---
abstract: 'In this paper, a lower bound is determined in the minimax sense for change point estimators of the first derivative of a regression function in the fractional white noise model. Similar minimax results presented previously in the area focus on change points in the derivatives of a regression function in the white noise model or consider estimation of the regression function in the presence of correlated errors.'
address: 'School of Mathematics & Statistics F07, The University of Sydney, NSW, 2006, Australia.'
author:
- Justin Rory Wishart
title: 'Minimax lower bound for kink location estimators in a nonparametric regression model with long-range dependence'
---

nonparametric regression ,long-range dependence ,kink ,minimax 62G08 ,62G05 ,62G20

Introduction {#Intro}
============

Nonparametric estimation of a kink in a regression function has been considered for Gaussian white noise models by @Cheng-Raimondo-2008 [@Goldenshluger-et-al-2008a; @Goldenshluger-et-al-2008b]. Recently, this was extended to the fractional Gaussian noise model by [@Wishart-2009]. The fractional Gaussian noise model assumes the regression structure, $$dY(x) = \mu(x)\,dx + \varepsilon^\alpha dB_H(x), \quad x \in \mathbb{R},
    \label{eq:fixednonparareg}$$ where $B_H$ is a fractional Brownian motion (fBm) and $\mu {\,{:}\,\mathbb{R}\!\longrightarrow\!\mathbb{R}}$ is the regression function. The level of error is controlled by $\varepsilon \asymp n^{-1/2}$ where the relation $a_n \asymp b_n$ means the ratio $a_n/b_n$ is bounded above and below by constants. The level of dependence in the error is controlled by the Hurst parameter $H \in (1/2,1)$ and $\alpha {\mathrel{\mathop:}=}2 - 2H$, where the i.i.d. model corresponds to $\alpha = 1$. The fractional Gaussian noise model was used by [@Johnstone-Silverman-1997; @Wishart-2009] among others to model regression problems with long-range dependent errors.

This paper is interested in the performance of estimators of a change-point in the first derivative of $\mu$ observed in model . This type of change point is called a kink and the location denoted by $\theta$. Let $\widehat \theta_n$ denote an estimator of $\theta$ given $n$ observations. A lower bound is established for the minimax rate of kink location estimation using the quadratic loss in the sense that, $$\liminf_{n \to \infty } \inf_{\widehat \theta_n} \sup_{\mu \in \mathscr F_s(\theta)} \rho_n^{-2}\mathbb{E} \left| \widehat \theta_n - \theta\right|^2  \ge C \qquad \text{for some constant $C>0$}.\label{eq:rate}$$ The main quantity of interest in this lower bound is the rate, $\rho_n$. In , $\inf_{\widehat \theta_n}$ denotes the infimum over all possible estimators of $\theta$. The class of functions under consideration for $\mu$ is denoted $\mathscr F_s(\theta)$ and defined below.

\[def:functionalclass\] Let $s \geq 2$ be an integer and $a \in \mathbb{R}\setminus \left\{ 0\right\}$. Then, we say that $\mu\in \mathscr F_s(\theta)$ if,

1.  The function $\mu $ has a kink at $\theta \in (0,1)$. That is, $$\lim_{x \downarrow \theta}\mu^{(1)}(x) - \lim_{x \uparrow \theta}\mu^{(1)}(x) = a \neq 0.$$

2.  The function $\mu \in \mathscr L_2\left(\mathbb{R}\right) \cap \mathscr L_1(\mathbb{R}) $, and satisfies the following condition, $$\int_\mathbb{R} |\widetilde \mu(\omega)||\omega|^s\,d\omega < \infty, \label{sobolev}$$ where $\widetilde \mu(\omega) {\mathrel{\mathop:}=}\int_\mathbb{R} e^{-2 \pi i \omega x}\mu(x)\, dx$ is the Fourier transform of $\mu$.

The minimax rate for the kink estimators has been discussed in the i.i.d. scenario by [@Cheng-Raimondo-2008; @Goldenshluger-et-al-2008a] and was shown to be $n^{-s/(2s+1)}$. An extension of the kink estimators to the long-range dependent scenario was considered in [@Wishart-2009] that built on the work of [@Cheng-Raimondo-2008]. An estimator of kink locations was constructed by [@Wishart-2009] and achieved the rate in the probabilistic sense, $$\left| \widehat \theta_n - \theta\right| = \mathcal O_p (n^{-\alpha s /(2s+\alpha)}),\label{eq:kinkrate}$$ which includes the result of [@Cheng-Raimondo-2008] as a special case with the choice $\alpha = 1$. Both [@Cheng-Raimondo-2008] and [@Wishart-2009] considered a comparable model in the indirect framework and used the results of [-@Goldenshluger-et-al-2006] to infer the minimax optimality of . However, the results of [@Cheng-Raimondo-2008] and [@Wishart-2009] require a slightly more restrictive functional class than $\mathscr F_s(\theta)$. The rate obtained by [@Cheng-Raimondo-2008] of $n^{-s/(2s+1)}$ was confirmed as the minimax rate by the work of [@Goldenshluger-et-al-2008a] who used the i.i.d. framework and a functional class similar to $\mathscr F_s(\theta)$.

The fBm concept is an extension of Brownian motion that can exhibit dependence among its increments which is typically controlled by the Hurst parameter, $H$ (see [-@Beran-1994; -@Doukhan-et-al-2003] for more detailed treatment on long-range dependence and fBm). The fBm process is defined below.

\[fBm\] The fractional Brownian motion $\left\{B_H(t) \right\}_{t \in \mathbb{R}}$ is a Gaussian process with mean zero and covariance structure, $$\mathbb{E} B_H(t)B_H(s) =\frac{1}{2}\left\{ |t|^{2H} + |s|^{2H} - |t-s|^{2H} \right\}.$$

We assume throughout the paper that $H\in (1/2,1)$, whereby the increments of $B_H$ are positively correlated and are long-range dependent.

In this paper a lower bound for the minimax convergence rate of kink estimation using the quadratic loss function will be shown explicitly on model . This is a stronger result in terms of a lower bound than the simple probabilistic result in given by [@Wishart-2009] and is applicable to a broader class of functions.

Lower bound {#lowerbound}
===========

The aim of the paper is to establish the following result.

\[thm:lowerboundK\] Suppose $\mu \in \mathscr F_s \left( \theta \right)$ is observed from the model and $0 < \alpha < 1$. Then, there exists a positive constant $C < \infty$ that does not depend on $n$ such that the lower rate of convergence for an estimator for the kink location $\theta$ with the square loss is of the form, $$\liminf_{n \to \infty} \inf_{\widehat{\theta}_n} \sup_{\mu \in \mathscr F_s(\theta)}  n^{ 2\alpha s/(2s + \alpha)} \mathbb{E}\left| \widehat{\theta}_n - \theta\right|^2 \ge C.$$

From one can see that the minimax rate for kink estimation in the i.i.d. case is recovered with the choice $\alpha = 1$ [see @Goldenshluger-et-al-2008a]. Also unsurprisingly, the level of dependence is detrimental to the rate of convergence. For instance as the increments become more correlated, and $\alpha \to 0$, the rate of convergence diminishes.

As will become evident in the proof of the Kullback-Leibler divergence is required between two measures involving modified fractional Brownian motions. To cater for this, some auxiliary definitions to precede the proof of are given in the next section.

Preliminaries
=============

In this paper, the functions under consideration are defined in the Fourier domain (see ). Among others, there are two representations for fBm that satisfy that are used in this paper. The first being the moving average representation of [@Mandelbrot-van-Ness-1968] in the time domain and second is the spectral representation given by [@Samorodnitsky-Taqqu-1994] in the Fourier domain. These both need to be considered since they are both used in the proof of the main result. Both representations have normalisation constants $C_{T,H}$ and $C_{F,H}$ for the time and spectral representations respectively to ensure the fBm satisfies . Start with the time domain representation.

\[fBmMVN\] The fractional Brownian motion $\left\{B_H(t) \right\}_{t \in \mathbb{R}}$ can be represented by, $$B_H(t) =\frac{1}{C_{T,H}}\int_\mathbb{R} \left((t-s)_+^{H- 1/2} - (-s)_+^{H-1/2}\right)dB(s),$$ where $C_{T,H} = \Gamma(H + 1/2)/\sqrt{2H \sin (\pi H) \Gamma(2H)}$ and $x_+ = x\mathbbm{1}_{\left\{ x > 0\right\}}(x).$

For the spectral representation a complex Gaussian measure $\breve B{\mathrel{\mathop:}=}B^{[1]} + i B^{[2]}$ is used where $B^{[1]}$ and $B^{[2]}$ are independent Gaussian measures such that for $i = 1,2;$ $B^{[i]}(A) = B^{[i]}(-A)$ for any Borel set $A$ of finite Lebesgue measure and $\mathbb{E} (B^{[i]}(A))^2 = \text{mesh}(A)/2$.

\[fBmST\] The fractional Brownian motion $\left\{B_H(t) \right\}_{t \in \mathbb{R}}$ can be represented by, $$B_H(t) =\frac{1}{C_{F,H}}\int_\mathbb{R} \frac{e^{i s t} - 1}{is}|s|^{-(H-1/2)}d\breve{B}(s),$$ where $C_{F,H} = \sqrt{\pi/(2H \sin (\pi H) \Gamma(2H))}$.

As will become evident in , to obtain the lower bound result for the minimax rate, it is crucial to know which functional class to consider for $\mu {\,{:}\,\mathbb{R}\!\longrightarrow\!\mathbb{R}}$ such that the process $ \int_\mathbb{R} \mu(x)\, dB_H(x)$ is a well defined random variable with finite variance. Two such classes of functions will be considered, $\mathcal H$ and $\widetilde{\mathcal{H}}$, which correspond to the time and spectral versions of fBm respectively. Begin with the moving average representation.

\[otherstochasticintegralfBmclass\] Let $H \in \left( 1/2, 1\right)$ be constant. Then the class $\mathcal H$ is defined by, $$\mathcal H = \left\{ \mu {\,{:}\,\mathbb{R}\!\longrightarrow\!\mathbb{R}} \Bigg|  \int_\mathbb{R}\int_\mathbb{R} \mu(x) \mu(y) | x-y|^{-\alpha}\, dy \, dx  < \infty \right\}.$$

Simlar to there is an inner product on the space $\mathcal H$ that satisfies the following. For all $f,g \in \mathcal H$, $$\mathbb{E} \left\{ \int_\mathbb{R} f(x) \, dB_H(x)\int_\mathbb{R} g(y) \, dB_H(y) \right\} = C_\alpha\int_\mathbb{R}\int_\mathbb{R} f(x) g(y)| x-y|^{-\alpha}\, dy \, dx {=\mathrel{\mathop:}}{ \langle f , g \rangle }_{\mathcal{H}},$$ where the constant $C_\alpha  = \tfrac{1}{2} (1-\alpha)(2-\alpha)$. The other functional class for the spectral representation is denoted by $\mathcal H$ and defined below.

\[stochasticintegralfBmclass\] Let $H \in \left( 1/2, 1\right)$ be constant. Then the class $\widetilde{\mathcal H}$ is defined by, $$\widetilde{\mathcal H} = \left\{ \mu {\,{:}\,\mathbb{R}\!\longrightarrow\!\mathbb{R}} \Bigg| \int_\mathbb{R} |\widetilde\mu(\omega)|^2 \left| \omega \right|^{-(1-\alpha)}\, d\omega < \infty \right\}.$$

On the space $\widetilde{\mathcal H}$, the stochastic integrals with respect to fBm are well defined and satisfy the following. For all $f,g \in \widetilde{\mathcal{H}}$, $$\mathbb{E} \left\{ \int_\mathbb{R} f(x) \, dB_H(x)\int_\mathbb{R} g(y) \, dB_H(y) \right\} = \frac{1}{C_{F,H}^2}\int_\mathbb{R} \widetilde{f}(\omega) \overline{\widetilde{g}(\omega) }| \omega|^{-(1-\alpha)}\, d\omega {=\mathrel{\mathop:}}{ \langle f , g \rangle }_{\widetilde{\mathcal{H}}},
    \label{eq:spectralExpectation}$$ where $\overline{\widetilde{g}}$ denotes the complex conjugate of $\widetilde{g}$.

These two classes of integrands were considered extensively in [@Pipiras-Taqqu-2000]. In this context of this paper the inner products can be used interchangeably because if $\mu \in \mathscr F_s(\theta)$ then $\mu \in \mathscr L_1(\mathbb{R})\cap \mathscr L_2(\mathbb{R})$ and by @Pipiras-Taqqu-2000 [Proposition 3.1] then $\mu \in \mathcal H$. Also, using @Pipiras-Taqqu-2000 [Proposition 3.2] with the isometry @Biagini-et-al-2008 [Lemma 3.1.2] and Parseval’s Theorem then $\mu \in \widetilde{\mathcal H}$ and consequently $\mu \in \mathcal H \cap \widetilde{\mathcal H}$.

Proof of Theorem 1 {#proof}
==================

The lower bound for the minimax rate is constructed by adapting the results of [@Goldenshluger-et-al-2006] to our framework. This requires obtaining the Kullback-Leibler divergence of two suitably chosen functions $\mu_0$ and $\mu_1$ from the functional class $\mathscr F_s(\theta)$. The main hurdle in determining the Kullback-Leibler divergence is the long-range dependent structure in the fBm increments. A summary of Girsanov type theorems for fBm have been established by @Biagini-et-al-2008 [Theorem 3.2.4]. Here however, the Radon-Nikodym derivative is the main focus. Once that is determined, the Kullback-Leibler divergence is linked to the lower rate of convergence using @Tsybakov-2009 [Theorem 2.2 (iii)]. Lastly, before proceeding to the proof, the quantity $C > 0$ denotes a generic constant that could possibly change from line to line.

Without loss of generality, consider a function $\mu_0 \in \mathscr F_s(\theta_0)$ where $\theta_0 \in (0,1/2]$ and define $\theta_1 = \theta_0 + \delta$ where $\delta \in (0, 1/2)$ (a symmetric argument can be setup to accommodate the case when $\theta_0 \in [1/2,1)$). Define the functions $v {\,{:}\,\mathbb{R}\!\longrightarrow\!\mathbb{R}}$ and $v_N {\,{:}\,\mathbb{R}\!\longrightarrow\!\mathbb{R}}$ such that $$\begin{aligned}
 v(x) &{\mathrel{\mathop:}=}a((\theta_1\wedge x) - \theta_0) \mathbbm{1}_{ (\theta_0,1] }(x), \qquad v_N(x) {\mathrel{\mathop:}=}\int_{-N}^N \widetilde{v}(\omega) e^{2 \pi i x \omega}\, d\omega,\end{aligned}$$ where $a$ is the size of the jump given in and $\widetilde v$ is the Fourier transform of $v$. Note that, $v_N(x)$ is close to $v(x)$ in the sense that it is the inverse Fourier transform of $\widetilde{v}(\omega)\mathbbm{1}_{|\omega| \le N}$ and $\widetilde v_N(\omega) = \widetilde v(\omega)\mathbbm{1}_{|\omega| \le N}$. With these definitions, the derivative takes the form, $ v^{(1)}(x) = a \mathbbm{1}_{[\theta_0, \theta_1] }(x)$ and the function $(\mu_0 - v)$ has a single kink at $\theta_1$. Then define $\mu_1 {\mathrel{\mathop:}=}\mu_0 - (v - v_N)$. The function $v_N$ is infinitely differentiable across the whole real line and smooth for finite $N$, which implies that $\mu_1 = \mu_0 - (v - v_N)$ has a single kink at $\theta_1$. It can be shown that, $$\left|\widetilde{v}(\omega)\right| \le a\delta/(2 \pi  \left|\omega\right|)^{-1}.\label{eq:vomegabound}$$ Further, if $N$ is chosen to be $N = \left( s\pi C/(a \delta) \right)^{1/s}$ then $\int_\mathbb{R} | \widetilde{v_N}(\omega)| | \omega|^s \,d\omega < \infty$ and consequently $\mu_1 \in \mathscr F_s(\theta_1)$.

To be able to determine the Radon-Nikodym derivative, define $\Delta {\mathrel{\mathop:}=}\mu_0 - \mu_1 = v - v_N$ and note that $\Delta {\,{:}\,\mathbb{R}\!\longrightarrow\!\mathbb{R}}$. The Radon-Nikodym derivative also needs a paired function $\underline{\Delta}{\,{:}\,\mathbb{R}\!\longrightarrow\!\mathbb{R}}$. Define such a function with a singular integral operator with $$\Delta(x) {\mathrel{\mathop:}=}\varepsilon^{-\alpha}C_\alpha \int_\mathbb{R} |x-y|^{-\alpha} \underline{\Delta}(y) \, dy = \frac{\Gamma(3-\alpha)}{2} \left( \mathcal D_-^{-(1-\alpha)}\underline\Delta(x) + \mathcal D_+^{-(1-\alpha)}\underline\Delta(x)\right),\label{eq:DeltaDecomp}$$ where, for $\nu \in (0,1) $, $\mathcal D_-^{-\nu}$ and $\mathcal D_+^{-\nu}$ are the left and right fractional Liouville integral operators defined by, $$\mathcal D_-^{-\nu} f(x)  {\mathrel{\mathop:}=}\frac{1}{\Gamma(\nu)}\int_{-\infty}^x (x-y)^{\nu -1}f(y)\, dy \qquad \mathcal D_+^{-\nu} f(x)  {\mathrel{\mathop:}=}\frac{1}{\Gamma(\nu)}\int_x^\infty (y-x)^{\nu -1}f(y)\, dy$$ This function $\underline \Delta$ has representation in the Fourier domain with $$\widetilde{\underline \Delta} (\omega)  \asymp \varepsilon^{-\alpha}|\omega|^{1-\alpha} \widetilde{\Delta}(\omega).$$ Furthermore, $\underline{\Delta} \in \mathcal H \cap \widetilde{\mathcal H}$. Indeed, by definition, $\Delta = \mu_0 - \mu_1$ with $\mu_0 \in \mathscr F_s(\theta_0)$ and $\mu_1 \in \mathscr F_s(\theta_1)$ which implies that $\Delta \in \mathscr L_1(\mathbb{R}) \cap \mathscr L_2(\mathbb{R})$ and $\widetilde \Delta (\omega) = o(\omega^{-s})$ due to . First, it will be shown that, $\underline \Delta \in \widetilde{\mathcal H}$. $$\begin{aligned}
    { \langle \underline \Delta , \underline \Delta \rangle }_{\widetilde{\mathcal H}} &\asymp  \int_\mathbb{R} | \widetilde \Delta (\omega)|^2 |\omega|^{1-\alpha}\, d\omega\nonumber\\
    &\le  C\left\{ \|\Delta\|_{1}^2\int_{|\omega|\le 1} |\omega|^{1-\alpha}\, d\omega +   \int_{|\omega|\ge 1} |\widetilde \Delta (\omega)|^2 |\omega|^{1-\alpha}\, d\omega\right\},\label{eq:uDeltainnerp}\end{aligned}$$ where $C > 0$ is some constant and $ \|\Delta\|_{1} = \int_\mathbb{R} | \Delta (x)| \, dx $. In , the first integral is finite since $\alpha \in (0,1)$ and the last integral is finite since $\widetilde \Delta (\omega) = o(\omega^{-s}) $ for $s \ge 2$, proving $\underline \Delta \in \widetilde{\mathcal H}$. Then apply the isometry in @Biagini-et-al-2008 [Lemma 3.1.2] with Plancherel and , it follows that $\underline \Delta \in \mathcal H$.

Now let $P_0$ and $P_1$ be the probability measures associated with model with $\mu = \mu_0$ and $ \mu = \mu_1$ respectively. Define, $\mathring{B}_H(x) {\mathrel{\mathop:}=}\varepsilon^{-\alpha} \int_0^x \Delta(x)\, dx + B_H(x)$. Then under the $P_0$ measure, $$\begin{aligned}
  dY_0(x) &= \mu_0(x)\,dx + \varepsilon^\alpha \, dB_H(x)
= \mu_1(x)\,dx + \varepsilon^\alpha \, d\mathring{B}_H(x).\end{aligned}$$ The Radon-Nikodym derivative between these measures takes the form, $$\begin{aligned}
 \frac{d P_1}{d P_0} &{\mathrel{\mathop:}=}\exp \left\{ - \int_\mathbb{R} \underline{\Delta}(x) \, dB_H(x) - \frac{1}{2}\mathbb{E}_{P_0} \left(  \int_\mathbb{R} \underline{\Delta}(x) \, dB_H(x) \right)^2\right\} . \label{eq:radnikderiv}\end{aligned}$$ Indeed to show is valid, for $\underline{\Delta} \in \mathcal H$ and $\psi \in \mathcal H$, use and apply @Biagini-et-al-2008 [Lemma 3.2.1] with the change of measure formula in to yield, $$\mathbb{E}_{P_1} \left[ \psi(\mathring{B}_H(x)) \right]= \mathbb{E}_{P_0} \left[\psi(\mathring{B}_H(x))\frac{d P_1}{d P_0} \right] = \mathbb{E}_{P_0} \Big[\psi(B_H(x))\Big].$$ So, using in , the Kullback-Leibler divergence between the two models can be evaluated, $$\mathcal K(P_0,P_1) {\mathrel{\mathop:}=}\mathbb{E} \ln \frac{d P_0}{d P_1} = \frac{1}{2 }  { \langle \underline{\Delta} , \underline{\Delta} \rangle }_{\widetilde{\mathcal H}}.\label{logstochasticexp}$$ To evaluate , obtain a finer bound on $|\widetilde{\underline{\Delta}}(\omega)|^2$ by recalling that $\Delta = v - v_N$ and using , $$|\widetilde{\underline{\Delta}}(\omega)|^2 \asymp \varepsilon^{-2\alpha} |\widetilde{v}(\omega)|^2 \mathbbm{1}_{\left\{ |\omega| \geq N \right\}} |\omega|^{2 - 2\alpha} \leq \frac{C^2a^2\delta^2}{4\pi^2} \varepsilon^{-2\alpha} |\omega|^{-2\alpha}    \mathbbm{1}_{\left\{ |\omega| \geq N \right\}}. \label{eq:DeltaModulus}$$ Apply the bound in to with the chosen $N = \left( s\pi C/(a \delta) \right)^{1/s}$, $$\begin{aligned}
 \mathcal{K}(P_0,P_1)   &= \frac{1}{2} \int_\mathbb{R} |\widetilde{\underline{\Delta}}(\omega)|^2|\omega|^{-(1-\alpha)}\, d\omega\\
  &\leq \frac{ Ca^2 \delta^2}{4 \pi^2} \varepsilon^{-2\alpha}  \int_{\left|\omega\right| \geq N } \left|\omega\right|^{-\alpha-1}\, d\omega\\
  &= Ca^2 \delta^2\varepsilon^{-2\alpha} \left( s /(a \delta) \right)^{-\alpha/s}\\
  &\asymp \delta^{(2s+\alpha)/s} \varepsilon^{-2\alpha} .\end{aligned}$$ Now choose $\delta \asymp \varepsilon^{2\alpha s/(2s + \alpha)}$ which guarantees that $\mathcal{K}(P_0,P_1) \leq K < \infty$ for some finite positive constant $K$. Then by @Tsybakov-2009 [Theorem 2.2 (iii)] combined with the fact that $\varepsilon \asymp n^{-1/2}$ it follows that the lower rate of convergence for the minimax risk is $\varepsilon^{2\alpha s/(2s+\alpha)}\asymp n^{- \alpha s/(2s+\alpha)}$. $\Box$

Acknowledgements {#acknowledgements .unnumbered}
================

The author would like to thank the editor and an anonymous referee for their comments and suggestions which lead to an improved version of this paper.

[13]{} natexlab\#1[\#1]{}url \#1[`#1`]{}urlprefix

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