & 0 & 1 & 2 & 3 & 4 & $\geq 5$ 
% & $\hat{\theta}$
% \\
% Observed & 23 & 3 & 0 & 1& 1& 0\\
% ML & 19.59 & 7.00 & 1.25 & 0.15 & 0.01 & -- & 0.3571\\
% ML+D & 24.95 & 2.88 & 0.17 & 0.01 & -- & -- & 0.1154\\
% {\rm LSD}$_{\lambda = 1~~\&~~\alpha = 0.1}$ & 14.90 & 9.40 & 2.97 & 0.62 & 0.10 & 0.01 & 0.6311\\
% {\rm LSD}$_{\lambda = -1~~\&~~\alpha = 0.1}$ & 25.95 & 1.98 & 0.07 & -- & -- & -- & 0.0762\\
% {\rm LSD}$_{\lambda = 1~~\&~~\alpha = 1}$ & 24.59 & 3.19 & 0.21 & 0.01 & -- & -- & 0.1297\\
% {\rm LSD}$_{\lambda =-1~~\&~~\alpha = 1}$ & 24.59 & 3.19 & 0.21 & 0.01 & -- & -- & 0.1297\\
% \hline
% % \hline 
% \end{tabular}
% % \hline\hline 
% \label{droso_table1}
%  \end{table}
%  \begin{table}
% 	\centering 
% 	\caption{Estimates of the parameter for drosophila data in Table {\ref{droso_table1}}}
% 		\begin{tabular}{c c c c c c c c c c c c} \hline
% $\lambda$ \textdownarrow $\alpha$ $\rightarrow$ 
% % & 0 & 0.1 & 0.2 & 0.3 & 0.4 & 0.5 & 0.6 & 0.7 & 0.8 & 0.9 & 1 \\ 
% & 0 & 0.2  & 0.4  &  0.6 & 0.8 & 0.9 & 1 \\ 
% % 0.1 & 0.3147 & 0.2283 & 0.1742 & 0.1483 & 0.1373 & 0.1327 & 0.1307 & 0.13 & 0.1297 & 0.1297 \\ 
% % 0.2 & 0.3666 & 0.2685 & 0.1958 & 0.157 & 0.1405 & 0.1338 & 0.1311 & 0.1301 & 0.1298 & 0.1297 \\ 
% % 0.3 & 0.4144 & 0.3112 & 0.223 & 0.1688 & 0.1447 & 0.1353 & 0.1317 & 0.1303 & 0.1298 & 0.1297 \\ 
% % 0.4 & 0.4572 & 0.3533 & 0.2546 & 0.1842 & 0.1501 & 0.1371 & 0.1322 & 0.1305 & 0.1298 & 0.1297 \\ 
% % 0.5 & 0.4952 & 0.3929 & 0.2885 & 0.2035 & 0.1571 & 0.1392 & 0.1329 & 0.1307 & 0.1299 & 0.1297 \\ 
% % 0.6 & 0.529 & 0.4296 & 0.3227 & 0.2262 & 0.1662 & 0.1419 & 0.1336 & 0.1308 & 0.13 & 0.1297 \\ 
% % 0.7 & 0.5589 & 0.4631 & 0.3561 & 0.2515 & 0.1774 & 0.1451 & 0.1345 & 0.1311 & 0.13 & 0.1297 \\ 
% % 0.8 & 0.5856 & 0.4936 & 0.388 & 0.2783 & 0.1912 & 0.1491 & 0.1355 & 0.1313 & 0.1301 & 0.1297 \\ 
% % 0.9 & 0.6096 & 0.5213 & 0.4179 & 0.3056 & 0.2072 & 0.1541 & 0.1366 & 0.1316 & 0.1301 & 0.1297 \\ 
% % 1 & 0.6311 & 0.5464 & 0.4458 & 0.3326 & -- & 0.1602 & 0.138 & 0.1319 & 0.1302 & 0.1297 \\ 
% % \hline
% % $-0.1$ & 0.2136 & 0.1687 & 0.1471 & 0.1372 & 0.1328 & 0.1308 & 0.13 & 0.1297 & 0.1297 & 0.1297 \\ 
% % $-0.2$ & 0.1764 & 0.1511 & 0.1393 & 0.1337 & 0.1312 & 0.1301 & 0.1297 & 0.1296 & 0.1296 & 0.1297 \\ 
% % $-0.3$ & 0.1514 & 0.1394 & 0.1337 & 0.1311 & 0.1299 & 0.1294 & 0.1294 & 0.1294 & 0.1296 & 0.1297 \\ 
% % $-0.4$ & 0.1356 & 0.1315 & 0.1296 & 0.1289 & 0.1288 & 0.1288 & 0.1291 & 0.1293 & 0.1296 & 0.1297 \\ 
% % $-0.5$ & 0.1252 & 0.1257 & 0.1264 & 0.1271 & 0.1278 & 0.1284 & 0.1288 & 0.1292 & 0.1295 & 0.1297 \\ 
% % $-0.6$ & 0.1174 & 0.1211 & 0.1237 & 0.1256 & 0.1269 & 0.1279 & 0.1286 & 0.1291 & 0.1294 & 0.1297 \\ 
% % $-0.7$ & 0.1105 & 0.117 & 0.1212 & 0.1242 & 0.1261 & 0.1274 & 0.1284 & 0.129 & 0.1294 & 0.1297 \\ 
% % $-0.8$ & 0.103 & 0.1126 & 0.1188 & 0.1228 & 0.1253 & 0.127 & 0.1281 & 0.1289 & 0.1294 & 0.1297 \\ 
% % $-0.9$ & 0.0931 & 0.1075 & 0.116 & 0.1212 & 0.1245 & 0.1266 & 0.1279 & 0.1288 & 0.1293 & 0.1297 \\ 
% % $-1$ & 0.0762 & 0.1003 & 0.1127 & 0.1196 & 0.1237 & 0.1262 & 0.1277 & 0.1287 & 0.1293 & 0.1297 \\ 
% \hline
% % $-1$ & 0 & 0.076 & 0.1 & 0.113 & 0.12 & 0.124 & 0.126 & 0.128 & 0.129 & 0.129 & 0.13 \\ 
% % $-0.9$ & 0.068 & 0.093 & 0.108 & 0.116 & 0.121 & 0.125 & 0.127 & 0.128 & 0.129 & 0.129 & 0.13 \\ 
% % $-0.8$ & 0.088 & 0.103 & 0.113 & 0.119 & 0.123 & 0.125 & 0.127 & 0.128 & 0.129 & 0.129 & 0.13 \\ 
% % $-0.7$ & 0.101 & 0.111 & 0.117 & 0.121 & 0.124 & 0.126 & 0.127 & 0.128 & 0.129 & 0.129 & 0.13 \\ 
% % $-0.6$ & 0.112 & 0.117 & 0.121 & 0.124 & 0.126 & 0.127 & 0.128 & 0.129 & 0.129 & 0.129 & 0.13 \\ 
% % $-0.5$ & 0.125 & 0.125 & 0.126 & 0.126 & 0.127 & 0.128 & 0.128 & 0.129 & 0.129 & 0.13 & 0.13 \\ 
% % $-0.4$ & 0.145 & 0.136 & 0.132 & 0.13 & 0.129 & 0.129 & 0.129 & 0.129 & 0.129 & 0.13 & 0.13 \\ 
% % $-0.3$ & 0.177 & 0.151 & 0.139 & 0.134 & 0.131 & 0.13 & 0.129 & 0.129 & 0.129 & 0.13 & 0.13 \\ 
% % $-0.2$ & 0.227 & 0.176 & 0.151 & 0.139 & 0.134 & 0.131 & 0.13 & 0.13 & 0.13 & 0.13 & 0.13 \\ 
% % $-0.1$ & 0.291 & 0.214 & 0.169 & 0.147 & 0.137 & 0.133 & 0.131 & 0.13 & 0.13 & 0.13 & 0.13 \\ 
% % 0 & 0 & 0.262 & 0.194 & 0.158 & 0.142 & 0.135 & 0.132 & 0.13 & 0.13 & 0.13 & 0.13 \\ 
% % 0.1 & 0.417 & 0.315 & 0.228 & 0.174 & 0.148 & 0.137 & 0.133 & 0.131 & 0.13 & 0.13 & 0.13 \\ 
% % 0.2 & 0.47 & 0.367 & 0.269 & 0.196 & 0.157 & 0.141 & 0.134 & 0.131 & 0.13 & 0.13 & 0.13 \\ 
% % 0.3 & 0.514 & 0.414 & 0.311 & 0.223 & 0.169 & 0.145 & 0.135 & 0.132 & 0.13 & 0.13 & 0.13 \\ 
% % 0.4 & 0.553 & 0.457 & 0.353 & 0.255 & 0.184 & 0.15 & 0.137 & 0.132 & 0.131 & 0.13 & 0.13 \\ 
% % 0.5 & 0.586 & 0.495 & 0.393 & 0.289 & 0.204 & 0.157 & 0.139 & 0.133 & 0.131 & 0.13 & 0.13 \\ 
% % 0.6 & 0.615 & 0.529 & 0.43 & 0.323 & 0.226 & 0.166 & 0.142 & 0.134 & 0.131 & 0.13 & 0.13 \\ 
% % 0.7 & 0.641 & 0.559 & 0.463 & 0.356 & 0.252 & 0.177 & 0.145 & 0.135 & 0.131 & 0.13 & 0.13 \\ 
% % 0.8 & 0.663 & 0.586 & 0.494 & 0.388 & 0.278 & 0.191 & 0.149 & 0.136 & 0.131 & 0.13 & 0.13 \\ 
% % 0.9 & 0.683 & 0.61 & 0.521 & 0.418 & 0.306 & 0.207 & 0.154 & 0.137 & 0.132 & 0.13 & 0.13 \\ 
% % 1 & 0.701 & 0.631 & 0.546 & 0.446 & 0.333 & 0 & 0.16 & 0.138 & 0.132 & 0.13 & 0.13 \\ 
% % $-1$ & -- & 0.1 & 0.12 & 0.126 & 0.129 & 0.129 & 0.13 \\ 
% $-0.9$ & 0.068 & 0.108 & 0.121 & 0.127 & 0.129 & 0.129 & 0.13 \\ 
% $-0.8$ & 0.088 & 0.113 & 0.123 & 0.127 & 0.129 & 0.129 & 0.13 \\ 
% $-0.7$ & 0.101 & 0.117 & 0.124 & 0.127 & 0.129 & 0.129 & 0.13 \\ 
% $-0.6$ & 0.112 & 0.121 & 0.126 & 0.128 & 0.129 & 0.129 & 0.13 \\ 
% $-0.5$ & 0.125 & 0.126 & 0.127 & 0.128 & 0.129 & 0.13 & 0.13 \\ 
% $-0.4$ & 0.145 & 0.132 & 0.129 & 0.129 & 0.129 & 0.13 & 0.13 \\ 
% $-0.3$ & 0.177 & 0.139 & 0.131 & 0.129 & 0.129 & 0.13 & 0.13 \\ 
% $-0.2$ & 0.227 & 0.151 & 0.134 & 0.13 & 0.13 & 0.13 & 0.13 \\ 
% $-0.1$ & 0.291 & 0.169 & 0.137 & 0.131 & 0.13 & 0.13 & 0.13 \\ 
% 0 & 0.357 & 0.194 & 0.142 & 0.132 & 0.13 & 0.13 & 0.13 \\ 
% 0.1 & 0.417 & 0.228 & 0.148 & 0.133 & 0.13 & 0.13 & 0.13 \\ 
% 0.2 & 0.47 & 0.269 & 0.157 & 0.134 & 0.13 & 0.13 & 0.13 \\ 
% 0.3 & 0.514 & 0.311 & 0.169 & 0.135 & 0.13 & 0.13 & 0.13 \\ 
% 0.4 & 0.553 & 0.353 & 0.184 & 0.137 & 0.131 & 0.13 & 0.13 \\ 
% 0.5 & 0.586 & 0.393 & 0.204 & 0.139 & 0.131 & 0.13 & 0.13 \\ 
% 0.6 & 0.615 & 0.43 & 0.226 & 0.142 & 0.131 & 0.13 & 0.13 \\ 
% 0.7 & 0.641 & 0.463 & 0.252 & 0.145 & 0.131 & 0.13 & 0.13 \\ 
% 0.8 & 0.663 & 0.494 & 0.278 & 0.149 & 0.131 & 0.13 & 0.13 \\ 
% 0.9 & 0.683 & 0.521 & 0.306 & 0.154 & 0.132 & 0.13 & 0.13 \\ 
% 1 & 0.701 & 0.546 & 0.333 & 0.16 & 0.132 & 0.13 & 0.13 \\ 
% % $-1$ & -- & 0.0762 & 0.1003 & 0.1127 & 0.1196 & 0.1237 & 0.1262 & 0.1277 & 0.1287 & 0.1293 & 0.1297 \\ 
% % $-0.9$ & 0.0677 & 0.0931 & 0.1075 & 0.116 & 0.1212 & 0.1245 & 0.1266 & 0.1279 & 0.1288 & 0.1293 & 0.1297 \\ 
% % $-0.8$ & 0.0879 & 0.103 & 0.1126 & 0.1188 & 0.1228 & 0.1253 & 0.127 & 0.1281 & 0.1289 & 0.1294 & 0.1297 \\ 
% % $-0.7$ & 0.1008 & 0.1105 & 0.117 & 0.1212 & 0.1242 & 0.1261 & 0.1274 & 0.1284 & 0.129 & 0.1294 & 0.1297 \\ 
% % $-0.6$ & 0.1121 & 0.1174 & 0.1211 & 0.1237 & 0.1256 & 0.1269 & 0.1279 & 0.1286 & 0.1291 & 0.1294 & 0.1297 \\ 
% % $-0.5$ & 0.1252 & 0.1252 & 0.1257 & 0.1264 & 0.1271 & 0.1278 & 0.1284 & 0.1288 & 0.1292 & 0.1295 & 0.1297 \\ 
% % $-0.4$ & 0.1446 & 0.1356 & 0.1315 & 0.1296 & 0.1289 & 0.1288 & 0.1288 & 0.1291 & 0.1293 & 0.1296 & 0.1297 \\ 
% % $-0.3$ & 0.1768 & 0.1514 & 0.1394 & 0.1337 & 0.1311 & 0.1299 & 0.1294 & 0.1294 & 0.1294 & 0.1296 & 0.1297 \\ 
% % $-0.2$ & 0.2272 & 0.1764 & 0.1511 & 0.1393 & 0.1337 & 0.1312 & 0.1301 & 0.1297 & 0.1296 & 0.1296 & 0.1297 \\ 
% % $-0.1$ & 0.2913 & 0.2136 & 0.1687 & 0.1471 & 0.1372 & 0.1328 & 0.1308 & 0.13 & 0.1297 & 0.1297 & 0.1297 \\ 
% % 0 & 0.3571 & 0.2618 & 0.1943 & 0.1583 & 0.142 & 0.1348 & 0.1317 & 0.1303 & 0.1298 & 0.1297 & 0.1297 \\ 
% % 0.1 & 0.4172 & 0.3147 & 0.2283 & 0.1742 & 0.1483 & 0.1373 & 0.1327 & 0.1307 & 0.13 & 0.1297 & 0.1297 \\ 
% % 0.2 & 0.4695 & 0.3666 & 0.2685 & 0.1958 & 0.157 & 0.1405 & 0.1338 & 0.1311 & 0.1301 & 0.1298 & 0.1297 \\ 
% % 0.3 & 0.5143 & 0.4144 & 0.3112 & 0.223 & 0.1688 & 0.1447 & 0.1353 & 0.1317 & 0.1303 & 0.1298 & 0.1297 \\ 
% % 0.4 & 0.5528 & 0.4572 & 0.3533 & 0.2546 & 0.1842 & 0.1501 & 0.1371 & 0.1322 & 0.1305 & 0.1298 & 0.1297 \\ 
% % 0.5 & 0.5862 & 0.4952 & 0.3929 & 0.2885 & 0.2035 & 0.1571 & 0.1392 & 0.1329 & 0.1307 & 0.1299 & 0.1297 \\ 
% % 0.6 & 0.6152 & 0.529 & 0.4296 & 0.3227 & 0.2262 & 0.1662 & 0.1419 & 0.1336 & 0.1308 & 0.13 & 0.1297 \\ 
% % 0.7 & 0.6407 & 0.5589 & 0.4631 & 0.3561 & 0.2515 & 0.1774 & 0.1451 & 0.1345 & 0.1311 & 0.13 & 0.1297 \\ 
% % 0.8 & 0.6633 & 0.5856 & 0.4936 & 0.388 & 0.2783 & 0.1912 & 0.1491 & 0.1355 & 0.1313 & 0.1301 & 0.1297 \\ 
% % 0.9 & 0.6834 & 0.6096 & 0.5213 & 0.4179 & 0.3056 & 0.2072 & 0.1541 & 0.1366 & 0.1316 & 0.1301 & 0.1297 \\ 
% % 1 & 0.7014 & 0.6311 & 0.5464 & 0.4458 & 0.3326 & -- & 0.1602 & 0.138 & 0.1319 & 0.1302 & 0.1297 \\ 
% \hline
% 		\end{tabular}
% \label{droso_estimate}	 %{tab:Dwo}
% \end{table}
% 
% % \subsection{Continuous Example}
% % % \paragraph{}
% % The above section gives an example on discrete case by fitting a Poisson model.
% % In this section we will discuss on implementing continuous model.
% % %  in an example.
% % % two separate 
% % % examples. These two examples are 
% % % This data is well documented in Stigler (1977) and regarded
% % % as early data set used in robust statistics. 
% % % \subsubsection{Short's Data}
% % % % \paragraph{}
% % % This data comes from the experiment done in $1761$ when 
% % % James Short took some observations of the transit of Venus which is 
% % % regarded as the first reliable estimate of the mean distance from earth to sun.
% % % The dataset is available in Stigler $(1977)$. 
% % % We have used 
% % % kernel density function to calculate LSD estimators for the result of 
% % % Tables [\ref{short_mean}] and [\ref{short_std}]. Maximum likelihood estimate of mean and standard 
% % % deviation for this data set under normal model
% % % are respectively as $8.6096$ and $0.6742$ respectively.
% % % After deleting the outliers the values respectively become
% % % as $8.6277$ and $0.6364$ respectively. 
% % % We have used $\lambda=0$
% % % for this data and thus the divergence becomes
% % % \begin{equation}
% % % %  {\rm LSD}_{(\alpha=1, \lambda)}(g,f_\theta)=\log \int f_\theta^2 
% % % % -2 \log \int f_\theta g + \log \int g^2,
% % %  {\rm LSD}_{(\alpha, \lambda=0)}(g,f_\theta)=\log \int f_\theta^{1+\alpha}
% % % -2 \log \int f_\theta^\alpha g + \log \int g^{1+\alpha}.
% % % \label{lsd_alpha1}
% % % \end{equation}
% % % % and it shows that is does not depend on the value of $\lambda$. 
% % % The third term in the above expression does not depend on
% % % $\theta$, so in estimating $\theta$ the third term will not 
% % % contribute. Rewriting (\ref{lsd_alpha1}) as the following:
% % % \begin{equation}
% % %  {\rm LSD}_{(\alpha=1, \lambda)}(g,f_\theta)=\log \int f_\theta^{1+\alpha} 
% % % -\left(1+\frac{1}{\alpha}\right) \log \sum \limits_{i=1}^n f_\theta^\alpha(X_i) + K,
% % % \label{lsd_alpha}
% % % \end{equation}
% % % where $X_1,\ldots,X_n$ be
% % % independent and identically distributed observations having density $g$
% % % and $K$ is independent of $\theta$. Table [\ref{short_mean}] gives 
% % % results for both Shorts's data and Newcomb data (described in next section).
% % % Results with suffix $s$ gives the result for Short's data. 
% % % \subsubsection{Newcomb Data}
% % % \paragraph{}
% % This data was due to A. Michelson and S. Newcomb (for details see Stigler (1977)) when they performed
% % experiments in $1879-1882$ in determining the speed of light. 
% % We have used 
% % kernel density function to calculate LSD estimators for the result of 
% % Tables [\ref{newcomb_mean}] and [\ref{newcomb_std}]. 
% % To calculate 
% %  LSD estimators under normal model and using normal kernel we have 
% % followed the following procedure. We have taken bandwidth as
% % $$
% % h_n = 1.06 \sigma n^{1/5},
% % $$
% % where $$
% % \sigma = Median_i {|X_i - Median_j{X_j}| }/ 0.6745. 
% % $$
% % Following Basu and Lindsay (1994) approach we have done 
% % smoothing on model itself to significantly reduce the 
% % effect of kernel.  Here our model  $f_\theta(x)$ is the $N(\mu, \sigma^2)$ 
% % density, where $\theta = (\mu, \sigma^2)$, and we consider the kernel to 
% % be the normal kernel with smoothing parameter $h$ (i.e., $K(x, y, h)$ 
% % is the $N(y, h^2)$ density at $x$). Then the smoothed density 
% % $f_\theta^* (x)$ is the normal $N(\mu, \sigma^2+h^2)$ density.
% % We will minimize the following divergence between $f_\theta^{*}$ and 
% % $g^*$ as
% % \begin{eqnarray}
% % \label{lsd_form_cont}
% % 		{\rm LSD}_{(\alpha, \lambda)}(g^*,f_\theta^{*}) 
% %  &=&  \frac{1}{A} ~ \log \int ~ f_\theta{^{*^{1+\alpha}}}  -   \frac{1+\alpha}{A B} ~ 
% % \log \int ~~ f_\theta^{*^{B}} g^{*^{A}}  + \frac{1}{B} ~ \log \int ~~ g^{*^{1+\alpha}},
% % \end{eqnarray}
% % where
% % \begin{equation}
% %  g^{*}(x)=\int k(x;y,h)\hspace{2mm}dF_n(y),
% % \end{equation}
% % and $F_n$ is the empirical distribution
% % function. Maximum likelihood estimate of mean and standard 
% % deviation for this data set under normal model
% % are respectively as $26.2121$ and $10.6636$ respectively.
% % After deleting the outliers the values respectively become
% % as $27.75$ and $5.0442$ respectively. Tables [\ref{newcomb_mean}] and [\ref{newcomb_std}]
% % are divided into four parts according to the values of $\alpha$ and $\lambda$
% % respectively. It is evident that for $\alpha \geq 0.6$ and irrespective of $\lambda$ 
% % the method gives robust estimate. For $\lambda<0$, lower value of $\alpha$ also gives 
% % robust estimate but for $\lambda>0$ and lower value of $\alpha$ the method does not perform 
% % to that extent. 
% % %  \begin{table}
% % % 	\centering 
% % % 	\caption{Estimates of the parameter for Newcomb and Short's data}
% % % 		\begin{tabular}{|c| c c c c c| c c c c c|} \hline
% % % $\lambda$ \textdownarrow $\alpha$ $\rightarrow$ & 0.1 & 0.2 & 0.3 & 0.4 & 0.5 & 0.6 & 0.7 & 0.8 & 0.9 & 1 \\ 
% % % \hline
% % % % 0.1 & 8.4295 & 0.7504 & 27.5993 & 5.3849 \\ 
% % % % 0.2 & 8.466 & 0.5838 & 27.6487 & 5.0693 \\ 
% % % % 0.3 & 8.4249 & 0.4089 & 27.6162 & 4.9676 \\ 
% % % % 0.4 & 8.4056 & 0.2908 & 27.5679 & 4.8972 \\ 
% % % % 0.5 & 8.4155 & 0.2549 & 27.515 & 4.822 \\ 
% % % % 0.6 & 8.4203 & 0.2294 & 27.4598 & 4.7364 \\ 
% % % % 0.7 & 8.4175 & 0.2123 & 27.4039 & 4.6415 \\ 
% % % % 0.8 & 8.4102 & 0.1997 & 27.3485 & 4.5405 \\ 
% % % % 0.9 & 8.3989 & 0.1862 & 27.2954 & 4.4381 \\ 
% % % % 1 & 8.3139 & 0.056 & 27.246 & 4.339 \\ 
% % % -0.1 & 8.6048 & 8.6015 & 8.5977 & 8.5991 & 8.5907 & 8.5875 & 8.5846 & 8.583 & 8.5808 & 8.5788 \\ 
% % % -0.2 & 8.6054 & 8.6015 & 8.5975 & 8.5938 & 8.5906 & 8.5874 & 8.5845 & 8.583 & 8.5808 & 8.5788 \\ 
% % % -0.3 & 8.6046 & 8.6014 & 8.5974 & 8.5939 & 8.5905 & 8.5874 & 8.5845 & 8.583 & 8.5808 & 8.5788 \\ 
% % % -0.4 & 8.6052 & 8.6067 & 8.5973 & 8.5938 & 8.5904 & 8.5873 & 8.5844 & 8.583 & 8.5808 & 8.5788 \\ 
% % % -0.5 & 8.6045 & 8.6012 & 8.5973 & 8.5937 & 8.5904 & 8.5872 & 8.5844 & 8.583 & 8.5808 & 8.5788 \\ 
% % % -0.6 & 8.6045 & 8.6011 & 8.5971 & 8.5936 & 8.5903 & 8.5872 & 8.5844 & 8.5829 & 8.5808 & 8.5788 \\ 
% % % -0.7 & 8.6044 & 8.601 & 8.6021 & 8.5935 & 8.5902 & 8.5871 & 8.5853 & 8.5829 & 8.5808 & 8.5788 \\ 
% % % -0.8 & 8.6043 & 8.6009 & 8.5964 & 8.5934 & 8.5901 & 8.587 & 8.5852 & 8.5829 & 8.5808 & 8.5788 \\ 
% % % -0.9 & 8.6043 & 8.6008 & 8.5982 & 8.5932 & 8.5905 & 8.587 & 8.5852 & 8.5829 & 8.5808 & 8.5788 \\ 
% % % -1 & 8.6042 & 8.6007 & 8.5965 & 8.5932 & 8.5904 & 8.5869 & 8.5851 & 8.5828 & 8.5808 & 8.5788 \\ 
% % % \hline
% % % 		\end{tabular}
% % % \label{short_mean}	 %{tab:Dwo}
% % % \end{table}
% % %  \begin{table}
% % % 	\centering 
% % % 	\caption{Estimates of the parameter for Newcomb and Short's data}
% % % 		\begin{tabular}{|c| c c c c c| c c c c c|} \hline
% % % $\lambda$ \textdownarrow $\alpha$ $\rightarrow$ & 0.1 & 0.2 & 0.3 & 0.4 & 0.5 & 0.6 & 0.7 & 0.8 & 0.9 & 1 \\ 
% % % \hline
% % % % 0.1 & 8.4295 & 0.7504 & 27.5993 & 5.3849 \\ 
% % % % 0.2 & 8.466 & 0.5838 & 27.6487 & 5.0693 \\ 
% % % % 0.3 & 8.4249 & 0.4089 & 27.6162 & 4.9676 \\ 
% % % % 0.4 & 8.4056 & 0.2908 & 27.5679 & 4.8972 \\ 
% % % % 0.5 & 8.4155 & 0.2549 & 27.515 & 4.822 \\ 
% % % % 0.6 & 8.4203 & 0.2294 & 27.4598 & 4.7364 \\ 
% % % % 0.7 & 8.4175 & 0.2123 & 27.4039 & 4.6415 \\ 
% % % % 0.8 & 8.4102 & 0.1997 & 27.3485 & 4.5405 \\ 
% % % % 0.9 & 8.3989 & 0.1862 & 27.2954 & 4.4381 \\ 
% % % % 1 & 8.3139 & 0.056 & 27.246 & 4.339 \\ 
% % % -0.1 & 0.6546 & 0.6415 & 0.634 & 0.6309 & 0.6148 & 0.6074 & 0.6014 & 0.595 & 0.5898 & 0.5852 \\ 
% % % -0.2 & 0.6546 & 0.6409 & 0.6337 & 0.624 & 0.6144 & 0.6072 & 0.6013 & 0.5949 & 0.5898 & 0.5852 \\ 
% % % -0.3 & 0.6522 & 0.6403 & 0.6333 & 0.6219 & 0.6141 & 0.6069 & 0.6011 & 0.5948 & 0.5898 & 0.5852 \\ 
% % % -0.4 & 0.6525 & 0.6386 & 0.6329 & 0.6214 & 0.6138 & 0.6067 & 0.601 & 0.5947 & 0.5897 & 0.5852 \\ 
% % % -0.5 & 0.6501 & 0.6392 & 0.6299 & 0.621 & 0.6136 & 0.6064 & 0.6008 & 0.5946 & 0.5897 & 0.5852 \\ 
% % % -0.6 & 0.6491 & 0.6387 & 0.6294 & 0.6205 & 0.6133 & 0.6062 & 0.6007 & 0.5945 & 0.5896 & 0.5852 \\ 
% % % -0.7 & 0.6481 & 0.6382 & 0.6336 & 0.6201 & 0.613 & 0.6059 & 0.5998 & 0.5944 & 0.5896 & 0.5852 \\ 
% % % -0.8 & 0.6472 & 0.6378 & 0.6281 & 0.6197 & 0.6127 & 0.6057 & 0.5996 & 0.5943 & 0.5895 & 0.5852 \\ 
% % % -0.9 & 0.6463 & 0.6373 & 0.6302 & 0.6193 & 0.6126 & 0.6054 & 0.5994 & 0.5942 & 0.5895 & 0.5852 \\ 
% % % -1 & 0.6443 & 0.6369 & 0.6279 & 0.6188 & 0.6123 & 0.6052 & 0.5993 & 0.5941 & 0.5894 & 0.5852 \\ 
% % % \hline
% % % 		\end{tabular}
% % % \label{short_std}	 %{tab:Dwo}
% % % \end{table}
% %  \begin{table}
% % 
% % 	\centering 
% % 	\caption{Newcomb Estimate for Mean using Kernel Density}
% % 		\begin{tabular}{|c| c c c c c| c c c c c|} \hline
% % $\lambda$ \textdownarrow $\alpha$ $\rightarrow$ & 0.1 & 0.2 & 0.3 & 0.4 & 0.5 & 0.6 & 0.7 & 0.8 & 0.9 & 1 \\ 
% % \hline
% % -0.1 & 27.5619 & 27.5332 & 27.5431 & 27.4537 & 27.4978 & 27.5293 & 27.5526 & 27.5703 & 27.5844 & 27.5954 \\ 
% % -0.2 & 27.5338 & 27.5326 & 27.3991 & 27.459 & 27.5002 & 27.5303 & 27.5531 & 27.5708 & 27.5846 & 27.5954 \\ 
% % -0.3 & 27.5247 & 27.5329 & 27.4093 & 27.4631 & 27.5025 & 27.5312 & 27.5536 & 27.571 & 27.5845 & 27.5954 \\ 
% % -0.4 & 27.5214 & 27.5309 & 27.4171 & 27.4672 & 27.5042 & 27.5324 & 27.5543 & 27.5712 & 27.5849 & 27.5954 \\ 
% % -0.5 & 27.5191 & 27.5342 & 27.4229 & 27.4701 & 27.506 & 27.5332 & 27.5546 & 27.5713 & 27.5849 & 27.5954 \\ 
% % -0.6 & 27.521 & 27.3702 & 27.4283 & 27.473 & 27.5076 & 27.5339 & 27.555 & 27.5715 & 27.5849 & 27.5954 \\ 
% % -0.7 & 27.3215 & 27.3776 & 27.433 & 27.4759 & 27.5091 & 27.5353 & 27.5556 & 27.5719 & 27.5849 & 27.5954 \\ 
% % -0.8 & 27.3248 & 27.3951 & 27.437 & 27.4784 & 27.5105 & 27.5357 & 27.5562 & 27.5721 & 27.585 & 27.5954 \\ 
% % -0.9 & 27.3238 & 27.4201 & 27.4409 & 27.4804 & 27.5118 & 27.5367 & 27.5563 & 27.5722 & 27.5849 & 27.5954 \\ 
% % -1 & 27.5297 & 27.486 & 27.4446 & 27.4831 & 27.5135 & 27.5374 & 27.5571 & 27.5727 & 27.5852 & 27.5954 \\
% % \hline 
% % 0.1 & 27.8521 & 27.5625 & 27.5433 & 27.4374 & 27.4919 & 27.5268 & 27.5514 & 27.57 & 27.5842 & 27.5954 \\ 
% % 0.2 & 28.1891 & 27.6257 & 27.5458 & 27.4236 & 27.488 & 27.525 & 27.5511 & 27.5696 & 27.584 & 27.5954 \\ 
% % 0.3 & 28.5733 & 27.7729 & 27.556 & 27.5542 & 27.4828 & 27.5236 & 27.5502 & 27.5694 & 27.584 & 27.5954 \\ 
% % 0.4 & 28.949 & 27.9787 & 27.5865 & 27.553 & 27.4759 & 27.5215 & 27.5496 & 27.5692 & 27.584 & 27.5954 \\ 
% % 0.5 & 29.2794 & 28.24 & 27.651 & 27.5534 & 27.4656 & 27.5195 & 27.5487 & 27.5691 & 27.5839 & 27.5954 \\ 
% % 0.6 & 29.5903 & 28.4791 & 27.7632 & 27.5589 & 27.4504 & 27.5171 & 27.5484 & 27.5686 & 27.5839 & 27.5954 \\ 
% % 0.7 & 29.8875 & 28.7295 & 27.9059 & 27.5169 & 27.5636 & 27.5139 & 27.5471 & 27.5686 & 27.5838 & 27.5954 \\ 
% % 0.8 & 26.6758 & 28.9379 & 28.0686 & 27.606 & 27.5617 & 27.5096 & 27.5463 & 27.5679 & 27.5836 & 27.5954 \\ 
% % 0.9 & -36.4015 & 29.1438 & 28.2444 & 27.6671 & 27.5597 & 27.5038 & 27.5454 & 27.5676 & 27.5836 & 27.5954 \\ 
% % 1 & -84.4704 & 29.3249 & 28.4014 & 27.75 & -- & 27.4949 & 27.5442 & 27.5676 & 27.5834 & 27.5954 \\ 
% % \hline
% % 		\end{tabular}
% % \label{newcomb_mean}	 %{tab:Dwo}
% % \end{table}
% %  \begin{table}
% % 
% % 	\centering 
% % 	\caption{Newcomb Estimate for Standard Deviation using Kernel Density}
% % 		\begin{tabular}{|c| c c c c c| c c c c c|} \hline
% % $\lambda$ \textdownarrow $\alpha$ $\rightarrow$ & 0.1 & 0.2 & 0.3 & 0.4 & 0.5 & 0.6 & 0.7 & 0.8 & 0.9 & 1 \\ 
% % \hline
% % -0.1 & 7.2353 & 6.3747 & 5.9915 & 5.28 & 5.2312 & 5.1985 & 5.1754 & 5.1585 & 5.1455 & 5.1356 \\ 
% % -0.2 & 6.7339 & 6.2053 & 5.3423 & 5.273 & 5.2282 & 5.1972 & 5.1748 & 5.1581 & 5.1454 & 5.1356 \\ 
% % -0.3 & 6.4759 & 6.1141 & 5.3285 & 5.2675 & 5.2255 & 5.1961 & 5.1742 & 5.1579 & 5.1454 & 5.1356 \\ 
% % -0.4 & 6.3452 & 6.0734 & 5.3182 & 5.2625 & 5.2233 & 5.1948 & 5.1736 & 5.1576 & 5.1452 & 5.1356 \\ 
% % -0.5 & 6.2607 & 6.0233 & 5.3103 & 5.2586 & 5.2211 & 5.1937 & 5.1731 & 5.1574 & 5.1451 & 5.1356 \\ 
% % -0.6 & 6.189 & 5.5169 & 5.3034 & 5.255 & 5.2192 & 5.1928 & 5.1726 & 5.1571 & 5.1451 & 5.1356 \\ 
% % -0.7 & 5.4335 & 5.5423 & 5.2974 & 5.2515 & 5.2174 & 5.1916 & 5.172 & 5.1568 & 5.145 & 5.1356 \\ 
% % -0.8 & 5.4272 & 5.5795 & 5.2922 & 5.2485 & 5.2157 & 5.1908 & 5.1715 & 5.1566 & 5.1449 & 5.1356 \\ 
% % -0.9 & 5.4207 & 5.6564 & 5.2874 & 5.2458 & 5.2141 & 5.1898 & 5.1711 & 5.1564 & 5.1449 & 5.1356 \\ 
% % -1 & 6.0522 & 5.8222 & 5.2829 & 5.2429 & 5.2124 & 5.1889 & 5.1705 & 5.1561 & 5.1447 & 5.1356 \\ 
% % \hline
% % 0.1 & 9.754 & 7.2157 & 6.2049 & 5.3025 & 5.2387 & 5.2015 & 5.1767 & 5.1589 & 5.1457 & 5.1356 \\ 
% % 0.2 & 11.4248 & 8.2105 & 6.4632 & 5.3224 & 5.2438 & 5.2035 & 5.1773 & 5.1593 & 5.1459 & 5.1356 \\ 
% % 0.3 & 12.867 & 9.5841 & 6.9368 & 5.9971 & 5.2508 & 5.2053 & 5.1782 & 5.1595 & 5.1459 & 5.1356 \\ 
% % 0.4 & 14.0592 & 10.9752 & 7.7377 & 6.1564 & 5.2603 & 5.2078 & 5.1789 & 5.1598 & 5.146 & 5.1356 \\ 
% % 0.5 & 15.0456 & 12.2181 & 8.8224 & 6.4361 & 5.2746 & 5.2105 & 5.1798 & 5.16 & 5.1461 & 5.1356 \\ 
% % 0.6 & 15.8745 & 13.2645 & 10.0034 & 6.9121 & 5.297 & 5.2138 & 5.1805 & 5.1605 & 5.1462 & 5.1356 \\ 
% % 0.7 & 16.5844 & 14.1598 & 11.1106 & 7.7248 & 5.9613 & 5.2181 & 5.1818 & 5.1607 & 5.1463 & 5.1356 \\ 
% % 0.8 & 15.6928 & 14.9248 & 12.0884 & 8.5145 & 6.1368 & 5.224 & 5.1828 & 5.1612 & 5.1464 & 5.1356 \\ 
% % 0.9 & 184.5052 & 15.5899 & 12.9444 & 9.4817 & 6.4233 & 5.2321 & 5.184 & 5.1615 & 5.1465 & 5.1356 \\ 
% % 1 & 317.0595 & 16.1616 & 13.6875 & 10.4092 & -- & 5.2443 & 5.1854 & 5.1617 & 5.1466 & 5.1356 \\ 
% % \hline
% % 		\end{tabular}
% % \label{newcomb_std}	 %{tab:Dwo}
% % \end{table}
In this article we have restricted ourselves 
to the simple null case. However the results may be extended 
to the case involving nuisance 
parameters following the same general approach. 
\section{A Two-Sample Example}
% {Two Sample Testing of Hypothesis Problem}
\label{SEC:testing:example}
Here we will discuss a two sample real data example which
is known to give rise to occasional spurious counts. This experiment
is available in  Woodruff et al.~(1984) and 
has been analyzed previously by 
Simpson (1989). 
This is a sex-linked recessive lethal experiment in drosophila (fruit flies)
to test chemical mutagenicity. Male flies were exposed either to 
$2000~\mu g$ butyraldehyde or to control conditions. The 
responses are the numbers of recessive lethal 
mutations observed among daughters of these flies. 
The data are given in Table \ref{tab:twoDdata}. 
We will use a Poisson model in this experiment
where the control responses 
are supposed to follow Poisson distribution with mean $\theta_0$
and and the treated responses follows a Poisson distribution with mean $\theta_1$. 
The two large counts for the treated group appears to be 
possible outliers. 
 We want to test
$H_0:\theta_0\geq\theta_1$ against $H_1:\theta_0<\theta_1.$  
The test statistic for testing this hypothesis is given by,
$$
^*S_{\beta,\gamma}\left( ^{(1)}\hat{\theta}_{\beta,\gamma}, 
^{(2)}\hat{\theta}_{\beta,\gamma}\right) = \frac{1}{\zeta(^{(0)}\hat{\theta}_{\beta,\gamma})} 
 \frac{2nm}{n+m} ~  {\rm LSD}_{\beta,\gamma}\left( p({^{(1)}\hat{\theta}_{\beta,\gamma}}) 
, p({^{(2)}\hat{\theta}_{\beta,\gamma}})\right),
$$
where
\begin{eqnarray}
    \zeta(^{(0)}\hat{\theta}_{\beta,\gamma}) 
&=& \frac{A_{\beta,\gamma}(^{(0)}\hat{\theta}_{\beta,\gamma}) 
K_{\beta,\gamma}(^{(0)}\hat{\theta}_{\beta,\gamma})}
{J_{\beta,\gamma}^2(^{(0)}\hat{\theta}_{\beta,\gamma})}. \nonumber \\
% A_\gamma(\theta) &=& (1+\gamma) \sum_{x=0}^\infty 
% \left(\frac{e^{-\theta} \theta^x}{x!}\right)^{1+\gamma} 
% \left(\frac{x}{\theta} - 1\right)^2 \nonumber \\
% J_\beta(\theta) &=&  \sum_{x=0}^\infty \left(\frac{e^{-\theta} 
% \theta^x}{x!}\right)^{1+\beta} \left(\frac{x}{\theta} - 1\right)^2 \nonumber \\
% V_\beta(\theta) &=& \sum_{x=0}^\infty \left(\frac{e^{-\theta} 
% \theta^x}{x!}\right)^{1+2\beta} \left(\frac{x}{\theta} - 1\right)^2  - 
% \left[ \sum_{x=0}^\infty \left(\frac{e^{-\theta} \theta^x}{x!}\right)^{1+\beta} 
% \left(\frac{x}{\theta} - 1\right) \right]^2\nonumber 
\end{eqnarray}
The asymptotic distribution of the statistic 
$^*S_{\gamma,\lambda}( ^{(1)}\hat{\theta}_\beta 
, ^{(2)}\hat{\theta}_\beta)$ is chi-square with one degree of freedom 
and the corresponding $p$-values are calculated and shown in the tables.
The statistic for $\beta = \gamma = 0$ case is not same as 
the likelihood ratio statistic but they are 
asymptotically same. Though the tests are different 
but the non-robust nature of the likelihood test can be seen 
under this set-up also.  
The results are shown in 
Tables \ref{tab:est2} and \ref{signed_test}. 
% Tables \ref{with_outlier} and \ref{witho_outlier}. 
From the results it is evident that for $\beta \geq 0.6$ and irrespective of $\gamma$ 
the presence and the absence of outliers 
has little impact on $p$-values.
% both cases give close $p$-values. 
For $\gamma<0$, lower values of $\beta$ also give 
close $p$-values but for $\gamma>0$, lower values of $\beta$ the method does not perform well. 
It is clear that for large values of $\beta~(\mbox{say}~\geq 0.5)$, 
the full data and the outlier deleted data basically lead to  the same conclusion 
and almost identical $p$-values irrespective of the value of $\gamma$. 
The situation changes when $\beta$ is a small positive value close to $0$. 
In this case the role of $\gamma$ becomes decisive. Large positive values 
of $\gamma$ and small values of $\beta$ lead to a highly unstable results. 
The outlier deleted $p$-values and full data $p$-values are far from 
close in these cases. However the negative values of $\gamma$ lead to
stable inference even when $\beta=0$ or in its neighborhood. 
On the whole it appears that the two large counts in the treated 
group indicate a false significance for the likelihood ratio 
test and some other members of our class, but the 
more robust members clearly recognize the significance 
to be false.  
% to that extent. 
\begin{table}
	\centering
	\caption{Frequencies of the number of recessive lethal daughters for drosophila data}
		\begin{tabular}{c c c c c c c c c } \hline
		$x$      &0 & 1  &2  &3  &4  &5  &6 & 7 \\ \hline
Observed (Control) & 159 & 15 & 3 & 0 & 0 & 0 & 0 & 0 \\
Observed (Treated) & 110 & 11 & 5 & 0 & 0 & 0 & 1 & 1 \\
 \hline
		\end{tabular}
	 \label{tab:twoDdata}
\end{table} 
\begin{table}
	\centering
	\caption{Estimated Poisson parameters for the two-sample drosophila example;
the numbers within the parentheses 
%  (the numbers 
% in the bracket 
show the corresponding estimates for the treated case after deleting 
the two outliers.
The parameter $\gamma$ is held at $0$  for the first part 
and the parameter $\beta$ is held at $0$  for the second part. 
%  ($\lambda=0$))
}
	\scalebox{0.85} {	\begin{tabular}
% {p{0.25\linewidth}p{0.25\linewidth}p{0.25\linewidth}}
{c c c c c c c c c c c } 
\hline
	$\beta$	        
% &    $ 0.1  $  
&    $0.2$
%  &    $ 0.3$   
&    $ 0.4 $   
% &    $0.5 $   
&    $0.6 $  
%  &    $ 0.7  $ 
&    $ 0.8 $   
% &      $0.9$  
& $1$ \\ \hline 
% \hline
% $^{(1)}\hat{\theta}_\alpha$  & 0.1135 & 0.1091 & 0.1055 & 0.1027 & 0.1006 
% & 0.099 & 0.0978 & 0.0969 & 0.0962 &  0.0957 \\ 
%  \hline 
% $^{(2)}\hat{\theta}_\alpha$  & 0.1818 & 0.153 & 0.1369 & 0.1266 
% & 0.1195 & 0.1143 & 0.1105 & 0.1077 & 0.1057 &  0.1042 \\ 
% %  \\		
% 				 & (0.1545) & (0.1432) & (0.1335) & (0.1255) & (0.1191) 
% & (0.1141) & (0.1105) & (0.1077) & (0.1057) &  (0.1042) \\ 
% \hline 
% $^{(0)}\hat{\theta}_\alpha$  & 0.1406 & 0.1264 & 0.1179 & 0.1122 
% & 0.108 & 0.1051 & 0.1029 & 0.1013 & 0.1 &  0.0991 \\ 
% 			 & (0.1302) & (0.1229) & (0.1167) & (0.1118) & (0.108) 
% & (0.105) & (0.1028) & (0.1013) & (0.1) &  (0.0991) \\ 
$^{(1)}\hat{\theta}_\beta$   & 0.1091 & 0.1027 & 0.099 & 0.0969 & 0.0957 \\ 
$^{(2)}\hat{\theta}_\beta$   & 0.153 & 0.1266 & 0.1143 & 0.1077 & 0.1042 \\ 
 & (0.1432) & (0.1255) & (0.1141) & (0.1105) & (0.1057) \\ 
$^{(0)}\hat{\theta}_\beta$   & 0.1264 & 0.1122 & 0.1051 & 0.1029 & 0.1 \\ 
 & (0.1229) & (0.1118) & (0.105) & (0.1028) & (0.1) \\ 



 \hline
	$\gamma$	        
% &    $ 0.1  $  
&    $0.2$
%  &    $ 0.3$   
&    $ 0.4 $   
% &    $0.5 $   
&    $0.6 $  
%  &    $ 0.7  $ 
&    $ 0.8 $   
% &      $0.9$  
& $1$ \\ \hline
$^{(1)}\hat{\theta}_\gamma$    & 0.1216 & 0.1245 & 0.1273 & 0.1303 & 0.133 \\ 
$^{(2)}\hat{\theta}_\gamma$    & 0.3227 & 0.4139 & 0.4916 & 0.5547 & 0.6059 \\ 
 & (0.1763) & (0.1854) & (0.1938) & (0.2015) & (0.2084) \\ 
$^{(0)}\hat{\theta}_\gamma$    & 0.2182 & 0.2851 & 0.3497 & 0.4055 & 0.4521 \\ 
  & (0.1444) & (0.1501) & (0.1555) & (0.1607) & (0.1655) \\ 
\hline
		\end{tabular}}
	 \label{tab:est2}
\end{table}
 \begin{table}

	\centering 
	\caption{The 
% (with $\gamma =\alpha$)
$p$-values for the two-sample drosophila data.
The outlier deleted 
% Outliers deleted 
$p$-values are given in the second line of 
each block
% result in
% the respective second lines)  
}
\scalebox{0.8}		{\begin{tabular}{c c c c c c c c c c c c} \hline
$\gamma$ \textdownarrow $\beta$ $\rightarrow$ 
& 0 & 0.1 & 0.2 & 0.3 & 0.4 & 0.5 & 0.6 & 0.7 & 0.8 & 0.9 & 1 \\ 

\hline
% $-0.8$ & 0.2622 & 0.3406 & 0.4263 & 0.5092 & 0.5815 & 0.6419 & 0.6945 & 0.7375 & 0.7752 & 0.806 & 0.8309 \\ 
%  & 0.2644 & 0.3428 & 0.4284 & 0.5111 & 0.5832 & 0.6434 & 0.6959 & 0.7387 & 0.7762 & 0.8069 & 0.8317 \\ 
% $-0.7$ & 0.2351 & 0.3247 & 0.4174 & 0.5056 & 0.5823 & 0.6441 & 0.6984 & 0.7398 & 0.7764 & 0.8065 & 0.8309 \\ 
%  & 0.2387 & 0.327 & 0.4196 & 0.5075 & 0.584 & 0.6456 & 0.6997 & 0.741 & 0.7773 & 0.8074 & 0.8317 \\ 
% $-0.6$ & 0.2184 & 0.3144 & 0.4118 & 0.5012 & 0.5792 & 0.6452 & 0.699 & 0.7424 & 0.7793 & 0.8069 & 0.8309 \\
%  & 0.2205 & 0.3167 & 0.4139 & 0.5032 & 0.5809 & 0.6467 & 0.7003 & 0.7435 & 0.7803 & 0.8078 & 0.8317 \\
% $-0.5$ & 0.2069 & 0.3063 & 0.4057 & 0.4985 & 0.5781 & 0.6456 & 0.7001 & 0.745 & 0.7788 & 0.8074 & 0.8309 \\
%  & 0.2111 & 0.3085 & 0.4079 & 0.5004 & 0.5798 & 0.6471 & 0.7014 & 0.7461 & 0.7798 & 0.8083 & 0.8317 \\  
% $-0.4$ & 0.1991 & 0.3007 & 0.4015 & 0.4955 & 0.579 & 0.647 & 0.7035 & 0.746 & 0.7818 & 0.8094 & 0.8309 \\ 
%  & 0.2079 & 0.3054 & 0.4036 & 0.4974 & 0.5808 & 0.6486 & 0.7048 & 0.7472 & 0.7828 & 0.8103 & 0.8317 \\ 
% $-0.3$ & 0.1874 & 0.2944 & 0.3977 & 0.4943 & 0.5783 & 0.6478 & 0.7022 & 0.7474 & 0.7817 & 0.8099 & 0.8309 \\ 
%  & 0.2107 & 0.3032 & 0.4025 & 0.4976 & 0.5801 & 0.6493 & 0.7035 & 0.7485 & 0.7826 & 0.8108 & 0.8317 \\ 
% $-0.2$ & 0.1491 & 0.284 & 0.395 & 0.4931 & 0.5781 & 0.6465 & 0.7048 & 0.7489 & 0.7847 & 0.8104 & 0.8309 \\
%  & 0.2195 & 0.3072 & 0.402 & 0.4964 & 0.5799 & 0.648 & 0.7061 & 0.75 & 0.7856 & 0.8112 & 0.8317 \\ 
% $-0.1$ & 0.0575 & 0.2505 & 0.3881 & 0.4911 & 0.5755 & 0.6479 & 0.7048 & 0.7492 & 0.7846 & 0.8109 & 0.8309 \\
%  & 0.2354 & 0.3154 & 0.4044 & 0.4968 & 0.5786 & 0.6508 & 0.7061 & 0.7503 & 0.7856 & 0.8118 & 0.8317 \\
% 0 & 0.003 & 0.1556 & 0.3672 & 0.4875 & 0.5751 & 0.6488 & 0.7068 & 0.7526 & 0.7862 & 0.8114 & 0.8309 \\ 
%   & 0.141 & 0.328 & 0.4109 & 0.4964 & 0.5781 & 0.6503 & 0.7081 & 0.7537 & 0.7872 & 0.8122 & 0.8317 \\ 
% 0.1 & 0 & 0.0303 & 0.3063 & 0.4789 & 0.5753 & 0.648 & 0.7065 & 0.7533 & 0.7878 & 0.8135 & 0.8309 \\ 
%  & 0.293 & 0.3454 & 0.4189 & 0.5001 & 0.5794 & 0.6495 & 0.7078 & 0.7544 & 0.7888 & 0.8143 & 0.8317 \\ 
% 0.2 & 0 & 0.0006 & 0.1721 & 0.4536 & 0.5717 & 0.6484 & 0.7068 & 0.7542 & 0.788 & 0.814 & 0.8309 \\ 
%  & 0.3377 & 0.3689 & 0.4288 & 0.5041 & 0.5801 & 0.6512 & 0.7081 & 0.7553 & 0.789 & 0.8148 & 0.8317 \\ 
% 0.3 & 0 & 0 & 0.0367 & 0.3908 & 0.5663 & 0.6476 & 0.7077 & 0.754 & 0.7898 & 0.8145 & 0.8309 \\ 
%  & 0.392 & 0.3977 & 0.444 & 0.5097 & 0.5825 & 0.6502 & 0.709 & 0.7551 & 0.7908 & 0.8154 & 0.8317 \\ 
% 0.4 & 0 & 0 & 0.0022 & 0.2609 & 0.55 & 0.6468 & 0.7092 & 0.7554 & 0.7902 & 0.8151 & 0.8309 \\ 
%  & 0.4598 & 0.4345 & 0.4618 & 0.5167 & 0.5845 & 0.6515 & 0.7104 & 0.7564 & 0.7911 & 0.8159 & 0.8317 \\ 
% 0.5 & 0 & 0 & 0.0001 & 0.101 & 0.5127 & 0.644 & 0.7087 & 0.757 & 0.7921 & 0.8157 & 0.8309 \\ 
%  & 0.5395 & 0.4778 & 0.483 & 0.5261 & 0.588 & 0.6517 & 0.7099 & 0.758 & 0.793 & 0.8165 & 0.8317 \\ 
% 0.6 & 0 & 0 & 0 & 0.0189 & 0.4357 & 0.6388 & 0.7089 & 0.7576 & 0.7926 & 0.8162 & 0.8309 \\ 
%  & 0.6293 & 0.5276 & 0.5084 & 0.5385 & 0.5922 & 0.6528 & 0.7113 & 0.7586 & 0.7935 & 0.817 & 0.8317 \\ 
% 0.7 & 0 & 0 & 0 & 0.0021 & 0.3002 & 0.6289 & 0.7085 & 0.7572 & 0.7933 & 0.8168 & 0.8309 \\ 
%  & 0.7266 & 0.5837 & 0.5379 & 0.5522 & 0.597 & 0.655 & 0.7119 & 0.7582 & 0.7942 & 0.8177 & 0.8317 \\ 
% 0.8 & 0.003 & 0 & 0 & 0.0002 & 0.1469 & 0.605 & 0.7088 & 0.7583 & 0.794 & 0.8174 & 0.8309 \\ 
%  & 0.8267 & 0.646 & 0.5713 & 0.5679 & 0.6032 & 0.6571 & 0.7121 & 0.7593 & 0.7949 & 0.8182 & 0.8317 \\ 
$-0.8$  & 0.262 & 0.341 & 0.426 & 0.509 & 0.582 & 0.642 & 0.695 & 0.738 & 0.775 & 0.806 & 0.831 \\ 
  & 0.264 & 0.343 & 0.428 & 0.511 & 0.583 & 0.643 & 0.696 & 0.739 & 0.776 & 0.807 & 0.832 \\ 
$-0.7$  & 0.235 & 0.325 & 0.417 & 0.506 & 0.582 & 0.644 & 0.698 & 0.74 & 0.776 & 0.807 & 0.831 \\ 
  & 0.239 & 0.327 & 0.42 & 0.508 & 0.584 & 0.646 & 0.7 & 0.741 & 0.777 & 0.807 & 0.832 \\ 
$-0.6$  & 0.218 & 0.314 & 0.412 & 0.501 & 0.579 & 0.645 & 0.699 & 0.742 & 0.779 & 0.807 & 0.831 \\ 
  & 0.221 & 0.317 & 0.414 & 0.503 & 0.581 & 0.647 & 0.7 & 0.744 & 0.78 & 0.808 & 0.832 \\ 
$-0.5$  & 0.207 & 0.306 & 0.406 & 0.499 & 0.578 & 0.646 & 0.7 & 0.745 & 0.779 & 0.807 & 0.831 \\ 
  & 0.211 & 0.309 & 0.408 & 0.5 & 0.58 & 0.647 & 0.701 & 0.746 & 0.78 & 0.808 & 0.832 \\ 
$-0.4$  & 0.199 & 0.301 & 0.402 & 0.496 & 0.579 & 0.647 & 0.704 & 0.746 & 0.782 & 0.809 & 0.831 \\ 
  & 0.208 & 0.305 & 0.404 & 0.497 & 0.581 & 0.649 & 0.705 & 0.747 & 0.783 & 0.81 & 0.832 \\ 
$-0.3$  & 0.187 & 0.294 & 0.398 & 0.494 & 0.578 & 0.648 & 0.702 & 0.747 & 0.782 & 0.81 & 0.831 \\ 
  & 0.211 & 0.303 & 0.403 & 0.498 & 0.58 & 0.649 & 0.704 & 0.749 & 0.783 & 0.811 & 0.832 \\ 
$-0.2$  & 0.149 & 0.284 & 0.395 & 0.493 & 0.578 & 0.647 & 0.705 & 0.749 & 0.785 & 0.81 & 0.831 \\ 
  & 0.22 & 0.307 & 0.402 & 0.496 & 0.58 & 0.648 & 0.706 & 0.75 & 0.786 & 0.811 & 0.832 \\ 
$-0.1$  & 0.058 & 0.251 & 0.388 & 0.491 & 0.576 & 0.648 & 0.705 & 0.749 & 0.785 & 0.811 & 0.831 \\ 
  & 0.235 & 0.315 & 0.404 & 0.497 & 0.579 & 0.651 & 0.706 & 0.75 & 0.786 & 0.812 & 0.832 \\ 
0 & 0.003 & 0.156 & 0.367 & 0.488 & 0.575 & 0.649 & 0.707 & 0.753 & 0.786 & 0.811 & 0.831 \\ 
   & 0.141 & 0.328 & 0.411 & 0.496 & 0.578 & 0.65 & 0.708 & 0.754 & 0.787 & 0.812 & 0.832 \\ 
0.1 & 0 & 0.03 & 0.306 & 0.479 & 0.575 & 0.648 & 0.707 & 0.753 & 0.788 & 0.814 & 0.831 \\ 
  & 0.293 & 0.345 & 0.419 & 0.5 & 0.579 & 0.65 & 0.708 & 0.754 & 0.789 & 0.814 & 0.832 \\ 
0.2 & 0 & 0.001 & 0.172 & 0.454 & 0.572 & 0.648 & 0.707 & 0.754 & 0.788 & 0.814 & 0.831 \\ 
  & 0.338 & 0.369 & 0.429 & 0.504 & 0.58 & 0.651 & 0.708 & 0.755 & 0.789 & 0.815 & 0.832 \\ 
0.3 & 0 & 0 & 0.037 & 0.391 & 0.566 & 0.648 & 0.708 & 0.754 & 0.79 & 0.815 & 0.831 \\ 
  & 0.392 & 0.398 & 0.444 & 0.51 & 0.583 & 0.65 & 0.709 & 0.755 & 0.791 & 0.815 & 0.832 \\ 
0.4 & 0 & 0 & 0.002 & 0.261 & 0.55 & 0.647 & 0.709 & 0.755 & 0.79 & 0.815 & 0.831 \\ 
  & 0.46 & 0.435 & 0.462 & 0.517 & 0.585 & 0.652 & 0.71 & 0.756 & 0.791 & 0.816 & 0.832 \\ 
0.5 & 0 & 0 & 0 & 0.101 & 0.513 & 0.644 & 0.709 & 0.757 & 0.792 & 0.816 & 0.831 \\ 
  & 0.54 & 0.478 & 0.483 & 0.526 & 0.588 & 0.652 & 0.71 & 0.758 & 0.793 & 0.817 & 0.832 \\ 
0.6 & 0 & 0 & 0 & 0.019 & 0.436 & 0.639 & 0.709 & 0.758 & 0.793 & 0.816 & 0.831 \\ 
  & 0.629 & 0.528 & 0.508 & 0.539 & 0.592 & 0.653 & 0.711 & 0.759 & 0.794 & 0.817 & 0.832 \\ 
0.7 & 0 & 0 & 0 & 0.002 & 0.3 & 0.629 & 0.709 & 0.757 & 0.793 & 0.817 & 0.831 \\ 
  & 0.727 & 0.584 & 0.538 & 0.552 & 0.597 & 0.655 & 0.712 & 0.758 & 0.794 & 0.818 & 0.832 \\ 
0.8 & 0.003 & 0 & 0 & 0 & 0.147 & 0.605 & 0.709 & 0.758 & 0.794 & 0.817 & 0.831 \\ 
  & 0.827 & 0.646 & 0.571 & 0.568 & 0.603 & 0.657 & 0.712 & 0.759 & 0.795 & 0.818 & 0.832 \\ 
% $-0.8$  & 0.26 & 0.34 & 0.43 & 0.51 & 0.58 & 0.64 & 0.7 & 0.74 & 0.78 & 0.81 & 0.83 \\ 
%   & 0.26 & 0.34 & 0.43 & 0.51 & 0.58 & 0.64 & 0.7 & 0.74 & 0.78 & 0.81 & 0.83 \\ 
% $-0.7$  & 0.24 & 0.33 & 0.42 & 0.51 & 0.58 & 0.64 & 0.7 & 0.74 & 0.78 & 0.81 & 0.83 \\ 
%   & 0.24 & 0.33 & 0.42 & 0.51 & 0.58 & 0.65 & 0.7 & 0.74 & 0.78 & 0.81 & 0.83 \\ 
% $-0.6$  & 0.22 & 0.31 & 0.41 & 0.5 & 0.58 & 0.65 & 0.7 & 0.74 & 0.78 & 0.81 & 0.83 \\ 
%   & 0.22 & 0.32 & 0.41 & 0.5 & 0.58 & 0.65 & 0.7 & 0.74 & 0.78 & 0.81 & 0.83 \\ 
% $-0.5$  & 0.21 & 0.31 & 0.41 & 0.5 & 0.58 & 0.65 & 0.7 & 0.75 & 0.78 & 0.81 & 0.83 \\ 
%   & 0.21 & 0.31 & 0.41 & 0.5 & 0.58 & 0.65 & 0.7 & 0.75 & 0.78 & 0.81 & 0.83 \\ 
% $-0.4$  & 0.2 & 0.3 & 0.4 & 0.5 & 0.58 & 0.65 & 0.7 & 0.75 & 0.78 & 0.81 & 0.83 \\ 
%   & 0.21 & 0.31 & 0.4 & 0.5 & 0.58 & 0.65 & 0.71 & 0.75 & 0.78 & 0.81 & 0.83 \\ 
% $-0.3$  & 0.19 & 0.29 & 0.4 & 0.49 & 0.58 & 0.65 & 0.7 & 0.75 & 0.78 & 0.81 & 0.83 \\ 
%   & 0.21 & 0.3 & 0.4 & 0.5 & 0.58 & 0.65 & 0.7 & 0.75 & 0.78 & 0.81 & 0.83 \\ 
% $-0.2$  & 0.15 & 0.28 & 0.4 & 0.49 & 0.58 & 0.65 & 0.71 & 0.75 & 0.79 & 0.81 & 0.83 \\ 
%   & 0.22 & 0.31 & 0.4 & 0.5 & 0.58 & 0.65 & 0.71 & 0.75 & 0.79 & 0.81 & 0.83 \\ 
% $-0.1$  & 0.06 & 0.25 & 0.39 & 0.49 & 0.58 & 0.65 & 0.71 & 0.75 & 0.79 & 0.81 & 0.83 \\ 
%   & 0.24 & 0.32 & 0.4 & 0.5 & 0.58 & 0.65 & 0.71 & 0.75 & 0.79 & 0.81 & 0.83 \\ 
% 0 & 0 & 0.16 & 0.37 & 0.49 & 0.58 & 0.65 & 0.71 & 0.75 & 0.79 & 0.81 & 0.83 \\ 
%    & 0.14 & 0.33 & 0.41 & 0.5 & 0.58 & 0.65 & 0.71 & 0.75 & 0.79 & 0.81 & 0.83 \\ 
% 0.1 & 0 & 0.03 & 0.31 & 0.48 & 0.58 & 0.65 & 0.71 & 0.75 & 0.79 & 0.81 & 0.83 \\ 
%   & 0.29 & 0.35 & 0.42 & 0.5 & 0.58 & 0.65 & 0.71 & 0.75 & 0.79 & 0.81 & 0.83 \\ 
% 0.2 & 0 & 0 & 0.17 & 0.45 & 0.57 & 0.65 & 0.71 & 0.75 & 0.79 & 0.81 & 0.83 \\ 
%   & 0.34 & 0.37 & 0.43 & 0.5 & 0.58 & 0.65 & 0.71 & 0.76 & 0.79 & 0.82 & 0.83 \\ 
% 0.3 & 0 & 0 & 0.04 & 0.39 & 0.57 & 0.65 & 0.71 & 0.75 & 0.79 & 0.82 & 0.83 \\ 
%   & 0.39 & 0.4 & 0.44 & 0.51 & 0.58 & 0.65 & 0.71 & 0.76 & 0.79 & 0.82 & 0.83 \\ 
% 0.4 & 0 & 0 & 0 & 0.26 & 0.55 & 0.65 & 0.71 & 0.76 & 0.79 & 0.82 & 0.83 \\ 
%   & 0.46 & 0.44 & 0.46 & 0.52 & 0.59 & 0.65 & 0.71 & 0.76 & 0.79 & 0.82 & 0.83 \\ 
% 0.5 & 0 & 0 & 0 & 0.1 & 0.51 & 0.64 & 0.71 & 0.76 & 0.79 & 0.82 & 0.83 \\ 
%   & 0.54 & 0.48 & 0.48 & 0.53 & 0.59 & 0.65 & 0.71 & 0.76 & 0.79 & 0.82 & 0.83 \\ 
% 0.6 & 0 & 0 & 0 & 0.02 & 0.44 & 0.64 & 0.71 & 0.76 & 0.79 & 0.82 & 0.83 \\ 
%   & 0.63 & 0.53 & 0.51 & 0.54 & 0.59 & 0.65 & 0.71 & 0.76 & 0.79 & 0.82 & 0.83 \\ 
% 0.7 & 0 & 0 & 0 & 0 & 0.3 & 0.63 & 0.71 & 0.76 & 0.79 & 0.82 & 0.83 \\ 
%   & 0.73 & 0.58 & 0.54 & 0.55 & 0.6 & 0.66 & 0.71 & 0.76 & 0.79 & 0.82 & 0.83 \\ 
% 0.8 & 0 & 0 & 0 & 0 & 0.15 & 0.61 & 0.71 & 0.76 & 0.79 & 0.82 & 0.83 \\ 
%   & 0.83 & 0.65 & 0.57 & 0.57 & 0.6 & 0.66 & 0.71 & 0.76 & 0.8 & 0.82 & 0.83 \\ 

\hline
		\end{tabular}}
\label{signed_test}	 %{tab:Dwo}
\end{table}
%  \begin{table}
% 
% 	\centering 
% 	\caption{The signed test 
% % (with $\gamma =\alpha$)
% $p$-values for the two sample data}
% 		\begin{tabular}{c c c c c c c c c c c c} \hline
% $\lambda$ \textdownarrow $\alpha$ $\rightarrow$ 
% % & 0 & 0.1 & 0.2 & 0.3 & 0.4 & 0.5 & 0.6 & 0.7 & 0.8 & 0.9 & 1 \\ 
% & 0 & 0.2  & 0.4  &  0.6 & 0.8 & 0.9 & 1 \\ 
% \hline
% % $-0.1$ & 0.2505 & 0.3881 & 0.4911 & 0.5755 & 0.6479 & 0.7048 & 0.7492 & 0.7846 & 0.8109 & 0.8309 \\ 
% % $-0.2$ & 0.284 & 0.395 & 0.4931 & 0.5781 & 0.6465 & 0.7048 & 0.7489 & 0.7847 & 0.8104 & 0.8309 \\ 
% % $-0.3$ & 0.2944 & 0.3977 & 0.4943 & 0.5783 & 0.6478 & 0.7022 & 0.7474 & 0.7817 & 0.8099 & 0.8309 \\ 
% % $-0.4$ & 0.3007 & 0.4015 & 0.4955 & 0.579 & 0.647 & 0.7035 & 0.746 & 0.7818 & 0.8094 & 0.8309 \\ 
% % $-0.5$ & 0.3063 & 0.4057 & 0.4985 & 0.5781 & 0.6456 & 0.7001 & 0.745 & 0.7788 & 0.8074 & 0.8309 \\ 
% % $-0.6$ & 0.3144 & 0.4118 & 0.5012 & 0.5792 & 0.6452 & 0.699 & 0.7424 & 0.7793 & 0.8069 & 0.8309 \\ 
% % $-0.7$ & 0.3247 & 0.4174 & 0.5056 & 0.5823 & 0.6441 & 0.6984 & 0.7398 & 0.7764 & 0.8065 & 0.8309 \\ 
% % $-0.8$ & 0.3406 & 0.4263 & 0.5092 & 0.5815 & 0.6419 & 0.6945 & 0.7375 & 0.7752 & 0.806 & 0.8309 \\ 
% % $-0.9$ & 0.3598 & 0.4361 & 0.5118 & 0.58 & 0.6429 & 0.6952 & 0.7374 & 0.7741 & 0.8057 & 0.8309 \\ 
% % $-1$ & 0.3923 & 0.4464 & 0.5174 & 0.583 & 0.6405 & 0.69 & 0.7355 & 0.7731 & 0.8035 & 0.8309 \\ 
% % \hline
% % 0.1 & 0.0303 & 0.3063 & 0.4789 & 0.5753 & 0.648 & 0.7065 & 0.7533 & 0.7878 & 0.8135 & 0.8309 \\ 
% % 0.2 & 0.0006 & 0.1721 & 0.4536 & 0.5717 & 0.6484 & 0.7068 & 0.7542 & 0.788 & 0.814 & 0.8309 \\ 
% % 0.3 & 0 & 0.0367 & 0.3908 & 0.5663 & 0.6476 & 0.7077 & 0.754 & 0.7898 & 0.8145 & 0.8309 \\ 
% % 0.4 & 0 & 0.0022 & 0.2609 & 0.55 & 0.6468 & 0.7092 & 0.7554 & 0.7902 & 0.8151 & 0.8309 \\ 
% % 0.5 & 0 & 0.0001 & 0.101 & 0.5127 & 0.644 & 0.7087 & 0.757 & 0.7921 & 0.8157 & 0.8309 \\ 
% % 0.6 & 0 & 0 & 0.0189 & 0.4357 & 0.6388 & 0.7089 & 0.7576 & 0.7926 & 0.8162 & 0.8309 \\ 
% % 0.7 & 0 & 0 & 0.0021 & 0.3002 & 0.6289 & 0.7085 & 0.7572 & 0.7933 & 0.8168 & 0.8309 \\ 
% % 0.8 & 0 & 0 & 0.0002 & 0.1469 & 0.605 & 0.7088 & 0.7583 & 0.794 & 0.8174 & 0.8309 \\ 
% % 0.9 & 0 & 0 & 0 & 0.0481 & 0.5604 & 0.7058 & 0.7584 & 0.7948 & 0.8181 & 0.8309 \\ 
% % 1 & 0.0005 & 0 & 0 & 0.0117 & -- & 0.7037 & 0.7599 & 0.7957 & 0.8201 & 0.8309 \\ 
% % $-1$ & -- & 0.3923 & 0.4464 & 0.5174 & 0.583 & 0.6405 & 0.69 & 0.7355 & 0.7731 & 0.8035 & 0.8309 \\
% % $-0.9$ & 0.3149 & 0.3598 & 0.4361 & 0.5118 & 0.58 & 0.6429 & 0.6952 & 0.7374 & 0.7741 & 0.8057 & 0.8309 \\ 
% % $-0.8$ & 0.2622 & 0.3406 & 0.4263 & 0.5092 & 0.5815 & 0.6419 & 0.6945 & 0.7375 & 0.7752 & 0.806 & 0.8309 \\ 
% % $-0.7$ & 0.2351 & 0.3247 & 0.4174 & 0.5056 & 0.5823 & 0.6441 & 0.6984 & 0.7398 & 0.7764 & 0.8065 & 0.8309 \\
% % $-0.6$ & 0.2184 & 0.3144 & 0.4118 & 0.5012 & 0.5792 & 0.6452 & 0.699 & 0.7424 & 0.7793 & 0.8069 & 0.8309 \\ 
% % $-0.5$ & 0.2069 & 0.3063 & 0.4057 & 0.4985 & 0.5781 & 0.6456 & 0.7001 & 0.745 & 0.7788 & 0.8074 & 0.8309 \\ 
% % $-0.4$ & 0.1991 & 0.3007 & 0.4015 & 0.4955 & 0.579 & 0.647 & 0.7035 & 0.746 & 0.7818 & 0.8094 & 0.8309 \\ 
% % $-0.3$ & 0.1874 & 0.2944 & 0.3977 & 0.4943 & 0.5783 & 0.6478 & 0.7022 & 0.7474 & 0.7817 & 0.8099 & 0.8309 \\
% % $-0.2$ & 0.1491 & 0.284 & 0.395 & 0.4931 & 0.5781 & 0.6465 & 0.7048 & 0.7489 & 0.7847 & 0.8104 & 0.8309 \\ 
% % $-0.1$ & 0.0575 & 0.2505 & 0.3881 & 0.4911 & 0.5755 & 0.6479 & 0.7048 & 0.7492 & 0.7846 & 0.8109 & 0.8309 \\ 
% % 0 & 0.0029 & 0.1556 & 0.3672 & 0.4875 & 0.5751 & 0.6488 & 0.7068 & 0.7526 & 0.7862 & 0.8114 & 0.8309 \\ 
% % 0.1 & 0 & 0.0303 & 0.3063 & 0.4789 & 0.5753 & 0.648 & 0.7065 & 0.7533 & 0.7878 & 0.8135 & 0.8309 \\ 
% % 0.2 & 0 & 0.0006 & 0.1721 & 0.4536 & 0.5717 & 0.6484 & 0.7068 & 0.7542 & 0.788 & 0.814 & 0.8309 \\ 
% % 0.3 & 0 & 0 & 0.0367 & 0.3908 & 0.5663 & 0.6476 & 0.7077 & 0.754 & 0.7898 & 0.8145 & 0.8309 \\ 
% % 0.4 & 0 & 0 & 0.0022 & 0.2609 & 0.55 & 0.6468 & 0.7092 & 0.7554 & 0.7902 & 0.8151 & 0.8309 \\ 
% % 0.5 & 0 & 0 & 0.0001 & 0.101 & 0.5127 & 0.644 & 0.7087 & 0.757 & 0.7921 & 0.8157 & 0.8309 \\ 
% % 0.6 & 0 & 0 & 0 & 0.0189 & 0.4357 & 0.6388 & 0.7089 & 0.7576 & 0.7926 & 0.8162 & 0.8309 \\ 
% % 0.7 & 0 & 0 & 0 & 0.0021 & 0.3002 & 0.6289 & 0.7085 & 0.7572 & 0.7933 & 0.8168 & 0.8309 \\ 
% % 0.8 & 0.003 & 0 & 0 & 0.0002 & 0.1469 & 0.605 & 0.7088 & 0.7583 & 0.794 & 0.8174 & 0.8309 \\ 
% % 0.9 & 0.1413 & 0 & 0 & 0 & 0.0481 & 0.5604 & 0.7058 & 0.7584 & 0.7948 & 0.8181 & 0.8309 \\ 
% % 1 & 0.9681 & 0.0005 & 0 & 0 & 0.0117 & -- & 0.7037 & 0.7599 & 0.7957 & 0.8201 & 0.8309 \\ 
% -0.1 & 0.0575 & 0.3881 & 0.5755 & 0.7048 & 0.7492 & 0.7846 & 0.8109 & 0.8309 \\ 
% -0.2 & 0.1491 & 0.395 & 0.5781 & 0.7048 & 0.7489 & 0.7847 & 0.8104 & 0.8309 \\ 
% -0.3 & 0.1874 & 0.3977 & 0.5783 & 0.7022 & 0.7474 & 0.7817 & 0.8099 & 0.8309 \\ 
% -0.4 & 0.1991 & 0.4015 & 0.579 & 0.7035 & 0.746 & 0.7818 & 0.8094 & 0.8309 \\ 
% -0.5 & 0.2069 & 0.4057 & 0.5781 & 0.7001 & 0.745 & 0.7788 & 0.8074 & 0.8309 \\ 
% -0.6 & 0.2184 & 0.4118 & 0.5792 & 0.699 & 0.7424 & 0.7793 & 0.8069 & 0.8309 \\ 
% -0.7 & 0.2351 & 0.4174 & 0.5823 & 0.6984 & 0.7398 & 0.7764 & 0.8065 & 0.8309 \\ 
% -0.8 & 0.2622 & 0.4263 & 0.5815 & 0.6945 & 0.7375 & 0.7752 & 0.806 & 0.8309 \\ 
% -0.9 & 0.3149 & 0.4361 & 0.58 & 0.6952 & 0.7374 & 0.7741 & 0.8057 & 0.8309 \\ 
% -1 &  & 0.4464 & 0.583 & 0.69 & 0.7355 & 0.7731 & 0.8035 & 0.8309 \\ 
% 0 &  & 0.3672 & 0.5751 & 0.7068 & 0.7526 & 0.7862 & 0.8114 & 0.8309 \\ 
% 0.1 & 0 & 0.3063 & 0.5753 & 0.7065 & 0.7533 & 0.7878 & 0.8135 & 0.8309 \\ 
% 0.2 & 0 & 0.1721 & 0.5717 & 0.7068 & 0.7542 & 0.788 & 0.814 & 0.8309 \\ 
% 0.3 & 0 & 0.0367 & 0.5663 & 0.7077 & 0.754 & 0.7898 & 0.8145 & 0.8309 \\ 
% 0.4 & 0 & 0.0022 & 0.55 & 0.7092 & 0.7554 & 0.7902 & 0.8151 & 0.8309 \\ 
% 0.5 & 0 & 0.0001 & 0.5127 & 0.7087 & 0.757 & 0.7921 & 0.8157 & 0.8309 \\ 
% 0.6 & 0 & 0 & 0.4357 & 0.7089 & 0.7576 & 0.7926 & 0.8162 & 0.8309 \\ 
% 0.7 & 0 & 0 & 0.3002 & 0.7085 & 0.7572 & 0.7933 & 0.8168 & 0.8309 \\ 
% 0.8 & 0.003 & 0 & 0.1469 & 0.7088 & 0.7583 & 0.794 & 0.8174 & 0.8309 \\ 
% 0.9 & 0.1413 & 0 & 0.0481 & 0.7058 & 0.7584 & 0.7948 & 0.8181 & 0.8309 \\ 
% 1 & 0.9681 & 0 & 0.0117 & 0.7037 & 0.7599 & 0.7957 & 0.8201 & 0.8309 \\ 
% 
% \hline
% 		\end{tabular}
% \label{with_outlier}	 %{tab:Dwo}
% \end{table}
% 
% %%%%%%%%%%
%  \begin{table}
% 
% 	\centering 
% 	\caption{The signed test 
% % (with $\gamma =\alpha$)
% $p$-values for the two sample data (Outliers deleted data)  }
% 		\begin{tabular}{c c c c c c c c c c c c} \hline
% $\lambda$ \textdownarrow $\alpha$ $\rightarrow$ 
% % & 0 & 0.1 & 0.2 & 0.3 & 0.4 & 0.5 & 0.6 & 0.7 & 0.8 & 0.9 & 1 \\ 
% & 0 & 0.2  & 0.4  &  0.6 & 0.8 & 0.9 & 1 \\ 
% \hline
% % -0.1 & 0.3154 & 0.4044 & 0.4968 & 0.5786 & 0.6508 & 0.7061 & 0.7503 & 0.7856 & 0.8118 & 0.8317 \\ 
% % -0.2 & 0.3072 & 0.402 & 0.4964 & 0.5799 & 0.648 & 0.7061 & 0.75 & 0.7856 & 0.8112 & 0.8317 \\ 
% % -0.3 & 0.3032 & 0.4025 & 0.4976 & 0.5801 & 0.6493 & 0.7035 & 0.7485 & 0.7826 & 0.8108 & 0.8317 \\ 
% % -0.4 & 0.3054 & 0.4036 & 0.4974 & 0.5808 & 0.6486 & 0.7048 & 0.7472 & 0.7828 & 0.8103 & 0.8317 \\ 
% % -0.5 & 0.3085 & 0.4079 & 0.5004 & 0.5798 & 0.6471 & 0.7014 & 0.7461 & 0.7798 & 0.8083 & 0.8317 \\ 
% % -0.6 & 0.3167 & 0.4139 & 0.5032 & 0.5809 & 0.6467 & 0.7003 & 0.7435 & 0.7803 & 0.8078 & 0.8317 \\ 
% % -0.7 & 0.327 & 0.4196 & 0.5075 & 0.584 & 0.6456 & 0.6997 & 0.741 & 0.7773 & 0.8074 & 0.8317 \\ 
% % -0.8 & 0.3428 & 0.4284 & 0.5111 & 0.5832 & 0.6434 & 0.6959 & 0.7387 & 0.7762 & 0.8069 & 0.8317 \\ 
% % -0.9 & 0.362 & 0.4382 & 0.5137 & 0.5817 & 0.6444 & 0.6965 & 0.7385 & 0.7752 & 0.8065 & 0.8317 \\ 
% % -1 & 0.3945 & 0.4485 & 0.5193 & 0.5847 & 0.642 & 0.6913 & 0.7366 & 0.7741 & 0.8044 & 0.8317 \\ 
% % \hline
% % 0.1 & 0.3454 & 0.4189 & 0.5001 & 0.5794 & 0.6495 & 0.7078 & 0.7544 & 0.7888 & 0.8143 & 0.8317 \\ 
% % 0.2 & 0.3689 & 0.4288 & 0.5041 & 0.5801 & 0.6512 & 0.7081 & 0.7553 & 0.789 & 0.8148 & 0.8317 \\ 
% % 0.3 & 0.3977 & 0.444 & 0.5097 & 0.5825 & 0.6502 & 0.709 & 0.7551 & 0.7908 & 0.8154 & 0.8317 \\ 
% % 0.4 & 0.4345 & 0.4618 & 0.5167 & 0.5845 & 0.6515 & 0.7104 & 0.7564 & 0.7911 & 0.8159 & 0.8317 \\ 
% % 0.5 & 0.4778 & 0.483 & 0.5261 & 0.588 & 0.6517 & 0.7099 & 0.758 & 0.793 & 0.8165 & 0.8317 \\ 
% % 0.6 & 0.5276 & 0.5084 & 0.5385 & 0.5922 & 0.6528 & 0.7113 & 0.7586 & 0.7935 & 0.817 & 0.8317 \\ 
% % 0.7 & 0.5837 & 0.5379 & 0.5522 & 0.597 & 0.655 & 0.7119 & 0.7582 & 0.7942 & 0.8177 & 0.8317 \\ 
% % 0.8 & 0.646 & 0.5713 & 0.5679 & 0.6032 & 0.6571 & 0.7121 & 0.7593 & 0.7949 & 0.8182 & 0.8317 \\ 
% % 0.9 & 0.7134 & 0.609 & 0.5856 & 0.6107 & 0.6593 & 0.7119 & 0.7595 & 0.7957 & 0.8189 & 0.8317 \\ 
% % 1 & 0.784 & 0.6493 & 0.6058 & 0.6195 & -- & 0.7131 & 0.761 & 0.7966 & 0.8209 & 0.8317 \\ 
% % $-1$ & -- & 0.3945 & 0.4485 & 0.5193 & 0.5847 & 0.642 & 0.6913 & 0.7366 & 0.7741 & 0.8044 & 0.8317 \\ 
% % $-0.9$ & 0.3171 & 0.362 & 0.4382 & 0.5137 & 0.5817 & 0.6444 & 0.6965 & 0.7385 & 0.7752 & 0.8065 & 0.8317 \\
% % $-0.8$ & 0.2644 & 0.3428 & 0.4284 & 0.5111 & 0.5832 & 0.6434 & 0.6959 & 0.7387 & 0.7762 & 0.8069 & 0.8317 \\
% % $-0.7$ & 0.2387 & 0.327 & 0.4196 & 0.5075 & 0.584 & 0.6456 & 0.6997 & 0.741 & 0.7773 & 0.8074 & 0.8317 \\ 
% % $-0.6$ & 0.2205 & 0.3167 & 0.4139 & 0.5032 & 0.5809 & 0.6467 & 0.7003 & 0.7435 & 0.7803 & 0.8078 & 0.8317 \\ 
% % $-0.5$ & 0.2111 & 0.3085 & 0.4079 & 0.5004 & 0.5798 & 0.6471 & 0.7014 & 0.7461 & 0.7798 & 0.8083 & 0.8317 \\ 
% % $-0.4$ & 0.2079 & 0.3054 & 0.4036 & 0.4974 & 0.5808 & 0.6486 & 0.7048 & 0.7472 & 0.7828 & 0.8103 & 0.8317 \\ 
% % $-0.3$ & 0.2107 & 0.3032 & 0.4025 & 0.4976 & 0.5801 & 0.6493 & 0.7035 & 0.7485 & 0.7826 & 0.8108 & 0.8317 \\ 
% % $-0.2$ & 0.2195 & 0.3072 & 0.402 & 0.4964 & 0.5799 & 0.648 & 0.7061 & 0.75 & 0.7856 & 0.8112 & 0.8317 \\ 
% % $-0.1$ & 0.2354 & 0.3154 & 0.4044 & 0.4968 & 0.5786 & 0.6508 & 0.7061 & 0.7503 & 0.7856 & 0.8118 & 0.8317 \\ 
% % 0 & 0.1413 & 0.328 & 0.4109 & 0.4964 & 0.5781 & 0.6503 & 0.7081 & 0.7537 & 0.7872 & 0.8122 & 0.8317 \\ 
% % 0.1 & 0.293 & 0.3454 & 0.4189 & 0.5001 & 0.5794 & 0.6495 & 0.7078 & 0.7544 & 0.7888 & 0.8143 & 0.8317 \\ 
% % 0.2 & 0.3377 & 0.3689 & 0.4288 & 0.5041 & 0.5801 & 0.6512 & 0.7081 & 0.7553 & 0.789 & 0.8148 & 0.8317 \\ 
% % 0.3 & 0.392 & 0.3977 & 0.444 & 0.5097 & 0.5825 & 0.6502 & 0.709 & 0.7551 & 0.7908 & 0.8154 & 0.8317 \\ 
% % 0.4 & 0.4598 & 0.4345 & 0.4618 & 0.5167 & 0.5845 & 0.6515 & 0.7104 & 0.7564 & 0.7911 & 0.8159 & 0.8317 \\ 
% % 0.5 & 0.5395 & 0.4778 & 0.483 & 0.5261 & 0.588 & 0.6517 & 0.7099 & 0.758 & 0.793 & 0.8165 & 0.8317 \\ 
% % 0.6 & 0.6293 & 0.5276 & 0.5084 & 0.5385 & 0.5922 & 0.6528 & 0.7113 & 0.7586 & 0.7935 & 0.817 & 0.8317 \\ 
% % 0.7 & 0.7266 & 0.5837 & 0.5379 & 0.5522 & 0.597 & 0.655 & 0.7119 & 0.7582 & 0.7942 & 0.8177 & 0.8317 \\ 
% % 0.8 & 0.8267 & 0.646 & 0.5713 & 0.5679 & 0.6032 & 0.6571 & 0.7121 & 0.7593 & 0.7949 & 0.8182 & 0.8317 \\ 
% % 0.9 & 0.9232 & 0.7134 & 0.609 & 0.5856 & 0.6107 & 0.6593 & 0.7119 & 0.7595 & 0.7957 & 0.8189 & 0.8317 \\ 
% % 1 & 0.999 & 0.784 & 0.6493 & 0.6058 & 0.6195 & -- & 0.7131 & 0.761 & 0.7966 & 0.8209 & 0.8317 \\ 
% -1 & -- & 0.4485 & 0.5847 & 0.6913 & 0.7741 & 0.8044 & 0.8317 \\ 
% -0.9 & 0.3171 & 0.4382 & 0.5817 & 0.6965 & 0.7752 & 0.8065 & 0.8317 \\ 
% -0.8 & 0.2644 & 0.4284 & 0.5832 & 0.6959 & 0.7762 & 0.8069 & 0.8317 \\ 
% -0.7 & 0.2387 & 0.4196 & 0.584 & 0.6997 & 0.7773 & 0.8074 & 0.8317 \\ 
% -0.6 & 0.2205 & 0.4139 & 0.5809 & 0.7003 & 0.7803 & 0.8078 & 0.8317 \\ 
% -0.5 & 0.2111 & 0.4079 & 0.5798 & 0.7014 & 0.7798 & 0.8083 & 0.8317 \\ 
% -0.4 & 0.2079 & 0.4036 & 0.5808 & 0.7048 & 0.7828 & 0.8103 & 0.8317 \\ 
% -0.3 & 0.2107 & 0.4025 & 0.5801 & 0.7035 & 0.7826 & 0.8108 & 0.8317 \\ 
% -0.2 & 0.2195 & 0.402 & 0.5799 & 0.7061 & 0.7856 & 0.8112 & 0.8317 \\ 
% -0.1 & 0.2354 & 0.4044 & 0.5786 & 0.7061 & 0.7856 & 0.8118 & 0.8317 \\ 
% 0 &  & 0.4109 & 0.5781 & 0.7081 & 0.7872 & 0.8122 & 0.8317 \\ 
% 0.1 & 0.293 & 0.4189 & 0.5794 & 0.7078 & 0.7888 & 0.8143 & 0.8317 \\ 
% 0.2 & 0.3377 & 0.4288 & 0.5801 & 0.7081 & 0.789 & 0.8148 & 0.8317 \\ 
% 0.3 & 0.392 & 0.444 & 0.5825 & 0.709 & 0.7908 & 0.8154 & 0.8317 \\ 
% 0.4 & 0.4598 & 0.4618 & 0.5845 & 0.7104 & 0.7911 & 0.8159 & 0.8317 \\ 
% 0.5 & 0.5395 & 0.483 & 0.588 & 0.7099 & 0.793 & 0.8165 & 0.8317 \\ 
% 0.6 & 0.6293 & 0.5084 & 0.5922 & 0.7113 & 0.7935 & 0.817 & 0.8317 \\ 
% 0.7 & 0.7266 & 0.5379 & 0.597 & 0.7119 & 0.7942 & 0.8177 & 0.8317 \\ 
% 0.8 & 0.8267 & 0.5713 & 0.6032 & 0.7121 & 0.7949 & 0.8182 & 0.8317 \\ 
% 0.9 & 0.9232 & 0.609 & 0.6107 & 0.7119 & 0.7957 & 0.8189 & 0.8317 \\ 
% 1 & 0.999 & 0.6493 & 0.6195 & 0.7131 & 0.7966 & 0.8209 & 0.8317 \\ 
% \hline
% 		\end{tabular}
% \label{witho_outlier}
% 	 %{tab:Dwo}
% \end{table}
%  \begin{table}
% 
% 	\centering 
% 	\caption{The signed test 
% % (with $\gamma =\alpha$)
% $p$-values for the two-sample drosophila data.
% The outlier deleted 
% % Outliers deleted 
% $p$-values are given in the second line of 
% each block
% % result in
% % the respective second lines)  
% }
% 		\begin{tabular}{c c c c c c c } \hline
% $\gamma$ \textdownarrow $\beta$ $\rightarrow$ 
% % & 0 & 0.1 & 0.2 & 0.3 & 0.4 & 0.5 & 0.6 & 0.7 & 0.8 & 0.9 & 1 \\ 
% & 0 & 0.2  & 0.4  &  0.6 & 0.8 & 1 \\ 
% \hline
% % $-1$ & - & 0.446 & 0.583 & 0.69 & 0.773 & 0.831 \\ 
% % $-1$ & - & 0.449 & 0.585 & 0.691 & 0.774 & 0.832 \\ 
% % $-0.9$ & 0.315 & 0.436 & 0.58 & 0.695 & 0.774 & 0.831 \\ 
% % $-0.9$ & 0.317 & 0.438 & 0.582 & 0.697 & 0.775 & 0.832 \\ 
% $-0.8$ & 0.262 & 0.426 & 0.582 & 0.695 & 0.775 & 0.831 \\ 
%  & 0.264 & 0.428 & 0.583 & 0.696 & 0.776 & 0.832 \\ 
% $-0.7$ & 0.235 & 0.417 & 0.582 & 0.698 & 0.776 & 0.831 \\ 
%  & 0.239 & 0.42 & 0.584 & 0.7 & 0.777 & 0.832 \\ 
% $-0.6$ & 0.218 & 0.412 & 0.579 & 0.699 & 0.779 & 0.831 \\ 
%  & 0.221 & 0.414 & 0.581 & 0.7 & 0.78 & 0.832 \\ 
% $-0.5$ & 0.207 & 0.406 & 0.578 & 0.7 & 0.779 & 0.831 \\ 
%  & 0.211 & 0.408 & 0.58 & 0.701 & 0.78 & 0.832 \\ 
% $-0.4$ & 0.199 & 0.402 & 0.579 & 0.704 & 0.782 & 0.831 \\ 
%  & 0.208 & 0.404 & 0.581 & 0.705 & 0.783 & 0.832 \\ 
% $-0.3$ & 0.187 & 0.398 & 0.578 & 0.702 & 0.782 & 0.831 \\ 
%  & 0.211 & 0.403 & 0.58 & 0.704 & 0.783 & 0.832 \\ 
% $-0.2$ & 0.149 & 0.395 & 0.578 & 0.705 & 0.785 & 0.831 \\ 
%  & 0.22 & 0.402 & 0.58 & 0.706 & 0.786 & 0.832 \\ 
% $-0.1$ & 0.058 & 0.388 & 0.576 & 0.705 & 0.785 & 0.831 \\ 
%  & 0.235 & 0.404 & 0.579 & 0.706 & 0.786 & 0.832 \\ 
% 0 & 0.003 & 0.367 & 0.575 & 0.707 & 0.786 & 0.831 \\ 
%  & 0.141 & 0.411 & 0.578 & 0.708 & 0.787 & 0.832 \\ 
% 0.1 & 0 & 0.306 & 0.575 & 0.707 & 0.788 & 0.831 \\ 
%  & 0.293 & 0.419 & 0.579 & 0.708 & 0.789 & 0.832 \\ 
% 0.2 & 0 & 0.172 & 0.572 & 0.707 & 0.788 & 0.831 \\ 
%  & 0.338 & 0.429 & 0.58 & 0.708 & 0.789 & 0.832 \\ 
% 0.3 & 0 & 0.037 & 0.566 & 0.708 & 0.79 & 0.831 \\ 
%  & 0.392 & 0.444 & 0.583 & 0.709 & 0.791 & 0.832 \\ 
% 0.4 & 0 & 0.002 & 0.55 & 0.709 & 0.79 & 0.831 \\ 
%  & 0.46 & 0.462 & 0.585 & 0.71 & 0.791 & 0.832 \\ 
% 0.5 & 0 & 0 & 0.513 & 0.709 & 0.792 & 0.831 \\ 
%  & 0.54 & 0.483 & 0.588 & 0.71 & 0.793 & 0.832 \\ 
% 0.6 & 0 & 0 & 0.436 & 0.709 & 0.793 & 0.831 \\ 
%  & 0.629 & 0.508 & 0.592 & 0.711 & 0.794 & 0.832 \\ 
% 0.7 & 0 & 0 & 0.3 & 0.709 & 0.793 & 0.831 \\ 
%  & 0.727 & 0.538 & 0.597 & 0.712 & 0.794 & 0.832 \\ 
% 0.8 & 0.003 & 0 & 0.147 & 0.709 & 0.794 & 0.831 \\ 
%  & 0.827 & 0.571 & 0.603 & 0.712 & 0.795 & 0.832 \\ 
% % 0.9 & 0.141 & 0 & 0.048 & 0.706 & 0.795 & 0.831 \\ 
% % 0.9 & 0.923 & 0.609 & 0.611 & 0.712 & 0.796 & 0.832 \\ 
% % 1 & 0.968 & 0 & 0.012 & 0.704 & 0.796 & 0.831 \\ 
% % 1 & 0.999 & 0.649 & 0.62 & 0.713 & 0.797 & 0.832 \\ 
% 
% \hline
% 		\end{tabular}
% \label{signed_test}
% 	 %{tab:Dwo}
% \end{table}
\section{Conclusion}
\label{Sec_Conclusion}
% 
% \paragraph{}
Logarithmic super divergence family acts as a super family of both LPD and LDPD family. 
The theoretical properties of this new family of divergences have been established
for discrete models and similar results under continuous set-up can be done 
in subsequent works.
% Its usage in both statistical estimation and testing of hypotheses have been studied.
% Computational exercises have shown that there exist a region of the parameter which 
% usually performs better where outliers are present in the observations.  

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\end{thebibliography}
\section*{Appendix}

% \\
{\bf Consistency Part:} Consider the behavior 
of ${\rm LSD}(r_n,f_{\theta})$ on a sphere $Q_a$ which has  radius 
$a$ and center at $\theta^g$. We will show, for sufficiently 
small $a$, the probability tends to one that 
% \\
$$
{\rm LSD}(r_n,f_\theta) > {\rm LSD}(r_n,f_{\theta^g}) ~~ \mbox{ for all } 
\theta ~~ \mbox{ on the surface of } Q_a
$$
so that the LSD has a local minimum with respect to $\theta$ 
in the interior of $Q_a$. At a local minimum, the estimating 
equations must be satisfied. Therefore, for any $a>0$ sufficiently 
small, the minimum LSD estimating equation have a solution $\theta_n$ 
within $Q_a$ with probability tending to one as $n \rightarrow \infty$.
% \\
	Now taking Taylor series expansion of ${\rm LSD}(r_n,f_{\theta})$ about $\theta = \theta^g$, we get 
% 
\begin{eqnarray}
    {\rm LSD}(r_n,f_{\theta^g}) - {\rm LSD}(r_n,f_\theta)
&=& - \sum_j (\theta_j - \theta_j^g)\nabla_j {\rm LSD}(r_n,f_\theta)|_{\theta = \theta^g} \nonumber \\
& & - 
\frac{1}{2} \sum_{j,k}  (\theta_j - \theta_j^g)(\theta_k - 
\theta_k^g)\nabla_{jk} {\rm LSD}(r_n,f_\theta)|_{\theta = \theta^g} \nonumber \\  
   			& & -      \frac{1}{6} \sum_{j,k,l} (\theta_j - 
\theta_j^g)(\theta_k - \theta_k^g)(\theta_l - \theta_l^g)\nabla_{jkl} 
{\rm LSD}(r_n,f_\theta)|_{\theta = \theta^*} \nonumber \\
& & = S_1 + S_2 + S_3, ~~~~~~~~ (\mbox{say}) \nonumber
\end{eqnarray}

% $$
%     {\rm LSD}(r_n,f_{\theta^g}) - {\rm LSD}(r_n,f_\theta)
% $$
% $$
% = - \sum_j (\theta_j - \theta_j^g)\nabla_j {\rm LSD}(r_n,f_\theta)|_{\theta = \theta^g} - 
% \frac{1}{2} \sum_{j,k}  (\theta_j - \theta_j^g)(\theta_k - 
% \theta_k^g)\nabla_{jk} {\rm LSD}(r_n,f_\theta)|_{\theta = \theta^g} 
%   $$
% $$    			 -      \frac{1}{6} \sum_{j,k,l} (\theta_j - 
% \theta_j^g)(\theta_k - \theta_k^g)(\theta_l - \theta_l^g)\nabla_{jkl} 
% {\rm LSD}(r_n,f_\theta)|_{\theta = \theta^*} 
% $$
% $$
% = S_1 + S_2 + S_3, ~~~~~~~~ (say)
% $$
where $\theta^*$ lies between $\theta^g$ and $\theta$. We will now consider each terms one-by-one.
% \\

	For the Linear term $S_1$, we consider
\begin{eqnarray}
    \nabla_j {\rm LSD}(r_n,f_\theta)
% |_{\theta = \theta^g} 
= 
\frac{(1+\beta)}{A} \left[\left(\sum_x f_\theta^{1+\beta}u_{j\theta}\right)
\left(\sum_x f_\theta^{1+\beta}\right)^{-1}
-\left(\sum_x r_n^A f_\theta^{B}u_{j\theta}\right)
\left(\sum_x r_n^A f_\theta^{B}\right)^{-1}\right]. \nonumber 
% -\frac{(1+\beta)}{A} ~ \sum_x K(\delta_n^g(x))f_{\theta^g}^{1+\beta}(x)w_{j\theta^g}(x)
\end{eqnarray}
To show  $$\nabla_j {\rm LSD}(r_n,f_\theta)
|_{\theta = \theta^g} \xrightarrow{P} 0,$$ we need to show 
  $$\sum_x K(\delta_n^g(x)) f_{\theta^g}^{1+\beta}(x) u_{j\theta^g}(x)
\xrightarrow{P} \sum_x K(\delta_g^g(x)) f_{\theta^g}^{1+\beta}(x) u_{j\theta^g}(x)$$
and 
  $$\sum_x K(\delta_n^g(x)) f_{\theta^g}^{1+\beta}(x) 
\xrightarrow{P} \sum_x K(\delta_g^g(x)) f_{\theta^g}^{1+\beta}(x) $$
as $n\rightarrow\infty$ and 
where $\delta_n^g(x)$ is the $\delta_n(x)$ evaluated at $\theta = \theta^g$. We will now 
show that 
\begin{eqnarray}
\label{EQ:k_delta_converge}
%     \sum_x K(\delta_n^g(x))f_{\theta^g}^{1+\beta}(x)
% w_{j\theta^g}(x) \xrightarrow{P} \sum_x K(\delta_g^g(x)) 
% f_{\theta^g}^{1+\beta}(x) w_{j\theta^g}(x). 
    \sum_x K(\delta_n^g(x))f_{\theta^g}^{1+\beta}(x)
u_{j\theta^g}(x) \xrightarrow{P} \sum_x K(\delta_g^g(x)) 
f_{\theta^g}^{1+\beta}(x) u_{j\theta^g}(x)
\end{eqnarray}
as $n \rightarrow \infty$. 
% and note that the right hand side of above is zero by definition of the minimum LSD estimator. 
% \\
Note that by assumption $7$ and the fact that $r_n(x) \rightarrow g(x) $ 
almost surely (a.s.) by Strong law of large numbers (SLLN), it follows that 
\begin{eqnarray}
\label{EQ:k_prime_bound}
	|K'(\delta)| = |A||\delta|^{A-1} < 2|A|C = C_1,  ~~~~~~ \mbox{(say)}
\end{eqnarray}
for any $\delta$ in between $\delta_n^g(x)$ and $\delta_g^g(x)$ 
(uniformly in $x$). So,  by using the one-term Taylor series expansion,
\begin{eqnarray}
 & &   |\sum_x K(\delta_n^g(x))f_{\theta^g}^{1+\beta}(x)u_{j\theta^g}(x) - 
\sum_x K(\delta_g^g(x)) f_{\theta^g}^{1+\beta}(x) u_{j\theta^g}(x) |  \nonumber \\
&\le & C_1 \sum_x |\delta_n^g(x) - \delta_g^g(x)| 
f_{\theta^g}^{1+\beta}(x) |u_{j\theta^g}(x) |. \nonumber \\ \nonumber 
\end{eqnarray}

	However, by Lemma \ref{lem:lamma1}(1), 
\begin{eqnarray}
     E\left[|\delta_n^g(x) - \delta_g^g(x)|\right] \le 
\frac{(g(x)(1-g(x))^{1/2}}{f_{\theta^g}(x) 
\sqrt n} \rightarrow 0  ~~~ as ~ ~ n \rightarrow \infty.
\end{eqnarray} 
	and, by Lemma \ref{lem:lamma1}(2), we have 
\begin{eqnarray}
 & &  E\left[ C_1 \sum_x |\delta_n^g(x) - \delta_g^g(x)| 
f_{\theta^g}^{1+\beta}(x) |u_{j\theta^g}(x) |\right] \nonumber \\
&\le& 2C_1 \sum_x g^{1/2}(x) f_{\theta^g}^{\beta}(x) |u_{j\theta^g}(x) |  ~ < ~ \infty     
 ~~~~~~~~~\mbox{(by Assumption $5$)}.
\end{eqnarray}
	Hence, by DCT, we get, 
\begin{eqnarray}
     E\left[|\sum_x K(\delta_n^g(x))f_{\theta^g}^{1+\beta}(x)
u_{j\theta^g}(x) - \sum_x K(\delta_g^g(x)) f_{\theta^g}^{1+\beta}(x) 
u_{j\theta^g}(x) |\right] \rightarrow 0 ~~~~ as ~ n \rightarrow \infty \nonumber
\end{eqnarray}
so that by Markov inequality we have the desired claim. 
% \noindent

By similar argument we can show 
\begin{eqnarray}
 \sum_x K(\delta_n^g(x)) f_{\theta^g}^{1+\beta}(x) \nonumber \\
\xrightarrow{P} \sum_x K(\delta_g^g(x)) f_{\theta^g}^{1+\beta}(x). 
\end{eqnarray}
Therefore, we have
\begin{eqnarray}
    \nabla_j {\rm LSD}(r_n,f_\theta)|_{\theta = \theta^g} \xrightarrow{P} 0.
\end{eqnarray}
	Thus, with probability tending to one, $|S_1| < p a^3$, where 
$p$ is the dimension of $\theta$ and $a$ is the radius of $Q_a$.
% \\

	Next we consider the quadratic term $S_2$. We have,
\begin{eqnarray}
\label{EQ:S2_form}
%     & &
\frac{A}{1+\beta}\nabla_{jk} {\rm LSD}(r_n, f_\theta) &=& 
% \frac{1+\beta}{A} 
% \left[
(1+\beta)\left(\sum_x f_\theta^{1+\beta} u_{j\theta}u_{k\theta}\right) 
\left(\sum_x f_\theta^{1+\beta} \right)^{-1} \nonumber \\ 
& & - (1+\beta)\left(\sum_x f_\theta^{1+\beta} u_{j\theta}\right)
\left(\sum_x f_\theta^{1+\beta} u_{k\theta}\right) 
\left(\sum_x f_\theta^{1+\beta} \right)^{-2}  \nonumber \\ 
& & -B \left(\sum_x r_n^A f_\theta^B u_{j\theta} u_{k\theta}\right)
\left(\sum_x r_n^A f_\theta^B\right)^{-1} \nonumber \\ 
& & +B \left( \sum_x r_n^A f_\theta^B u_{j\theta}\right)\left(\sum_x r_n^A f_\theta^B u_{k\theta}\right)
\left(\sum_x r_n^A f_\theta^B \right)^{-2} 
% \right] \\
% \nabla_k \left( -\frac{(1+\beta)}{A} ~ \sum_x K(\delta_n(x))
% f_{\theta}^{1+\beta}(x)w_{j\theta}(x)|_{\theta=\theta^g} \right)\nonumber \\
% &=& -\frac{(1+\beta)}{A} \left[\sum_x K'(\delta_n^g(x)) 
% \delta_n^g(x) f_{\theta^g}^{1+\beta}(x) w_{j\theta^g}(x) 
% w_{k\theta^g}(x) + \sum_x K(\delta_n^g(x))f_{\theta^g}^{1+\beta}(x)
% w_{jk\theta^g}(x) \right] \nonumber \\
%  &=& -\frac{(1+\beta)}{A} \left[\sum_x K'(\delta_n^g(x)) \delta_n^g(x) 
% f_{\theta^g}^{1+\beta}(x) w_{j\theta^g}(x) w_{k\theta^g}(x) 
% \right]. \\ \nonumber  
\end{eqnarray}
We will now show that 
\begin{eqnarray}
\nabla_{jk} {\rm LSD}(r_n, f_\theta) |_{\theta=\theta^g} \xrightarrow{P} - J_g^{j,k}. 
\end{eqnarray}
To show this we will show that 
\begin{eqnarray}
\label{EQ:k_ujk}
  \sum_x K(\delta_n^g(x)) f_{\theta^g}^{1+\beta}(x) u_{j\theta^g}(x) u_{k\theta^g}(x) \nonumber \\
\rightarrow \sum_x K(\delta_g^g(x)) f_{\theta^g}^{1+\beta}(x) u_{j\theta^g}(x) u_{k\theta^g}(x).
\end{eqnarray}
Note that by assumption $7$ and using the fact that $r_n(x) \rightarrow g(x) ~~ a.s. $ by SLLN
and by (\ref{EQ:k_prime_bound}) we get the following expansion,
% , it follows that 
% \begin{eqnarray}
% \label{EQ:k_prime_bound}
% 	|K'(\delta)| = |A||\delta|^{A-1} < 2|A|C = C_1,  ~~~~~~ \mbox{(say)}
% \end{eqnarray}
% for any $\delta$ in between $\delta_n^g(x)$ and $\delta_g^g(x)$ 
% (uniformly in $x$). 
% So,  by using the one-term Taylor series expansion,
\begin{eqnarray}
 & &   |\sum_x K(\delta_n^g(x))f_{\theta^g}^{1+\beta}(x)u_{j\theta^g}(x)u_{k\theta^g}(x)  - 
\sum_x K(\delta_g^g(x)) f_{\theta^g}^{1+\beta}(x) u_{j\theta^g}(x)u_{k\theta^g}(x)|  \nonumber \\
&\le & C_1 \sum_x |\delta_n^g(x) - \delta_g^g(x)| 
f_{\theta^g}^{1+\beta}(x) |u_{j\theta^g}(x)u_{k\theta^g}(x)|. \nonumber \\ \nonumber 
\end{eqnarray}

	However, by Lemma \ref{lem:lamma1}(1), 
\begin{eqnarray}
     E\left[|\delta_n^g(x) - \delta_g^g(x)|\right] \le 
\frac{(g(x)(1-g(x))^{1/2}}{f_{\theta^g}(x) 
\sqrt n} \rightarrow 0  ~~~ as ~ ~ n \rightarrow \infty.
\end{eqnarray} 
	and, by Lemma \ref{lem:lamma1}(2), we have 
\begin{eqnarray}
 & &  E\left[ C_1 \sum_x |\delta_n^g(x) - \delta_g^g(x)| 
f_{\theta^g}^{1+\beta}(x) |u_{j\theta^g}(x) u_{k\theta^g}(x)|\right] \nonumber \\
&\le& 2C_1 \sum_x g^{1/2}(x) f_{\theta^g}^{\beta}(x) |u_{j\theta^g}(x) u_{k\theta^g}(x)|  ~ < ~ \infty     
 ~~~~~~~~~\mbox{(by Assumption $5$)}. \nonumber
\end{eqnarray}
	Hence, by DCT, we get, 
\begin{eqnarray}
\label{EQ:s2_first}
%      E\left[
|\sum_x K(\delta_n^g(x))f_{\theta^g}^{1+\beta}(x)
u_{j\theta^g}(x) u_{k\theta^g}(x) \nonumber  \\  - \sum_x K(\delta_g^g(x)) f_{\theta^g}^{1+\beta}(x) 
u_{j\theta^g}(x) u_{k\theta^g}(x)|
% \right]  
& & \rightarrow 0 
\end{eqnarray}
 as $n \rightarrow \infty$.
so that by Markov inequality we have the desired claim. 
and along with we will use 
\begin{eqnarray}
\label{EQ:s2_second}
 \sum_x K(\delta_n^g(x)) f_{\theta^g}^{1+\beta}(x) u_{j\theta^g}(x)
\xrightarrow{P} \sum_x K(\delta_g^g(x)) f_{\theta^g}^{1+\beta}(x) u_{j\theta^g}(x)
\end{eqnarray}
and 
\begin{eqnarray}
\label{EQ:s2_third}
\sum_x K(\delta_n^g(x)) f_{\theta^g}^{1+\beta}(x) 
\xrightarrow{P} \sum_x K(\delta_g^g(x)) f_{\theta^g}^{1+\beta}(x) 
\end{eqnarray}
as $n\rightarrow\infty$ from the previous part of the proof. 
% 	We will now show that 
% \begin{eqnarray}
% \label{EQ:kw_prime_converge}
%      - \sum_x K'(\delta_n^g(x)) \delta_n^g(x) 
% f_{\theta^g}^{1+\beta}(x) w_{j\theta^g}(x) w_{k\theta^g}(x) \xrightarrow{P}  
% - \sum_x K'(\delta_g^g(x)) \delta_g^g(x) f_{\theta^g}^{1+\beta}(x) 
% w_{j\theta^g}(x) w_{k\theta^g}(x). 
% \end{eqnarray}
% Note that as in (\ref{EQ:k_prime_bound}), we have
% \begin{eqnarray}
% \label{EQ:k_dprime_bound}
%     |K''(\delta)\delta| = |A(A-1)||\delta|^{(A-1)} < C_2, ~~~ \mbox{(say)},
% \end{eqnarray}
% for every $\delta$ lying in between $\delta_n^g(x)$ and 
% $\delta_g^g(x)$ (uniformly in $x$). So,  by using the 
% one-term Taylor series expansion, 
% \begin{eqnarray}
%    |K'(\delta_n^g)\delta_n^g - K'(\delta_g^g)\delta_g^g|  
% & \le & |\delta_n^g -\delta_g^g||K''(\delta_n^*)\delta_n^* + K'(\delta_n^*)| \nonumber \\
% & \le &  |\delta_n^g -\delta_g^g| (C_2 +C_1). \nonumber \\  \nonumber 
% \end{eqnarray}
% Thus, we get 
% \begin{eqnarray}
%  & &      | \sum_x K'(\delta_n^g(x)) \delta_n^g(x) 
% f_{\theta^g}^{1+\beta}(x) w_{j\theta^g}(x) w_{k\theta^g}(x)  
% - \sum_x K'(\delta_g^g(x)) \delta_g^g(x) f_{\theta^g}^{1+\beta}(x) 
% w_{j\theta^g}(x) w_{k\theta^g}(x) | \nonumber \\
% & \le & (C_1 +C_2) \sum_x |\delta_n^g -\delta_g^g| 
% f_{\theta^g}^{1+\beta}(x) | w_{j\theta^g}(x) 
% w_{k\theta^g}(x) |. \nonumber \\ \nonumber    
% \end{eqnarray}
% Since by assumption (A5), we have $\sum_x g^{1/2}(x)
% f_{\theta^g}^{1+\beta}(x) |w_{j\theta^g}(x) 
% w_{k\theta^g}(x) | < \infty$, the desired result 
% (\ref{EQ:kw_prime_converge}) follows by the similar proof for proving (\ref{EQ:k_delta_converge}) above.
% 
% 	Similarly we also get that
% \begin{eqnarray}
% \label{EQ:kw_converge}
%   \sum_x K(\delta_n^g(x)) f_{\theta^g}^{1+\beta}(x) w_{jk\theta^g}(x)  
% &\xrightarrow{P} & \sum_x K(\delta_g^g(x))  f_{\theta^g}^{1+\beta}(x) 
% w_{jk\theta^g}(x), 
% \end{eqnarray}
% \begin{eqnarray}
% \label{EQ:kww_converge}
%  \sum_x K(\delta_n^g(x)) f_{\theta^g}^{1+\beta}(x) w_{j\theta^g}(x) 
% w_{k\theta^g}(x) &\xrightarrow{P}&   \sum_x K(\delta_g^g(x)) 
% f_{\theta^g}^{1+\beta}(x) w_{j\theta^g}(x) w_{k\theta^g}(x).  \\ \nonumber    
% \end{eqnarray}
	Thus, combining 
(\ref{EQ:s2_first}), (\ref{EQ:s2_second}) and (\ref{EQ:s2_third}), 
% (\ref{EQ:kw_prime_converge}), (\ref{EQ:kw_converge}) and (\ref{EQ:kww_converge}), 
we get that,
\begin{eqnarray}
\nabla_{jk} {\rm LSD}(r_n, f_\theta) |_{\theta=\theta^g} \xrightarrow{P} - J_g^{j,k}.
%     \nabla_k \left( \sum_x K(\delta_n(x))f_{\theta}^{1+\beta}(x)w_{j\theta}(x)|_{\theta=\theta^g} \right) 
% \xrightarrow{P} -J_g^{j,k}.
\end{eqnarray}
 Therefore,
\begin{eqnarray}
\label{EQ:2s2_form}
    2S_2 &=& \frac{1+\beta}{A} \sum_{j,k} 
\left\{ \nabla_{jk} {\rm LSD}(r_n, f_\theta) |_{\theta=\theta^g} 
% \left\{ \nabla_k \left( \sum_x K(\delta_n(x)) 
% f_{\theta}^{1+\beta}(x) w_{j\theta}(x)|_{\theta=\theta^g} \right)
 - 
( - J_g^{j,k})\right\} (\theta_j - \theta_j^g)(\theta_k - \theta_k^g) \nonumber \\
 & &  ~~~~~~~ + \sum_{j,k} \left\{ - \left(\frac{(1+\beta)}{A} 
J_g^{j,k} \right) (\theta_j - \theta_j^g)(\theta_k - \theta_k^g) \right\}. \\ \nonumber
\end{eqnarray}
	Now the absolute value of the first term in above (\ref{EQ:2s2_form}) is 
$ < p^2 a^3$ with probability tending to one. And, the second term in 
(\ref{EQ:2s2_form}) is a negative definite quadratic form in the variables 
$(\theta_j - \theta_j^g)$. Letting $\gamma_1$ be the largest 
eigenvalue of $\frac{(1+\beta)}{A} J_g$, the quadratic form is 
$ < \gamma_1 a^2$. Combining the two terms, we see that there 
exists $c > 0$ and $a_0 > 0$ such that for $a < a_0$, we have  
$ S_2 < -c a^2$ with probability tending to one.
% \\
% 	Finally, considering the cubic term $S_3$, we have
% 
% 
% \begin{eqnarray}
%     & &\nabla_{jkl} {\rm LSD}(r_n, f_\theta)|_{\theta=\theta^*} = 
% \nabla_{kl} \left( -\frac{(1+\beta)}{A} ~ \sum_x K(\delta_n(x))
% f_{\theta}^{1+\beta}(x)w_{j\theta}(x)|_{\theta=\theta^*} \right)\nonumber \\
%  & = & -\frac{(1+\beta)}{A}  \nabla_{l} \left(  - \sum_x 
% K'(\delta_n(x)) \delta_n(x) f_{\theta}^{1+\beta}(x) w_{j\theta}(x) 
% w_{k\theta}(x)  + \sum_x K(\delta_n(x)) f_{\theta}^{\beta}(x) 
% \nabla_{jk}f_\theta(x) \right) |_{\theta=\theta^*} \nonumber \\ 
% \end{eqnarray}
% or,
% \begin{eqnarray}
% \label{EQ:lsd_jkl}
%   & &  - \frac{A}{1+\beta} \nabla_{jkl} 
% {\rm LSD}(r_n, f_\theta)|_{\theta=\theta^*} = 
% \sum_x K''(\delta_n^*(x)) \delta_n^*(x)^2 
% f_{\theta^*}^{1+\beta}(x) w_{j\theta^*}(x) 
% w_{k\theta^*}(x) w_{l\theta^*}(x)  \nonumber \\
% & & ~~~ - \sum_x K'(\delta_n^*(x)) \delta_n^*(x) 
% f_{\theta^*}^{1+\beta}(x) w_{j\theta^*}(x) 
% w_{k\theta^*}(x) w_{l\theta^*}(x) - \sum_x K'(\delta_n^*(x)) 
% \delta_n^*(x) f_{\theta^*}^{1+\beta}(x) w_{j\theta^*}(x) 
% w_{kl\theta^*}(x) \nonumber \\
% & &~~~ - \sum_x K'(\delta_n^*(x)) \delta_n^*(x) 
% f_{\theta^*}^{1+\beta}(x)  w_{k\theta^*}(x) w_{jl\theta^*}(x)  
% - \sum_x K'(\delta_n^*(x)) \delta_n^*(x) f_{\theta^*}^{1+\beta}(x) 
% \frac{\nabla_{jk}f_{\theta^*(x)}}{f_{\theta^*(x)}}  
% w_{l\theta^*}(x)   \nonumber \\
%  & & ~~~ + \sum_x K(\delta_n^*(x)) f_{\theta^*}^{1+\beta}(x) 
% \frac{\nabla_{jkl}f_{\theta^*(x)}}{f_{\theta^*(x)}}  + \sum_x 
% K(\delta_n^*(x)) f_{\theta^*}^{1+\beta}(x) \frac{\nabla_{jk}
% f_{\theta^*(x)}}{f_{\theta^*(x)}}  w_{l\theta^*}(x).  \\ \nonumber 
% \end{eqnarray}
% where $\delta_n^*(x) = \frac{r_n(x)}{f_{\theta^*}(x)}$.	We will 
% now show that all the terms in the RHS of above (\ref{EQ:lsd_jkl}) 
% are bounded. Let us name the terms by (i) to (vii) respectively.
% % \\
% 	For the first term (i), we use 
% % (\ref{EQ:k_dprime_bound}) 
% to get
% \begin{eqnarray}
%      |\sum_x K''(\delta_n^*(x)) \delta_n^*(x)^2 f_{\theta^*}^{1+\beta}(x) 
% w_{j\theta^*}(x) w_{k\theta^*}(x) w_{l\theta^*}(x)| &\le & 
% C_2 ~ \sum_x |\delta_n^*(x)| M_{j,k,l}(x)f_{\theta^*}(x) \nonumber \\
%  & =&  C_2 ~ \sum_x r_n^*(x) M_{j,k,l}(x) ~~~~ \mbox{(by CLT)} \nonumber \\
%  & \rightarrow &  C_2 ~ E_g[ M_{j,k,l}(X)]  < \infty  ~~~~ \mbox{(by assumption (A6))}.  \nonumber \\ 
% \end{eqnarray}
% 	Thus term (i) is bounded. Now for the second term (ii), 
% we again use (\ref{EQ:k_prime_bound}) to get 
% \begin{eqnarray}
%      |\sum_x K'(\delta_n^*(x)) \delta_n^*(x) 
% f_{\theta^*}^{1+\beta}(x) w_{j\theta^*}(x) 
% w_{k\theta^*}(x) w_{l\theta^*}(x)| &\le & 
% C_1 ~ \sum_x |\delta_n^*(x)| M_{j,k,l}(x)f_{\theta^*}(x) \nonumber \\
%  & =&  C_1 ~ \sum_x r_n^*(x) M_{j,k,l}(x) ~~~~ \mbox{(by CLT)} \nonumber \\
%  & \rightarrow &  C_1 ~ E_g[ M_{j,k,l}(X)]  
% < \infty  ~~~~ \mbox{(by assumption (A6))}.  \nonumber \\ 
% \end{eqnarray}
% so that term (ii) is also bounded. Similarly the terms (iii),(iv) 
% and (v) are bounded as in case of term (ii) and using (\ref{EQ:k_prime_bound}) 
% and assumption (A6). Next for the term (vi), we will consider
% \begin{eqnarray}
%     |K(\delta)| = |\int_0^\delta K'(\delta) d\delta| \le C_1 |\delta|
% \end{eqnarray}
% so that 
% \begin{eqnarray}
%     |K(\delta_n^*(x))| \le C_1 \frac{r_n(x)}{f_{\theta^*}(x)}.
% \end{eqnarray}
% Also
% \begin{eqnarray}
%     \frac{\nabla_{jkl} f_{\theta^*}(x)}{f_{\theta^*}(x)} 
% &=& w_{jkl\theta^*}(x) + w_{jk\theta^*}(x)w_{l\theta^*}(x) 
% + w_{jl\theta^*}(x)w_{k\theta^*}(x) \nonumber \\
%  & &  + w_{j\theta^*}(x)w_{kl\theta^*}(x) + w_{j\theta^*}(x)
% w_{k\theta^*}(x)w_{l\theta^*}(x).
% \end{eqnarray}
% 	So,
% \begin{eqnarray}
%      |\sum_x K(\delta_n^*(x)) f_{\theta^*}^{1+\beta}(x) \frac{\nabla_{jkl} 
% f_{\theta^*}(x)}{f_{\theta^*}(x)}| &\le & C_1 ~ \sum_x r_n(x)  
% f_{\theta^*}^{\beta}(x) |\frac{\nabla_{jkl} f_{\theta^*}(x)}{f_{\theta^*}(x)}| \nonumber \\
%  & =&  C_1 ~ \sum_x r_n^*(x) M(x) ~~~~ \mbox{(by CLT)} \nonumber \\
%  & \rightarrow &  C_1 ~ E_g[ M(X)]  < \infty  ~~~~ \mbox{(by assumption (A6))}  \nonumber \\ 
% \end{eqnarray}
% where $M(x) = M_{jkl}(x) + M_{jk,l}(x) + M_{jl,k}(x) 
% + M_{j,kl}(x) + M_{j,k,l}(x) $. Thus the term (vi) 
% is bounded and also similarly the term (vii) is bounded.	
% \\
	Finally, considering the cubic term $S_3$, we have
\begin{align*}
 \frac{A}{1+\beta} \nabla_{jkl} {\rm LSD}(r_n, f_\theta)
% |_{\theta=\theta^*} 
= (1+\beta)^2 \left(\sum_x f_\theta^{1+\beta} u_{j\theta}u_{k\theta}u_{l\theta}\right)
\left(\sum_x f_\theta^{1+\beta}\right)^{-1} \nonumber \\
+(1+\beta)\left(\sum_x f_\theta^{1+\beta}
u_{jl\theta}u_{k\theta}\right) 
\left(\sum_x f_\theta^{1+\beta}\right)^{-1}
+(1+\beta)\left(\sum_x f_\theta^{1+\beta}
u_{j\theta}u_{kl\theta}\right) 
\left(\sum_x f_\theta^{1+\beta}\right)^{-1} \nonumber \\
-(1+\beta)^2 \left(\sum_x f_\theta^{1+\beta}u_{j\theta}u_{k\theta}\right)
\left(\sum_x f_\theta^{1+\beta}u_{l\theta}\right)\left(\sum_x f_\theta^{1+\beta}\right)^{-2}
+(1+\beta)\left(\sum_x f_\theta^{1+\beta}u_{l\theta}u_{jk\theta}\right)
\left(\sum_x f_\theta^{1+\beta}\right)^{-1}\nonumber \\
+\left(\sum_x f_\theta^{1+\beta}u_{jkl\theta}\right)\left(\sum_x f_\theta^{1+\beta}\right)^{-1}
-(1+\beta)\left(\sum_x f_\theta^{1+\beta}u_{jk\theta}\right)
\left(\sum_x f_\theta^{1+\beta}u_{l\theta}\right)\left(\sum_x f_\theta^{1+\beta}\right)^{-2}\nonumber \\
-(1+\beta)^2\left(\sum_x f_\theta^{1+\beta}u_{j\theta}u_{l\theta}\right)
\left(\sum_x f_\theta^{1+\beta}u_{k\theta}\right)\left(\sum_x f_\theta^{1+\beta}\right)^{-2}\nonumber \\
-(1+\beta)\left(\sum_x f_\theta^{1+\beta}u_{jl\theta}\right)\left(\sum_x f_\theta^{1+\beta}u_{k\theta}\right)
\left(\sum_x f_\theta^{1+\beta}\right)^{-2}\nonumber \\
-(1+\beta)^2\left(\sum_x f_\theta^{1+\beta}u_{j\theta}\right)\left(\sum_x f_\theta^{1+\beta}u_{k\theta}u_{l\theta}\right)
\left(\sum_x f_\theta^{1+\beta}\right)^{-2}\nonumber \\
-(1+\beta)\left(\sum_x f_\theta^{1+\beta}u_{j\theta}\right)\left(\sum_x f_\theta^{1+\beta}u_{kl\theta}\right)
\left(\sum_x f_\theta^{1+\beta}\right)^{-2}\nonumber \\
+2(1+\beta)^2\left(\sum_x f_\theta^{1+\beta}u_{j\theta}\right)\left(\sum_x f_\theta^{1+\beta}u_{k\theta}\right)
\left(\sum_x f_\theta^{1+\beta}u_{l\theta}\right)\left(\sum_x f_\theta^{1+\beta}\right)^{-3}\nonumber \\
-B^2\left(\sum_x r_n^A f_\theta^B u_{j\theta}u_{k\theta}u_{l\theta}\right)
\left(\sum_x r_n^A f_\theta^B\right)^{-1}
-B\left(\sum_x r_n^A f_\theta^B u_{jl\theta}u_{k\theta}\right)\left(\sum_x r_n^A f_\theta^B\right)^{-1}\nonumber \\
-B\left(\sum_x r_n^A f_\theta^B u_{j\theta}u_{kl\theta}\right)\left(\sum_x r_n^A f_\theta^B\right)^{-1}
+B^2\left(\sum_x r_n^Af_\theta^B u_{j\theta}u_{k\theta}\right)\left(\sum_x r_n^Af_\theta^B u_{l\theta}\right)
\left(\sum_x r_n^A f_\theta^B\right)^{-2}\nonumber \\
-B\left(\sum_x r_n^Af_\theta^B u_{jk\theta}u_{l\theta}\right)\left(\sum_x r_n^Af_\theta^B u_{jkl\theta}\right)
\left(\sum_x r_n^A f_\theta^B\right)^{-2}\nonumber \\
+B\left(\sum_x r_n^Af_\theta^B u_{jk\theta}\right)\left(\sum_x r_n^Af_\theta^B u_{l\theta}\right)
\left(\sum_x r_n^A f_\theta^B\right)^{-2}\nonumber \\
+B^2\left(\sum_x r_n^Af_\theta^B u_{j\theta}u_{l\theta}\right)\left(\sum_x r_n^Af_\theta^B u_{k\theta}\right)
\left(\sum_x r_n^A f_\theta^B\right)^{-2}\nonumber \\
+B\left(\sum_x r_n^Af_\theta^B u_{jl\theta}\right)\left(\sum_x r_n^Af_\theta^B u_{k\theta}\right)
\left(\sum_x r_n^A f_\theta^B\right)^{-2}\nonumber \\
+B^2\left(\sum_x r_n^Af_\theta^B u_{k\theta}u_{l\theta}\right)\left(\sum_x r_n^Af_\theta^B u_{j\theta}\right)
\left(\sum_x r_n^A f_\theta^B\right)^{-2}\nonumber \\\displaybreak[3]\nonumber \\
+B\left(\sum_x r_n^Af_\theta^B u_{kl\theta}\right)\left(\sum_x r_n^Af_\theta^B u_{j\theta}\right)
\left(\sum_x r_n^A f_\theta^B\right)^{-2}\nonumber \\
-2B^2\left(\sum_x r_n^Af_\theta^B u_{j\theta}\right)\left(\sum_x r_n^Af_\theta^B u_{k\theta}\right)
\left(\sum_x r_n^Af_\theta^B u_{l\theta}\right)
\left(\sum_x r_n^A f_\theta^B\right)^{-3}.\nonumber \\
\end{align*}
Using the assumptions and $r_n(x)\rightarrow g(x)$ we can show the 
cubic term $S_3$ is also bounded. 
% (detailed expression of the cubic term 
% has been put in Appendix).
Hence, we have $|S_3|< b a^3$ on the sphere $Q_a$ with probability tending to one. 
% \\
	Combining the three inequality we get that 
$$
max(S_1 + S_2 + S_3 ) < - c a^2 + (b + p) a^3   
~~\left[~ < 0 ~~\mbox{  for } ~~ a< \frac{c}{b+p} \right].
$$
Thus, for any sufficiently small $a$, there 
exists a sequence of roots $\theta_n = \theta_n(a)$ to 
the minimum LSD estimating equation such that 
$P(||\theta_n - \theta^g||_2 < a)$ converges to one, 
where $||.||_2$ denotes the $L_2-$norm.
	It remains to show that we can determine such a 
sequence independent of $a$. For let $\theta_n^*$ be the 
root which is closest to $\theta^g$. This exists because 
the limit of a sequence of roots is again a root by the 
continuity of the LSD. Hence proved the consistency part.
% \\
% \\
% %\newpage 
% \noindent
% 
% $$    		S_3=	 -      \frac{1}{6} \sum_{j,k,l} (\theta_j - 
% \theta_j^g)(\theta_k - \theta_k^g)(\theta_l - \theta_l^g)\nabla_{jkl} 
% {\rm LSD}(r_n,f_\theta)|_{\theta = \theta^*} 
% $$
% and
% 
%\bibliographystyle{alpha}
%\bibliography{references.bib}


\end{document}

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