[**Robust estimator of distortion risk premiums for heavy-tailed losses**]{}

Brahim Brahimi[[^1] and ]{}Zoubir Kenioua

[*Laboratory of Applied Mathematics, Mohamed Khider University, Biskra, Algeria*]{}

**Abstract**

We use the so-called t-Hill tail index estimator proposed by [Fab01]{}, rather than Hill’s one, to derive a robust estimator for the distortion risk premium of loss. Under the second-order condition of regular variation, we establish its asymptotic normality. By simulation study, we show that this new estimator is more robust than of [@NeMe09] both for small and large samples.

**Keywords:** Distortion risk premiums; Extreme values; Tail; Robustness.

**AMS 2010 Subject Classification:** 62G10; 62G32.

**Introduction\[sec1\]**
========================

In many important applications in Finance, Actuarial Science, Hydrology, Insurance, one of most crucial topics is the determination of the amounts of losses of a heavy-tailed risks. In literature there are many possible definitions of risk according to the investment objectives, so in some sense risk itself is a subjective concept as well as the risk measure for an investor. Thus, the existence of a unique risk that solves the investor’s problems is not confirmed. However, one must distinguish between risks. The concept of coherence due to the paper of [@ArDeEbHe99] which categorize risks by good or bad risks. For this reason and to improve the performance of investor’s strategies we may identify those risk measures and the appearance of the risk himself heavy tails, asymmetries,... Most of this risk measures, used are special cases of Wang’s distortion premium [@Wang96], defined as follows $$\Pi \lbrack \psi ;F]=\int_{0}^{\infty }\psi (1-F(x))dx,  \label{eq1}$$ where $\psi :[0,1]\rightarrow \lbrack 0,1]$ is a non-decreasing function called distortion function, such that $\psi (0)=0$ and $\psi (1)=1.$ The distortion functions $\psi $ are concave, which makes the corresponding distortion premiums $\Pi \lbrack \psi ;F]$ coherent [@ArDeEbHe99] as proved by [@WiHa99]. In this paper, we suppose that the distortion functions $t\longmapsto \psi \left( t^{-1}\right) $ is regularly varying at infinity with index of regular variation $\rho \geq 1,$ such that $$\psi \left( t^{-1}\right) =t^{-1/\rho }\mathcal{L}_{\psi }\left( t\right) ,
    \label{g}$$ where $t\longmapsto \mathcal{L}_{\psi }\left( t\right) $ is slowly varying as infinity, that is $\mathcal{L}_{\psi }\left( tx\right) /\mathcal{L}_{\psi
    }\left( t\right) \rightarrow 1$ as $t\rightarrow \infty ,$ for any $x>0.$ Note that a non negative random variable (rv) $X$ with finite mean and a cumulative distribution function (cdf) $F$ is called heavy-tailed, if $1-F$ is regularly varying with index $-1/\gamma <0$ ( notation: $F\in \mathcal{RV}    _{\left( -1/\gamma \right) })$, that is $$\lim_{t\rightarrow \infty }\frac{1-F\left( tx\right) }{1-F\left( t\right) }    =x^{-1/\gamma },\text{ for }x>0.  \label{first-condition}$$ In particular, the proportional-hazards premium (see, [@Roletal99], page 82). $$\Pi \lbrack \rho ;F]=\int_{0}^{\infty }\left( 1-F(x)\right) ^{1/\rho }dx,
    \label{Psi-ro}$$ with the concave distortion function $\psi (t)=t^{1/\rho }$ for every $\rho
    \geq 1.$ Since we are concerned with with heavy-tailed losses with infinite second moment, then by following [@BrMeNeRi], we assume that $\gamma \in
    (1/2,1)$ and $\rho \gamma \in \left( 0,1\right) ,$ thus we will work with $$1/2<\gamma <1/\rho .  \label{condition gamabeta}$$ Suppose that we have an independent and identically distributed (iid) sample $X_{1},...,X_{n}$ of rv $X$ of size $n$ with a cdf $F$ satisfying condition ( \[first-condition\]) and let denote by $X_{1:n}\leq ...\leq X_{n:n}$ the corresponding order statistics. Also, let $1<k=k_{n}$ be the number of extreme observations used in the computation of the tail index. We assume that $k$ satisfies the conditions $$1<k<n,\text{ }k\rightarrow \infty \text{ and }k/n\rightarrow 0\text{ as }    n\rightarrow \infty .  \label{K}$$ [@NeMe09] proposed an alternative estimator of (\[Psi-ro\]) and establish its asymptotic normality by using the Weissman’s estimator of the high quantile $q_{t}=F^{\leftarrow }\left( 1-t\right) $ defined by $$\widehat{q}_{t}=\left( k/n\right) ^{\widehat{\gamma }^{H}}X_{n-k:n}t^{-        \widehat{\gamma }^{H}},\text{ }t\downarrow 0,$$ where $F^{\leftarrow }$ denotes the generalized inverse of $F$ and $$\widehat{\gamma }^{H}=\widehat{\gamma }^{H}\left( k\right) :=\frac{1}{k}    \sum\limits_{i=1}^{k}\log X_{n-i+1:n}-\log X_{n-k:n},  \label{Hill}$$ is the well-known Hill estimator [@Hill75] of the tail index $    \gamma .$ For a fixed aversion parameter $\rho ,$ their estimator is given by $$\widehat{\Pi }_{n}\left( \widehat{\gamma }^{H},k\right) :=\left( k/n\right)
    ^{1/\rho }\frac{X_{n-k:n}}{1-\widehat{\gamma }^{H}\rho }+\sum_{i=k+1}^{n}    \left( \left( i/n\right) ^{1/\rho }-\left( \left( i-1\right) /n\right)
    ^{1/\rho }\right) X_{n-i+1:n}.  \label{NeMe}$$ The Hill estimator is a pseudo-maximum likelihood estimator based on the exponential approximation of normalized log-spacings $Y_{j}=j\left( \log
    X_{j:n}-\log X_{j+1:n}\right) $ for $j=1,...,k.$ In practice, the Hill estimator depends on the choice of the sample fraction $k$ and is inherently not very robust to large values $Y_{j},$ which makes the estimator proposed by [@NeMe09] sensitive. This constitutes a serious problem in terms of bias and root mean squared error (RMSE). To improve the quality of $\widehat{        \Pi }_{n}\left( \widehat{\gamma }^{H},k\right) $, instead of Hill’s one, we propose to estimate the tail index $\gamma $ by the so-called t-Hill estimator, proposed by [@Fab01], given by its harmonic mean $$\widehat{\gamma }=\widehat{\gamma }\left( k\right) :=\left( \frac{1}{k}    \sum_{j=1}^{k}\frac{X_{n-k:n}}{X_{n-j+1:n}}\right) ^{-1}-1  \label{t-Hill}$$ known as score moment estimation (t-score or t-estimation method). The latter is more robust than the classical Hill estimator $\widehat{\gamma }    ^{H}$ defined in (\[Hill\]) (see [@StFaS12] and the asymptotic normality is given in Theorem 2 of [@BeScSt14]). For other robust estimators for $\gamma $ we referred to [@PeWe01], [@JuSc04], [VaBeChHu07]{} and [@KiLe08]. The rest of the paper is organized as follows, in Section \[sec2\] we present a construction of a robust estimator of $\Pi \lbrack \psi ;F]$ in the case of heavy-tailed losses. In Section \[sec3\] we establish its asymptotic normality. In Section [sec4]{} we carry out a simulation study to illustrate empirical performance and robustness of the estimator. Concluding notes are given in Section [sec5]{}. Proofs are gathered in Section \[sec6\].

Throughout the paper, we use the standard notation $\overset{P}{        \rightarrow }$ for the convergence in probability and $\mathcal{N}\left( \mu
    ,\sigma \right) $ to denote a normal rv with mean $\mu $ and variance $    \sigma .$

**Defining the estimator\[sec2\]**
==================================

By using the generalized inverse $F^{\leftarrow }$ and for a fixed distortion function $\psi ,$ we may rewrite (\[eq1\]) into $$\Pi _{\psi }[X]:=-\int_{0}^{1}\psi \left( s\right) dF^{\leftarrow }\left(
    1-s\right) .  \label{DRMQ}$$ The empirical estimator of the risk premium $\Pi _{\psi }[X]$ is obtained by substituting $F^{\leftarrow }$ on the right-hand side of equation (\[DRMQ\] ) by its empirical counterpart $F_{n}^{\leftarrow }{\left( s\right) :=\inf
        \{x\in \mathbb{R}:}$ ${F}_{n}{\left( x\right) \geq s\},}$ $0<s\leq 1,$ associated to the empirical cdf defined on the real line, defined by $    {\normalsize F_{n}\left( x\right) :=n}^{-1}\#\left\{ X_{i}\leq x,1\leq i\leq
    n\right\} $ where $\#A$ denote the cardinality of a set $A.$ After straightforward computations, we obtain the formula $$\Pi _{n}[X]:=\int_{0}^{1}F_{n}^{\leftarrow }\left( 1-s\right) d\psi \left(
    s\right)$$ which may be rewritten, in terms of $X_{1:n},...,X_{n:n},$ as an $L$ -statistic $$\Pi _{n}[X]=\sum_{i=1}^{n}c_{i,n}\left( \psi \right) X_{n-i+1:n},
    \label{psin}$$ where $$c_{i,n}\left( \psi \right) \equiv \psi \left( i/n\right) -\psi \left( \left(
    i-1\right) /n\right) .  \label{ai}$$ The form (\[psin\]) is a linear combinations of the order statistics [see, @SW86 page 260]. The limit behavior was discussed by many authors: [@Chall67], [@Stigler74], [@Mason81], [@JoZi03] (see its Theorem 3.2 in the case that $X$ is not heavy-tailed) and in [BrMeNeRi]{} (in heavy-tailed case).

**Heavy-tailed losses case**
----------------------------

Let $X$ be a non-negative rv with cdf $F\in \mathcal{RV}_{\left(
        -1/\gamma \right) }.$ The condition (\[first-condition\]) is equivalent to $$\lim_{t\rightarrow 0}\frac{F^{\leftarrow }\left( 1-tx\right) }{F^{\leftarrow
        }\left( 1-t\right) }=x^{-\gamma },\text{ for every }x>0.
    \label{First-Cond-Qua}$$ We say that the function $s\rightarrow F^{\leftarrow }\left( 1-s\right) $ satisfying condition (\[First-Cond-Qua\]) is regularly varying at zero with the index $(-\gamma )<0.$ The parameter $\gamma $ is called the tail index or extreme value index (EVI). A various tail index estimators have been suggested in the literature, based for instance of the conventional maximum likelihood method, moment estimation, ... (see, e.g. [@Hill75] , [@Pick75], [@Dekeretal89]. [@Csorgoetal85] and [@Drees95]). For the robustness and bias reduction (see, i.e. [@PenQui04] and [@StFaS12]). The regular-variation condition itself is not sufficient for establishing asymptotic distributions. To this end, we suppose that cdf $F$ satisfy the well-known by the second-order condition of regular variation with second-order parameter $\tau \leq 0$, that is: there exists a function $t\rightarrow a(t)$ with constant sign at infinity and converges to $0$ as $t\rightarrow \infty $ such that $$\underset{t\rightarrow \infty }{\lim }\dfrac{\overline{F}\left( tx\right) /        \overline{F}\left( t\right) -x^{-1/\gamma }}{a\left( t\right) }=x^{-1/\gamma
    }\dfrac{x^{\tau /\gamma }-1}{\gamma \tau },  \label{second-order}$$ for every $x>0.$ When $\tau =0,$ then the ratio $\dfrac{x^{\tau /\gamma }-1}{        \gamma \tau }$ should be interpreted as $\log x.$ In terms of the quantile function $F^{\leftarrow },$ condition (\[second-order\]) is equivalent to the following one $$\lim_{t\rightarrow 0}\frac{\dfrac{F^{\leftarrow }\left( 1-tx\right) }{            F^{\leftarrow }\left( 1-t\right) }-x^{-\gamma }}{A\left( t\right) }    =x^{-\gamma }\frac{x^{\tau }-1}{\tau },  \label{second-cond}$$ for every $x>0,$ where $A\left( t\right) :=\gamma ^{2}a\left( F^{\leftarrow
    }\left( 1-t\right) \right) ,$ (see @deHS96, [-@deHS96] or Theorem 3.2.9 in @deHF06, [-@deHF06 page 48]). The Weissman estimator [@Weis78] of high quantiles $F^{\leftarrow }$ is given by $$F_{n}^{\leftarrow \left( W\right) }(1-s):=(k/n)^{\widehat{\gamma }    }X_{n-k:n}s^{-\widehat{\gamma }},\text{ }s\downarrow 0.  \label{wies}$$ The formula (\[DRMQ\]) can be split into $$\Pi _{\psi }[X]=-\int_{0}^{k/n}\psi \left( s\right) dF^{\leftarrow
    }(1-s)-\int_{k/n}^{1}\psi \left( s\right) dF^{\leftarrow }(1-s).  \label{pi}$$ By using an integration by part to the second integral yields $$\begin{aligned}
        \Pi _{\psi } &=&\psi \left( k/n\right) F^{\leftarrow
        }(1-k/n)-\int_{0}^{k/n}\psi \left( s\right) dF^{\leftarrow
    }(1-s)+\int_{k/n}^{1}F^{\leftarrow }(1-s)d\psi \left( s\right)  \\
    &:&=\Pi _{\psi }^{\left( 1\right) }+\Pi _{\psi }^{\left( 2\right) }+\Pi
    _{\psi }^{\left( 3\right) }.\end{aligned}$$A simple estimator of $\Pi _{\psi }^{\left( 1\right) }$ is $$\Pi _{\psi ,n}^{\left( 1\right) }:=\psi \left( k/n\right) X_{n-k,n}.
\label{pi1}$$To estimate $\Pi _{\psi }^{\left( 2\right) }$, we note that $\widehat{\gamma 
}$ is a consistent estimator for $\gamma $ [@StFaS12] and since $\rho <1/\gamma $, by substituting $F_{n}^{\leftarrow \left( W\right) }(1-s)$ given in (\[wies\]) instead of $F^{\leftarrow }(1-s)$ and integrating yield the following estimator$$\Pi _{\psi ,n}^{\left( 2\right) }:=\widehat{\gamma }\left( k/n\right) ^{    \widehat{\gamma }}X_{n-k:n}\int_{0}^{k/n}s^{-\widehat{\gamma }-1}\psi \left(
s\right) ds.  \label{pi2}$$Finally, by plugging $F_{n}^{\leftarrow }$ instead of $F^{\leftarrow }$ on second integral of Equation (\[pi\]) we obtain the estimator $$\Pi _{\psi ,n}^{\left( 3\right) }:=\sum_{i=k+1}^{n}c_{i,n}\left( \psi
\right) X_{n-i+1:n},  \label{pi3}$$of $\Pi _{\psi }^{\left( 3\right) },$ where $F_{n}^{\leftarrow }{\normalsize     \left( s\right) :=\inf \left\{ x\in \mathbb{R}:F_{n}\left( x\right) \geq
    s\right\} ,\;0<s\leq 1,}$ denote the sample quantile function associated to the empirical cdf defined on the real line by ${\normalsize F_{n}\left(
    x\right) :=n}^{-1}\sum\nolimits_{i=1}^{n}\mathbb{I}\left( X_{i}\leq x\right)
,$[ ]{}with $\mathbb{I}\left( \cdot \right) $ being theindicator function and the coefficients $c_{i,n}\left( \psi \right) $ are given in (\[ai\]). The final form of our estimator$$\widetilde{\Pi }_{\psi ,n}:=X_{n-k,n}\left( \psi \left( k/n\right) +\widehat{    \gamma }\left( k/n\right) ^{\widehat{\gamma }}\int_{0}^{k/n}s^{-\widehat{        \gamma }-1}\psi \left( s\right) ds\right) +\sum_{i=k+1}^{n}c_{i,n}\left(
\psi \right) X_{n-i+1:n}.$$To establish the asymptotic normality of our estimator and compared with the estimator proposed by @NeMe09 given in Equation (\[NeMe\]), we use the same function $\psi \left( t\right) =t^{1/\rho },$ for $\rho \geq 1$ used in [@NeMe09]. In this case our estimator have the following form$$\widetilde{\Pi }_{n}\left( \widehat{\gamma },k\right) :=\left( k/n\right)
^{1/\rho }\frac{X_{n-k,n}}{1-\widehat{\gamma }\rho }+\sum_{i=k+1}^{n}\left(
\left( i/n\right) ^{1/\rho }-\left( \left( i-1\right) /n\right) ^{1/\rho
}\right) X_{n-i+1:n}.  \label{Br}$$

**Asymptotic distribution\[sec3\]**
===================================

We will begin to expose our results as asymptotic representations theorems in the lines of [@BeScSt14]. For that purpose, we need to describe the probability theory on which they hold. Indeed, we use the so-called Hungarian construction of [@CsCsHM86]. For this we define $\left\{ U_{n}(s),0\leq s\leq 1\right\} $, the uniform empirical distribution function and we consider the order statistics $U_{1:n}\leq ...\leq U_{n:n}$ pertaining to the independent standard uniform rv’s $U_{1},U_{2},...,$ we introduce the uniform empirical quantile function $\left\{ V_{n}(s),0\leq
s\leq 1\right\} $ based on the $n\geq 1$ first observations of $U_{1},U_{2},...,$ on $(0,1)$ such that$$V_{n}\left( s\right) =U_{i,n}\quad \text{for}\ \left( i-1\right) /n<s\leq
i/n,\text{ }i=1,...,n,\text{ and }V_{n}\left( 0\right) =U_{1,n}.$$Let $\beta _{n}\left( s\right) =\sqrt{n}\left( s-V_{n}\left( s\right)
\right) ,0\leq s\leq 1$, be the corresponding quantile empirical process, for $n\geq 1.$

We use the well-known by Gaussian approximation given in [CsCsHM86]{} Corollary 2.1. It says that: on the probability space $\left(
\Omega ,\mathcal{A},\mathbb{P}\right) ,$ there exists a sequence of Brownian bridges $\left\{ \mathbb{B}_{n}\left( s\right) ;\text{ }0\leq s\leq
1\right\} $ has the representation $$\left\{ \mathbb{B}_{n}\left( s\right) ;\text{ }0\leq s\leq 1\right\} \overset{D}{=}\left\{ W_{n}\left( s\right) -sW_{n}\left( 1\right) ;\text{ }0\leq
s\leq 1\right\} ,$$where $W_{n}$ is a standard Wiener process such that for every $0\leq \zeta
<1/2,$$$\sup_{1/n\leq s\leq 1-1/n}\frac{n^{\zeta }\left\vert \beta _{n}\left(
    s\right) -\mathbb{B}_{n}\left( s\right) \right\vert }{\left( s\left(
    1-s\right) \right) ^{1/2-\zeta }}=O_{\mathbb{P}}\left( n^{-\zeta }\right) .
\label{approxi}$$Our main result is the following

\[Theorem2\]Let $F$ be a df satisfying (\[second-cond\]) with $\gamma
    >1/2$ and suppose that $F^{\leftarrow }\left( \cdot \right) $ is continuously differentiable on $[0,1).$ Let $k=k_{n}$ satisfying (\[K\]) such that $\sqrt{k}A\left( n/k\right) \rightarrow 0$ as $n\rightarrow \infty
    .$ For any $1\leq \rho <1/\gamma ,$ we have $$\frac{\sqrt{n}\left( \widetilde{\Pi }_{\psi ,n}-\Pi _{\psi }\right) }{        (k/n)^{1/\rho -1/2}F^{\leftarrow }(1-k/n)}\overset{d}{\rightarrow }\mathcal{N    }\left( 0,\sigma ^{2}\left( \gamma ,\rho \right) \right) ,\text{ as }    n\rightarrow \infty ,$$ where $$\begin{aligned}
        \sigma ^{2}\left( \gamma ,\rho \right) &=&\gamma ^{2}+\frac{\gamma ^{2}\rho
            \left( \rho -2\rho \gamma ^{2}+2\gamma \right) }{\left( \gamma \rho
            -1\right) ^{2}}+\frac{2\gamma ^{2}}{\left( \rho +\gamma \rho -1\right)
            \left( \rho +2\gamma \rho -2\right) } \\
        &&+\frac{2\gamma }{2\gamma -1}-\frac{2\gamma \rho \left( \rho \gamma
            ^{2}-\rho \gamma +1\right) }{\left( \gamma \rho -1\right) \left( \rho
            +\gamma \rho -1\right) }.
    \end{aligned}$$

**Simulation study**\[sec4\]
============================

**Performance and comparative study of** $\widetilde{\Pi }    _{n}$ **and** $\widehat{\Pi }_{n}$
------------------------------------------------------------------------------------------------

In this simulation study we examine the performance of our estimator $\widetilde{\Pi }_{n}\left( \widehat{\gamma },k\right) $ given in (\[Br\]) and compare it with that of $\widehat{\Pi }_{n}\left( \widehat{    \gamma }^{H},k\right) $ given in (\[NeMe\]). Thus we follow the steps below. 

**Step 1:** We generate $1000$ pseudorandom samples of size $n=100,200,500$ and $1000$ from Pareto cdf with $\gamma =0.6.$  

**Step 2:** We estimate the tail index parameter by Hill and t-Hill estimators $\widehat{\gamma }^{H}(k_{1}^{\ast })$ and $\widehat{    \gamma }(k_{2}^{\ast }),$ respectively given in (\[Hill\]) and ([t-Hill]{}). We adopt the Reiss and Thomas algorithm (see @ReTo07, [-@ReTo07 page 137]), for choosing the optimal numbers of upper extremes $k_{1}$ and $k_{2}$. By this methodology, we define the optimal sample fraction of upper order statistics $k_{j}^{\ast }$ by$$k_{j}^{\ast }:=\arg \min_{k}\frac{1}{k}\sum_{i=1}^{k}i^{\theta }\left\vert 
\widehat{\gamma }_{j}\left( i\right) -\text{median}\left\{ \widehat{\gamma }_{j}\left( 1\right) ,...,\widehat{\gamma }_{j}\left( k\right) \right\}
\right\vert ,j=1,2$$where $\widehat{\gamma }_{1}=\widehat{\gamma }^{H}$ and $\widehat{\gamma }_{2}=\widehat{\gamma }.$ On the light of our simulation study,** **we obtained reasonable results by choosing $\theta =0.3.$

**Step 3:** We fix the distortion parameter with respect to Condition (\[condition gamabeta\]) by $\rho =1.12,$ then we compute the bias and RMSE of the four estimators $\widehat{\gamma }^{H}(k_{1}^{\ast }),$ $\widehat{\gamma }(k_{2}^{\ast }),$ $\widetilde{\Pi }_{n}\left( \widehat{    \gamma },k_{1}^{\ast }\right) $ and $\widehat{\Pi} _{n}\left( \widehat{    \gamma }^{H},k_{2}^{\ast }\right) $. The results are summarized in Table [A1]{}. We see that when dealing with large samples our estimator performs better.

  ----------------- ----------------- -------------------- ------------------- --------------------------- ------------------- ---------------- -------------------- --------------------------- ------------------- -------------------
                     $k_{1}^{\ast }$                                                 $k_{2}^{\ast }$                                                                                                                 
  ${\small n}$                                bias                RMSE                    bias                    RMSE                                  bias                    RMSE                    bias                RMSE
  ${\small 100}$      ${\small 10}$    ${\small -0.0733}$   ${\small 0.2511}$   $        {\small 0.3618}$   ${\small 0.5199}$   ${\small 17}$    ${\small -0.1641}$   $        {\small 0.2865}$   ${\small 0.4096}$   ${\small 0.7332}$
  ${\small 200}$      ${\small 23}$    ${\small -0.0571}$   ${\small 0.1821}$   $        {\small 0.3562}$   ${\small 0.5147}$   ${\small 34}$    ${\small -0.0993}$   $        {\small 0.2350}$   ${\small 0.3918}$   ${\small 0.7185}$
  ${\small 500}$      ${\small 62}$    ${\small -0.0299}$   ${\small 0.1142}$   $        {\small 0.3404}$   ${\small 0.4820}$   ${\small 86}$    ${\small -0.0301}$   $        {\small 0.0739}$   ${\small 0.3639}$   ${\small 0.6936}$
  ${\small 1000}$    ${\small 129}$    ${\small -0.0147}$   ${\small 0.0798}$   $        {\small 0.1966}$   ${\small 0.2687}$   ${\small 169}$   ${\small -0.0181}$       ${\small 0.0545}$       ${\small 0.2827}$   ${\small 0.5279}$
                                                                                                                                                                                                                     
  ----------------- ----------------- -------------------- ------------------- --------------------------- ------------------- ---------------- -------------------- --------------------------- ------------------- -------------------

  :  $\widehat{\gamma }(k_{1}^{\ast }),$ $\widehat{\gamma }    ^{H}(k_{2}^{\ast })$ $\widetilde{\Pi }_{n}\left( \widehat{\gamma},k_{1}^{\ast }\right) $ and $\widehat{\Pi}
          _{n}\left( \widehat{\gamma},k_{2}^{\ast }\right)$ estimators based on 1000 samples of Pareto-distributed claim amounts with tail index 0.6 and distortion parameter $\rho=1.12$. The exact value of the premium is 2.0487.[]{data-label="A1"}

**Comparative robustness study**
--------------------------------

In this subsection we study the sensitivity to outliers of $\widetilde{\Pi }_{n}\left( \widehat{\gamma },k_{1}^{\ast }\right) $ and compare it with that of $\widehat{\Pi}_{n}\left( \widehat{\gamma }^{H},k_{2}^{\ast}\right).$ We consider an $\epsilon $-contaminated model known by mixture of Pareto distributions$$F_{\gamma _{1},\gamma _{2},\epsilon }\left( x\right) =1-\left( 1-\epsilon
\right) x^{-1/\gamma _{1}}+\epsilon x^{-1/\gamma _{2}},\text{ }
\label{mixture}$$where $\gamma _{1},\gamma _{2}>0$ and $0<\epsilon <0.5$ is the fraction of contamination. Note that for $\epsilon =0,$ $\widehat{\gamma }^{H}$ and $\widehat{\gamma}$ are asymptotically unbiased. Therefore, for $\epsilon >0,$ the effect of contamination becomes immediately apparent. If $\gamma_{1}<\gamma _{2}$ and $\epsilon >0,$ (\[mixture\]) corresponds to a Pareto distribution contaminated by a longer tailed distribution. For the implementation of mixtures models to the outliers study one refers, for instance, to [@BaLe95 page 43]. In this context, we proceed our study as follows.

First, we consider $\gamma _{1}=0.6,$ $\gamma _{2}=2$ to have the contaminated model and let $\rho =1.12.$ Then we consider four contamination scenarios according to $\epsilon=5\%,$ $10\%,$ $15\%,$ $25\%.$

For each value $\epsilon ,$ we generate $1000$ samples of size $n=100,$ $200$ and $1000$ from the model (\[mixture\]). Finally, we compare the $\widetilde{\Pi}_{n}\left(\widehat{\gamma},k_{1}^{\ast }\right)$ and $\widehat{\Pi}_{n}\left(\widehat{\gamma}^{H},k_{2}^{\ast}\right)$ estimators with this true value, by computing for each estimator, the appropriate bias and RMSE and summarize the results in Table \[A2\].

  ----------------- -------------------- ------------------- ------------------- -------------------------------- ---------------------------
                                                                                                                  
  ${\small n}$       $\%$ contamination         bias                RMSE                       bias                          RMSE
  ${\small 100}$        ${\small 5}$      ${\small 0.4043}$   ${\small 0.6664}$     $        {\small -0.4286}$         ${\small 1.4727}$
                       ${\small 10}$      ${\small 0.4389}$   ${\small 0.6862}$         ${\small -0.7291}$             ${\small 1.9123}$
                       ${\small 15}$      ${\small 0.4598}$   ${\small 0.7464}$         ${\small -1.2786}$             ${\small 2.1247}$
                       ${\small 25}$      ${\small 1.0578}$   ${\small 1.1305}$         ${\small -1.2103}$             ${\small 2.1828}$
  $200$                 ${\small 5}$      ${\small 0.3831}$   ${\small 0.5532}$   ${\small             -0.4713}$       ${\small 1.5118}$
                       ${\small 10}$      ${\small 0.3964}$   ${\small 0.5675}$         ${\small -1.2496}$             ${\small 2.1907}$
                       ${\small 15}$      ${\small 0.4508}$   ${\small 0.6870}$             $-1.4355$              $        {\small 2.3107}$
                       ${\small 25}$      ${\small 0.9470}$   ${\small 1.0197}$         ${\small -1.7366}$             ${\small 2.3274}$
  ${\small 1000}$       ${\small 5}$      ${\small 0.2124}$   ${\small 0.3211}$     $        {\small -0.3794}$         ${\small 2.1222}$
                       ${\small 10}$      ${\small 0.2329}$   ${\small 0.3349}$         ${\small -1.0662}$             ${\small 2.3978}$
                       ${\small 15}$      ${\small 0.2931}$   ${\small 0.3749}$         ${\small -1.2501}$             ${\small 2.0355}$
                       ${\small 25}$      ${\small 0.8124}$   ${\small 0.9291}$         ${\small -1.5238}$             ${\small 2.3596}$
                                                                                                                  
  ----------------- -------------------- ------------------- ------------------- -------------------------------- ---------------------------

  : $\widetilde{\Pi }_{n}\left( \widehat{\gamma},k_{1}^{\ast }\right)
          $ and $\widehat{\Pi}_{n}\left( \widehat{\gamma }^{H},k_{2}^{\ast}\right)$ are based on 1000 samples of mixture of Pareto distributions with tail index $0.6,$ $\epsilon=5\%, 10\%, 15\%, 25\%$ and distortion parameter $\rho=1.12$. The exact value of the premium is 2.0487.[]{data-label="A2"}

As expected, the estimator $\widehat{\Pi }_{n}\left( \widehat{    \gamma }^{H},k_{2}^{\ast }\right) $ as well as $\widetilde{\Pi }_{n}\left( 
\widehat{\gamma },k_{1}^{\ast }\right) $ turn out to be more sensitive to this type of contaminations. For example, in $0\%$ contamination for $n=200$, the couple (bias, RMSE) for $\widehat{\Pi }_{n}\left( \widehat{\gamma }^{H},k_{2}^{\ast }\right) $ take the values $\left( 0.3918,0.7185\right) $, while for $15\%$ contamination the bias and the RMSE are given by the couple $\left( -1.4355,2.3107\right) $. We may conclude that the bias and RMSE of $\widehat{\Pi }_{n}\left( \widehat{\gamma }^{H},k_{2}^{\ast }\right) $ estimator are more sensitive (or note robust) to outliers. However, for $0\%$ contamination the (bias, RMSE) of $\widetilde{\Pi }_{n}\left( \widehat{    \gamma },k_{1}^{\ast }\right) $ is $\left( 0.3562,0.5147\right) ,$ while for $15\%$ contamination is $\left( 0.4508,0.6870\right) .$ Both the bias and the RMSE of $\widetilde{\Pi }_{n}\left( \widehat{\gamma },k_{1}^{\ast
}\right) $ estimation are note sensitive to outliers. Then we may conclude that is the better estimator.

**Concluding notes\[sec5\]**
============================

We showed that the new estimator of premium based on t-Hill estimator is more robust and performs better than the one based on Hill estimator proposed by [@NeMe09]. Our estimator $\widetilde{\Pi }_{n}\left( \widehat{\gamma },k\right) $ is based on Weissman’s estimation of high quantiles, so we would lead to improve our result to use one of several bias-reduced estimators have been proposed (see for example [@MaBe03]).

Proofs\[sec6\]
==============

To establish the asymptotic normality of $\widetilde{\Pi }_{\psi
    ,n}$ we need the asymptotic approximation of $\widehat{\gamma }$ with the same sequence of Brownian bridges as $\widetilde{\Pi }_{\psi ,n},$ for this reason we give the following results.

\[Cor\]Assume that the second order condition (\[second-cond\]) holds with $\gamma >1/2$ and let $k=k_{n}$ be an integer sequence satisfying ( [K]{}) and $\sqrt{k}A\left( n/k\right) \rightarrow 0.$ Then, there exists a sequence of Brownian bridges $\left\{ \mathbb{B}_{n}\left( s\right) ,\text{ }    0\leq s\leq 1\right\} $ such that $$\sqrt{k}\left( \widehat{\gamma }-\gamma \right) =\gamma \left( \gamma
    +1\right) ^{2}\int_{0}^{1}s^{\gamma -1}\mathbb{B}_{n}\left( s\right)
    ds+o_{p}\left( 1\right) ,$$ leading to $$\sqrt{k}\left( \widehat{\gamma }-\gamma \right) \overset{d}{\rightarrow }    \mathcal{N}\left( 0,\frac{\gamma ^{2}\left( 1+\gamma \right) ^{2}}{\left(
        1+2\gamma \right) }\right) ,\text{ as }n\rightarrow \infty ,$$

Our proofs are conducted in the probability space described in Section [sec3]{}. Then we are entitled to write $$S_{k}:=\frac{1}{k}\sum_{j=1}^{k}\frac{X_{n-k:n}}{X_{n-j+1:n}}$$ then $\widehat{\gamma }=S_{k}^{-1}-1$. The asymptotic normality of $\widehat{        \gamma }$ is established in [@BeScSt14] by given their Wiener process representation. Here we suppose that $\sqrt{k}A\left( n/k\right) \rightarrow
    \lambda =0.$ So $$\sqrt{k}\left( S_{k}-\frac{1}{\gamma +1}\right) =\frac{\gamma }{\gamma +1}    W_{n}\left( 1\right) -\gamma \int_{0}^{1}t^{\gamma -1}W_{n}\left( t\right)
    dt+o_{p}\left( 1\right) .$$ Note that $$\mathbb{B}_{n}\left( s\right) \overset{d}{=}W_{n}\left( s\right)
    -sW_{n}\left( 1\right) ,$$ it follows that $$\sqrt{k}\left( S_{k}-\frac{1}{\gamma +1}\right) =-\gamma
    \int_{0}^{1}s^{\gamma -1}\mathbb{B}_{n}\left( s\right) dt+o_{p}\left(
    1\right) .$$ Using the map $g\left( x\right) =1/x-1,$ since $g\left( 1/\left( \gamma
    +1\right) \right) =\gamma $ and applying the delta method yields $$\sqrt{k}\left( \widehat{\gamma }-\gamma \right) =\left( \gamma +1\right)
    ^{2}\gamma \int_{0}^{1}s^{\gamma -1}\mathbb{B}_{n}\left( s\right)
    ds+o_{p}\left( 1\right) .$$ It is clear that $\sqrt{k}\left( \widehat{\gamma }-\gamma \right) $ is a Gaussian rv with mean $0$ and variance $\frac{\gamma ^{2}\left( 1+\gamma
        \right) ^{2}}{\left( 1+2\gamma \right) }.$ This completes the proof of Proposition \[Cor\].

**Proof of Theorem \[Theorem2\]**
---------------------------------

Making use of Proposition \[Cor\] and from [@NeMeMe07] we showe that under the assumptions of Theorem \[Theorem2\], there exists a sequence of Brownian bridges $\left\{ \mathbb{B}_{n}\left( s\right) ,0\leq
s\leq 1\right\} $ such that, for all large $n$$$\frac{\left( \Pi _{\psi ,n}^{\left( 1\right) }-\Pi _{\psi }^{\left( 1\right)
    }\right) }{(k/n)^{1/\rho }F^{\leftarrow }\left( 1-k/n\right) }=-\gamma
\left( n/k\right) ^{1/2}\mathbb{B}_{n}\left( 1-k/n\right) +o_{p}\left(
1\right) .$$Let $\mathbb{U}$ be the left-continuous inverse of $1/(1-F).$ Note that $\mathbb{U}(t)$ is defined for $t>1.$ Let $Y_{1},Y_{2}....$ be independent and identically distributed rv’s with cdf $1-1/y,y>1,$ and let $Y_{1,n}\leq
Y_{2,n}\leq ...\leq Y_{n,n}$ be the associated order statistics. Then, for $\rho \geq 1,$ we may rewrite the statistic $\Pi _{\psi ,n}^{\left( 2\right)
} $ as$$\Pi _{\psi ,n}^{\left( 2\right) }=(k/n)^{1/\rho }\frac{\widehat{\gamma }\rho 
}{1-\widehat{\gamma }\rho }X_{n-k:n}.$$ Then$$\Pi _{\psi ,n}^{\left( 2\right) }=(k/n)^{1/\rho }\frac{\widehat{\gamma }\rho 
}{1-\widehat{\gamma }\rho }\mathbb{U}\left( Y_{n-k:n}\right) ,$$where $\widehat{\gamma }$ is the t-Hill estimator of $\gamma .$ So we have $$\frac{\sqrt{k}\left( \Pi _{\psi ,n}^{\left( 2\right) }-\Pi _{\psi }^{\left(
        2\right) }\right) }{(k/n)^{1/\rho }\mathbb{U}\left( n/k\right) }:=\sum_{j=1}^{4}\Delta _{jn},$$where$$\begin{aligned}
    \Delta _{1n}:= &&\sqrt{k}\frac{\widehat{\gamma }\rho }{1-\widehat{\gamma }        \rho }\left( \frac{\mathbb{U}\left( Y_{n-k:n}\right) }{\mathbb{U}\left(
        n/k\right) }-\left( \frac{Y_{n-k:n}}{n/k}\right) ^{\gamma }\right) , \\
    \Delta _{2n}:= &&\sqrt{k}\frac{\widehat{\gamma }\rho }{1-\widehat{\gamma }        \rho }\left( \left( \frac{Y_{n-k:n}}{n/k}\right) ^{\gamma }-1\right) , \\
    \Delta _{3n}:= &&\sqrt{k}\left( \frac{\widehat{\gamma }\rho }{1-\widehat{            \gamma }\rho }-\frac{\gamma \rho }{1-\gamma \rho }\right)\end{aligned}$$and$$\Delta _{4n}:=\sqrt{k}(k/n)^{-1/\rho }\left( \frac{\frac{(k/n)^{1/\rho }\rho 
    }{\left( 1/\gamma -\rho \right) }\mathbb{U}\left( n/k\right) -\Pi _{\psi
}^{\left( 2\right) }}{\mathbb{U}\left( n/k\right) }\right) .$$As showed in [@NeMeMe07], we have : $\Delta _{1n}\rightarrow 0$ and $\Delta _{4n}\rightarrow 0$ as $n\rightarrow \infty .$

Next, we show that $\Delta _{2n}+\Delta _{3n}$ is asymptotically normal. Assume, without loss of generality, that the rv’s $(Y_{n})_{n\geq 1}$are defined on a probability space $(\Omega ,\mathcal{A},\mathbb{P})$ which carries the sequence $(U_{n})_{n\geq 1}$ in such a way that $Y_{n}=\left(
1-U_{n}\right) ^{-1}$ for $n=1,2,...$ and $Y_{i,n}=\left( 1-U_{i,n}\right)
^{-1},$ $i=1,...,n,$.$\ $Then, this allows us to write $Y_{n-i+1,n}=\left(
1-V_{n}\left( 1-s\right) \right) ^{-1},$ for $\dfrac{i-1}{n}<s\leq \dfrac{i}{    n},$ $i=1,...,n$. From [@NeMeMe07] we have For $\Delta _{2n}$  $$\Delta _{2n}=-\left( n/k\right) ^{1/2}\frac{\rho \gamma ^{2}}{1-\gamma \rho }\left( 1+o_{p}\left( 1\right) \right) \mathbb{B}_{n}\left( 1-k/n\right) .$$For $\Delta _{3n}$ and by using the map $h\left( \theta \right) =\rho
/\left( \frac{1}{\theta }-\rho \right) \ $and applying the delta method yields: $$\Delta _{3n}=\frac{\rho }{\left( \rho \gamma -1\right) ^{2}}\sqrt{k}\left(
\gamma -\widehat{\gamma }\right) .$$From Corollary \[Cor\] we get$$\Delta _{3n}=\frac{\gamma \rho \left( \gamma +1\right) ^{2}}{\left( \rho
    \gamma -1\right) ^{2}}\int_{0}^{1}s^{\gamma -1}\mathbb{B}_{n}\left( s\right)
ds+o_{p}\left( 1\right) .$$Finally we have$$\frac{\sqrt{k}\left( \Pi _{\psi ,n}^{\left( 2\right) }-\Pi _{\psi }^{\left(
        2\right) }\right) }{(k/n)^{1/\rho }F^{\leftarrow }\left( 1-k/n\right) }=\frac{\gamma \rho \left( \gamma +1\right) ^{2}}{\left( \rho \gamma -1\right)
    ^{2}}\int_{0}^{1}s^{\gamma -1}\mathbb{B}_{n}\left( s\right) ds-\left(
n/k\right) ^{1/2}\frac{\rho \gamma ^{2}}{1-\gamma \rho }\mathbb{B}_{n}\left(
1-k/n\right) +o_{p}\left( 1\right) .$$From [@NeMeMe07] we have$$\frac{\sqrt{k}\left( \Pi _{\psi ,n}^{\left( 3\right) }-\Pi _{\psi }^{\left(
        3\right) }\right) }{(k/n)^{1/\rho }F^{\leftarrow }\left( 1-k/n\right) }=\frac{\int_{k/n}^{1}s^{1/\rho -1}\mathbb{B}_{n}\left( 1-s\right)
    F^{\leftarrow \prime }\left( 1-s\right) ds}{\rho F^{\leftarrow }\left(
    1-k/n\right) \left( k/n\right) ^{1/\rho -1/2}}+o_{p}\left( 1\right) .$$Then $$\frac{\sqrt{n}\left( \widetilde{\Pi }_{\psi ,n}-\Pi _{\psi }\right) }{    (k/n)^{1/\rho -1/2}F^{\leftarrow }(1-k/n)}=\Lambda \left( \gamma ,\rho
\right) +o_{p}\left( 1\right)$$where$$\Lambda \left( \gamma ,\rho \right) :=W_{n1}+W_{n2}+W_{n3}+o_{p}\left(
1\right)$$and$$\begin{aligned}
    W_{n1} &:=&-\left( n/k\right) ^{1/2}\gamma \mathbb{B}_{n}\left( 1-k/n\right)
    , \\
    W_{n2} &:=&\frac{\gamma \rho \left( \gamma +1\right) ^{2}}{\left( \rho
        \gamma -1\right) ^{2}}\int_{0}^{1}s^{\gamma -1}\mathbb{B}_{n}\left( s\right)
    ds-\left( n/k\right) ^{1/2}\frac{\rho \gamma ^{2}}{1-\gamma \rho }\mathbb{B}    _{n}\left( 1-k/n\right) , \\
    W_{n3} &:=&\frac{\int_{k/n}^{1}s^{1/\rho -1}\mathbb{B}_{n}\left( 1-s\right)
        F^{\leftarrow \prime }\left( 1-s\right) ds}{\rho F^{\leftarrow }\left(
        1-k/n\right) \left( k/n\right) ^{1/\rho -1/2}}.\end{aligned}$$It is clear that $\Lambda \left( \gamma ,\rho \right) $ is a Gaussian rv with mean 0 and variance$$\begin{aligned}
    E\left( \Lambda \left( \gamma ,\rho \right) \right) ^{2} &=&E\left(
    W_{n1}^{2}\right) +E\left( W_{n2}^{2}\right) +E\left( W_{n3}^{2}\right)
    +2E\left( W_{n1}W_{n2}\right) \\
    &&+2E\left( W_{n1}W_{n3}\right) +2E\left( W_{n2}W_{n3}\right) .\end{aligned}$$An elementary calculation gives, we get $$\begin{aligned}
    E\left( W_{n1}^{2}\right) &=&\gamma ^{2}+o\left( 1\right) , \\
    E\left( W_{n2}^{2}\right) &=&\frac{\gamma ^{2}\rho ^{2}}{\left( 1-\gamma
        \rho \right) ^{2}}+\frac{2\gamma }{2\gamma -1}+o\left( 1\right) , \\
    E\left( W_{n3}^{2}\right) &=&\frac{2\gamma ^{2}}{\left( \rho +\gamma \rho
        -1\right) \left( \rho +2\gamma \rho -2\right) }+o\left( 1\right) , \\
    E\left( W_{n1}W_{n2}\right) &=&\frac{\rho \gamma ^{3}}{1-\gamma \rho }    +o\left( 1\right) , \\
    E\left( W_{n1}W_{n3}\right) &=&\frac{\gamma \rho }{\rho +\gamma \rho -1}    +o\left( 1\right)\end{aligned}$$and$$E\left( W_{n2}W_{n3}\right) =-\frac{\gamma ^{3}\rho ^{2}}{\left( \gamma \rho
    -1\right) \left( \rho +\gamma \rho -1\right) }+o\left( 1\right) .$$The proof of Theorem \[Theorem2\] is completed by combining all the preceding results.$\square $

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[^1]: [`brah.brahim@gmail.com`]{}
