Bandwidth Selection for the Wolverton-Wagner Estimator

TL;DR

This paper investigates bandwidth selection methods for the Wolverton-Wagner estimator, providing theoretical controls on its mean squared and integrated mean squared errors, and introduces adaptive estimators for the smoothness parameter.

Contribution

It introduces new adaptive bandwidth selection techniques for the Wolverton-Wagner estimator using Goldenshluger-Lepski and Lacour-Massart-Rivoirard methods, with numerical validation.

Findings

Theoretical bounds on MSE and MISE for the estimator.

Development of adaptive estimators for the smoothness parameter.

Numerical experiments demonstrating the effectiveness of the proposed methods.

Abstract

For $n$ independent random variables having the same H\"older continuous density, this paper deals with controls of the Wolverton-Wagner's estimator MSE and MISE. Then, for a bandwidth $h_n(\beta)$, estimators of $\beta$ are obtained by a Goldenshluger-Lepski type method and a Lacour-Massart-Rivoirard type method. Some numerical experiments are provided for this last method.

Tables

Table 1. Table 1. 100 × \times MISE with 100 × \times std in parenthesis, computed over 200 simulations.

		LMR for WW				Original LMR
	$n=$	$K_1$	$K_3$	$K_5$	$K_7$	$K_1$	$K_3$	$K_5$	$K_7$	ks
$f_1$	$250$	0.442	0.318	0.285	0.256	0.412	0.315	0.290	0.268	0.285
		(0.252)	(0.213)	(0.193)	(0.162)	(0.241)	(0.214)	(0.205)	(0.193)	(0.174)
	$1000$	0.144	0.091	0.080	0.075	0.133	0.088	0.079	0.076	0.101
		(0.079)	(0.065)	(0.061)	(0.059)	(0.076)	(0.064)	(0.062)	(0.061)	(0.059)
$f_{m,1}$	$250$	0.400	0.316	0.287	0.255	0.387	0.327	0.291	0.256	1.115
		(0.204)	(0.189)	(0.176)	(0.162)	(0.208)	(0.202)	(0.179)	(0.170)	(0.150)
	1000	0.141	0.101	0.090	0.084	0.135	0.101	0.094	0.091	0.585
		(0.0623)	(0.051)	(0.049)	(0.046)	(0.062)	(0.053)	(0.051)	(0.050)	(0.076)
$f_2$	250	3.586	2.141	1.840	1.709	1.865	1.343	1.221	1.178	1.272
		(1.403)	(1.230)	(1.155)	(1.116)	(1.108)	(0.930)	(0.884)	(0.885)	(0.789)
	1000	1.056	0.646	0.555	0.515	0.602	0.429	0.382	0.372	0.506
		(0.394)	(0.306)	(0.283)	(0.270)	(0.312)	(0.270)	(0.250)	(0.235)	(0.282)
$f_{m,2}$	$250$	3.071	2.040	1.778	1.654	1.825	1.362	1.217	1.157	8.912
		(0.851)	(0.743)	(0.706)	(0.681)	(0.655)	(0.584)	(0.605)	(0.565)	(0.909)
	$1000$	0.905	0.593	0.508	0.476	0.657	0.438	0.389	0.358	4.876
		(0.246)	(0.201)	(0.187)	(0.182)	(0.257)	(0.188)	(0.159)	(0.163)	(0.367)
$f_3$	$250$	0.449	0.358	0.340	0.326	0.419	0.356	0.343	0.327	0.298
		(0.263)	(0.236)	(0.221)	(0.198)	(0.259)	(0.241)	(0.224)	(0.201)	(0.202)
	$1000$	0.174	0.132	0.124	0.121	0.162	0.130	0.126	0.126	0.125
		(0.085)	(0.071)	(0.067)	(0.065)	(0.081)	(0.071)	(0.071)	(0.076)	(0.065)
$f_{m,3}$	$250$	1.257	1.129	1.106	1.103	1.140	1.117	1.138	1.162	4.089
		(0.597)	(0.555)	(0.537)	(0.532)	(0.564)	(0.562)	(0.568)	(0.576)	(0.355)
	$1000$	0.491	0.448	0.444	0.446	0.449	0.441	0.454	0.466	3.172
		(0.171)	(0.158)	(0.158)	(0.160)	(0.168)	(0.174)	(0.189)	(0.204)	(0.201)
$f_4$	$250$	0.683	0.642	0.642	0.649	0.663	0.680	0.706	0.708	0.519
		(0.353)	(0.318)	(0.301)	(0.294)	(0.347)	(0.343)	(0.339)	(0.322)	(0.260)
	$1000$	0.281	0.254	0.254	0.258	0.273	0.268	0.278	0.284	0.242
		(0.135)	(0.122)	(0.120)	(0.122)	(0.141)	(0.147)	(0.163)	(0.172)	(0.105)

Table 2. Table 2. Mean (and std) of γ ~ n subscript ~ 𝛾 𝑛 \widetilde{\gamma}_{n} for different values of n 𝑛 n and functions f 1 , … , f 4 subscript 𝑓 1 … subscript 𝑓 4 f_{1},\dots,f_{4} , 200 repetitions

$n$	250	1000	2000	4000
$f_1$	0.017	0.044	0.054	0.059
	(0.023)	(0.016)	(0.012)	(0.019)
$f_2$	0.331	0.304	0.291	0.281
	(0.020)	(0.014)	(0.011)	(0.013)
$f_3$	0.026	0.067	0.076	0.084
	(0.027)	(0.020)	(0.017)	(0.018)
$f_4$	0.070	0.120	0.136	0.146
	(0.040)	(0.039)	(0.039)	(0.043)

Table 3. Table 3. Comparison of 100 × \times MISE (with 100 × \times std in parenthesis) for the estimators: adaptive LMR-WW, and updated recrusively WW with "frozen" γ 𝛾 \gamma , first sample n 0 = 500 subscript 𝑛 0 500 n_{0}=500 , update sample with n 1 = 500 subscript 𝑛 1 500 n_{1}=500 additional observations

Function $f_1$		Function $f_2$		Function $f_3$		Function $f_4$
WW	Update	WW	Update	WW	Update	WW	Update
0.146	0.112	0.879	0.414	0.188	0.131	0.393	0.303
(0.105)	(0.082)	(0.502)	(0.214)	(0.115)	(0.076)	(0.186)	(0.146)

Equations

$${\hat{f}}_{n,h}(x):=\frac{1}{nh}\sum _{k=1}^{n}K\left(\frac{X_k-x}{h}\right)\text{ ; }x\in \mathbb{R},$$

$${\hat{f}}_{n,{\mathbf{h}}_n}(x):=\frac{1}{n}\sum _{k=1}^n\frac{1}{h_k}K\left(\frac{X_k-x}{h_k}\right),$$

$${\hat{f}}_{n+1,{\mathbf{h}}_{n+1}}(x)=\frac{n}{n+1}{\hat{f}}_{n,{\mathbf{h}}_n}(x)+\frac{1}{(n+1)h_{n+1}}K\left(\frac{X_{n+1}-x}{h_{n+1}}\right).$$

$$\Vert \phi {\Vert }_{\alpha }:=\underset{x,y\in \mathbb{T}:x\ne y}{\sup }\frac{|\phi (y)-\phi (x)|}{|y-x{|}^{\alpha }}\text{ ; }\forall \phi \in C^{\alpha }(\mathbb{T}).$$

$${\left[{\int }_{-\infty }^{\infty }{\left({\phi }^{(\ell )}(x+t)-{\phi }^{(\ell )}(x)\right)}^{2}dx\right]}^{1/2}⩽L|t{|}^{\beta -\ell },\forall t\in \mathbb{R}.$$

$$(f\ast g)(x):={\int }_{-\infty }^{\infty }f(x-y)g(y)dy\text{ ; }x\in \mathbb{R}.$$

$${\int }_{-\infty }^{\infty }K(y)dy=1,{\int }_{-\infty }^{\infty }{K}^2(y)dy<+\infty ,{\int }_{-\infty }^{\infty }|z{|}^{\beta }|K(z)|dz:=C_{\beta }(K)<+\infty ,$$

$$\text{mbox}and{\int }_{-\infty }^{\infty }{y}^{i}K(y)dy=0,\forall i\in \{1,\dots ,l\}.$$

$$\mathbb{E}(|{\hat{f}}_{n,{\mathbf{h}}_n}(x)-f(x){|}^2)⩽\frac{c}{n^2}\left({\left|\sum _{k=1}^n\frac{h_k^{\beta }}{l!}\right|}^2+\sum _{k=1}^n\frac{1}{h_k}\right),$$

$${\int }_{-\infty }^{\infty }\mathbb{E}(|{\hat{f}}_{n,{\mathbf{h}}_n}(x)-f(x){|}^2)dx⩽\frac{c}{n^2}\left({\left|\sum _{k=1}^n\frac{h_k^{\beta }}{(l-1)!}\right|}^2+\sum _{k=1}^n\frac{1}{h_k}\right),$$

$$h_k=k^{-\gamma }\text{ ; }k\in \{1,\dots ,n\}$$

$${\mathbb{B}}_n(\gamma ):=\frac{1}{n^2}{\left|\sum _{k=1}^n{h}_k^{\beta }\right|}^2\text{mbox}and{\mathbb{V}}_n(\gamma ):=\frac{1}{n^2}\sum _{k=1}^n\frac{1}{h_k}$$

$${\mathbb{B}}_n(\gamma )=O\left(\frac{1}{n^2}\right)\text{mbox}if\gamma \beta >1\text{mbox}and{\mathbb{B}}_n(\gamma )=O\left(\frac{1}{n^{2\gamma \beta }}\right)\text{mbox}if\gamma \beta <1$$

$${\mathbb{B}}_n(\gamma )=O\left(\frac{\log (n)}{n^2}\right)\text{mbox}if\gamma \beta =1.$$

$$\begin{gathered} {\mathbb{B}}_n(\gamma )={\left|\frac{1}{n}\sum _{k=1}^n{k}^{-\gamma \beta }\right|}^2=n^{-2\gamma \beta }{\left|\frac{1}{n}\sum _{k=1}^n{\left(\frac{k}{n}\right)}^{-\gamma \beta }\right|}^2 \\ \sim \frac{n^{-2\gamma \beta }}{(1-\gamma \beta {)}^2}. \end{gathered}$$

$${\mathbb{V}}_n(\gamma )=O(n^{\gamma -1}).$$

$$h_k=k^{-\gamma }\text{ ; }k\in \{0,1,\dots ,n\},\text{mbox}with\gamma =\frac{1}{2\beta +1}$$

$$\underset{\{f\in \mathcal{H}(\beta ,L),f⩾0,\int f=1\}}{\sup }{\int }_{-\infty }^{\infty }\mathbb{E}(|{\hat{f}}_{n,{\mathbf{h}}_n}(x)-f(x){|}^2)dx⩽c{n}^{-\frac{2\beta }{2\beta +1}},$$

$${\phi }_n(\gamma ):=n^{-2\gamma \beta }+n^{\gamma -1}.$$

$$\frac{\partial {\phi }_n(\gamma )}{\partial \gamma }=\log (n)(-2\beta e^{-2\gamma \beta \log (n)}+e^{(\gamma -1)\log (n)}).$$

$$\gamma =\frac{1}{2\beta +1}+\frac{\log (2\beta )}{\log (n)(1+2\beta )}\sim \frac{1}{2\beta +1}.$$

$$h_k=h_k(\gamma )\text{ ; }\forall k\in \{1,\dots ,n\},$$

$$0<h_n({\gamma }^{\prime })<\cdots <h_1({\gamma }^{\prime }).$$

$${\mathbf{h}}_n(\gamma ):=(h_1(\gamma ),\dots ,h_n(\gamma ))$$

$$0<{\gamma }_1<\cdots <{\gamma }_{N(n)}⩽h_n^{-1}(1/n).$$

$${\hat{f}}_{n,\gamma ,{\gamma }^{\prime }}(x):=\frac{1}{n}\sum _{k=1}^n(K_{h_k({\gamma }^{\prime })}\ast K_{h_k(\gamma )})(X_k-x),$$

$$\underset{\gamma \in {\Gamma }_n}{\min }(A_n(\gamma )+V_n(\gamma )),$$

$$A_n(\gamma ):=\underset{{\gamma }^{\prime }\in {\Gamma }_n}{\sup }(\Vert {\hat{f}}_{n,{\mathbf{h}}_n({\gamma }^{\prime })}-{\hat{f}}_{n,\gamma ,{\gamma }^{\prime }}{\Vert }_2^2-V_n({\gamma }^{\prime }){)}_{+},V_n({\gamma }^{\prime }):=\upsilon \frac{\Vert K{\Vert }_2^2\Vert K{\Vert }_1^2}{n{\mathfrak{h}}_n({\gamma }^{\prime })}$$

$$\frac{1}{{\mathfrak{h}}_n({\gamma }^{\prime })}:=\frac{1}{n}\sum _{k=1}^n\frac{1}{h_k({\gamma }^{\prime })}.$$

$$\underset{n\in {\mathbb{N}}^{\ast }}{\sup }\sum _{{\gamma }^{\prime }\in {\Gamma }_n}\exp (-c/{\mathfrak{h}}_n({\gamma }^{\prime }{)}^r)<\infty .$$

Full text

1. 1 Introduction

2. 2 Bounds on the MSE and the MISE of Wolverton-Wagner’s estimator

3. 3 Goldenshluger-Lepski’s method for Wolverton-Wagner’s estimator

4. 4 The Lacour-Massart-Rivoirard (LMR) estimator

5. 4.1 Estimator and main result

6. 4.2 Simulation experiments

7. 4.3 Back to recursivity

8. 5 Concluding remarks

9. 6 Proofs

10. 6.1 Proof of Proposition 2.3

11. 6.2 Proof of Proposition 2.5

12. 6.3 Proof of Proposition 3.3

13. 6.4 Proof of Proposition 4.2

14. 6.4.1 Steps of the proof

15. 6.4.2 Proof of Lemma 6.2

16. 6.4.3 Proof of Lemma 6.3

17. 6.4.4 Proof of Lemma 6.4

1. Introduction

Consider $n\in\mathbb{N}^{*}$ independent random variables $X_{1},\dots,X_{n}$ having the same probability distribution of density $f$ with respect to Lebesgue’s measure.

The usual Parzen [11] - Rosenblatt [12] kernel estimator of $f$ is defined by

[TABLE]

where $h>0$ and $K:\mathbb{R}\rightarrow\mathbb{R}_{+}$ is a kernel. In 1969, Wolverton and Wagner introduced in [15] a variant of $\widehat{f}_{n,h}(x)$ defined by

[TABLE]

where $\mathbf{h}_{n}=(h_{1},\dots,h_{n})$ and $0<h_{n}<\dots<h_{1}$. Thanks to its recursive form, this type of estimator is well-suited to online treatment of data: by denoting $\mathbf{h}_{n+1}=(h_{1},\dots,h_{n},h_{n+1})$,

[TABLE]

Thus, up-dating the estimator when new observations are available is easy and fast.

We can mention here that several variants or generalizations of the Wolverton and Wagner (WW) estimator have been proposed: see Yamato [16], Wegman and Davies [14], Hall and Patil [5]. They were studied from almost sure convergence point of view, or asymptotic rates of convergence under fixed regularity assumptions. We choose to focus on Wolverton and Wagner estimator but our results and discussions may be applied to these.

Theoretical developments concerning either classical Parzen-Rosenblatt or WW recursive kernels estimators occurred recently following different and independent roads.

On the one hand, several recent works are dedicated to efficient and data-driven bandwidth selection, see Goldenshluger and Lespki [4] and several companion papers by these authors, or Lacour et al. [8] who proposed a modification of the method. The original Goldenshluger and Lepski (GL) method was difficult to implement because it turned out to be numerically consuming and with calibration difficulties, see Comte and Rebafka [3]. This is why the improvement proposed in Lacour et al. [8] has both theoretical and practical interest.

On the other hand, the increase of computer speed and of data sets sizes made fast up-dating of estimators mandatory. The theoretical developments in this context are in the field of stochastic algorithms (see e.g. Mokkadem et al. [10]) or in view of specific applications (see Bercu et al. [2]).

Bandwidths have to be chosen for WW estimators as for Parzen-Rosenblatt ones, and this choice is crucial to obtain good performances. This is why we propose to extend to this context general risk study as described in Tsybakov [13] and the GL method as improved by Lacour et al. [8]. More precisely, considering for instance $h_{k}=k^{-\gamma}$ for a parameter $\gamma>0$ in formula (1), we study adaptive selection of $\gamma$. We prove risk bounds for the Mean Integrated Squares Error (MISE) of the resulting estimator $\widehat{f}_{n,\widehat{\mathbf{h}}_{n}}$ where $\widehat{\mathbf{h}}_{n}=(\widehat{h}_{1},\dots,\widehat{h}_{n})$ and $\widehat{h}_{k}=k^{-\widetilde{\gamma}}$.

Amiri [1] proved that for $f$ with regularity 2 and an adequate choice of the bandwidth, Parzen-Rosenblatt’s estimator had asymptotical smaller risk than the WW estimator. We propose an empirical finite sample study of this question, together with an interesting insight on the gain brought by higher order kernels.

Now, clearly, plugging $\widetilde{\gamma}_{n}=\widetilde{\gamma}(X_{1},\dots,X_{n})$ in the estimator makes the recursivity fail. Therefore, an adequate strategy is required, either with initial estimation of $\gamma$ on the first $n$-sample and recursive up-dating relying on this "frozen" value on the following $N$-sample,or with adequate matrix updating for $\widetilde{\gamma}_{n+1}?$ selection. This is what is experimented in our final section, and empirically illustrated and discussed.

This paper provides in Section 2 controls of the MSE and of the MISE of the estimator $\widehat{f}_{n,\mathbf{h}_{n}}$ under general regularity conditions on $f$. Then, in Section 3, the well-known Goldenshluger-Lepski’s bandwidth selection method for Parzen-Rosenblatt’s estimator is extended to Wolverton-Wagner’s estimator. Lastly, an estimator in the spirit of Lacour et al. [8] is studied from both theoretical and practical point of view in Section 4. In particular, a recursive global strategy is proposed. Concluding remarks are given in Section 5. Proofs are relegated in Section 6.

Notation.

1. (1)

Consider $\alpha\in(0,1)$. The space of $\alpha$-Hölder continuous functions from ${\mathbb{T}}$ into $\mathbb{R}$ is denoted by $C^{\alpha}(\mathbb{T})$ and equipped with the $\alpha$-Hölder semi-norm $\|.\|_{\alpha}$ defined by

[TABLE] 2. (2)

Let $\beta,L>0$, and ${\mathbb{T}}$ an interval of ${\mathbb{R}}$. The Hölder class $\Sigma(\beta,L)$ on ${\mathbb{T}}$ is the set of functions $\varphi~{}:{\mathbb{T}}\rightarrow{\mathbb{R}}$ such that $\varphi^{(\ell)}$, where $\ell:=\lfloor\beta\rfloor$, is the greatest integer less than or equal to $\beta$, exists and satisfies $\|\varphi\|_{\beta-\ell}\leqslant L$. 3. (3)

Let $\beta,L>0$. The Nikol’ski class ${\mathcal{H}}(\beta,L)$ is the set of function $\varphi:{\mathbb{R}}\rightarrow{\mathbb{R}}$ such that $\varphi^{(\ell)}$ exists and satisfies

[TABLE] 4. (4)

For every square integrable function $f,g:\mathbb{R}\rightarrow\mathbb{R}$, $\|f\|_{2}^{2}=\int_{-\infty}^{+\infty}f^{2}(x)dx$, $\langle f,g\rangle=\int_{-\infty}^{+\infty}f(x)g(x)dx$, and

[TABLE] 5. (5)

$K_{\varepsilon}:=(1/\varepsilon)K(\cdot/\varepsilon)$ for every $\varepsilon>0$.

The definitions of $\Sigma(\beta,L)$ and ${\mathcal{H}}(\beta,L)$ can be found in Tsybakov (2009, Chapter 1).

2. Bounds on the MSE and the MISE of Wolverton-Wagner’s estimator

Consider $\beta>0$ and $l:=\lfloor\beta\rfloor$. Throughout this section, the map $K$ fulfills the following assumption.

Assumption 2.1**.**

The map $y\in\mathbb{R}\mapsto y^{i}K(y)$ is integrable for every $i\in\{0,1,\dots,l\}$,

[TABLE]

[TABLE]

Let us establish a control of the MSE of Wolverton-Wagner’s estimator under the following condition on $f$.

Assumption 2.2**.**

The map $f$ belongs to the Hölder ball $\Sigma(\beta,L)$.

Proposition 2.3**.**

Under Assumptions 2.1 and 2.2, there exists a constant $c>0$, such that for all $x\in{\mathbb{T}}$

[TABLE]

where $c$ depends on $\beta,L,\|K\|_{2}$ and $C_{\beta}(K)$ (but not on $n$ and $h_{1},\dots,h_{n}$).

Now, let us establish a control of the MISE of Wolverton-Wagner’s estimator under Nikolski’s condition on $f$.

Assumption 2.4**.**

*The map $f$ belongs to the Nikol’ski ball ${\mathcal{H}}(\beta,L)$. *

Proposition 2.5**.**

Under Assumptions 2.1 and 2.4, there exists a constant $c>0$, such that

[TABLE]

where $c$ depends on $\beta,L,\|K\|_{2}$ and $C_{\beta}(K)$ (but not on $n$ and $h_{1},\dots,h_{n}$).

Remark. Assumptions 2.1, 2.2 and 2.4 are standard for density estimation, see Tsybakov [13]. Moreover, if we set $h_{k}=h$, we recover the results stated in Section 1.2.1 for Proposition 2.3 and in Theorem 1.3 for Proposition 2.5 in Tsybakov ([13] (that is a squared bias term of order $h^{2\beta}$ and a variance term of order $1/(nh)$).

The estimator is consistent if the risk tends to zero when $n$ grows to infinity, that is if $(1/n^{2})\left|\sum_{k=1}^{n}h_{k}^{\beta}\right|^{2}$ and $(1/n^{2})\sum_{k=1}^{n}\frac{1}{h_{k}}$ tend to 0 when $n$ tends to infinity.

Let us consider

[TABLE]

and, for this collection of bandwidths, set

[TABLE]

with $\gamma\in(0,1)$ (otherwise $\mathbb{B}_{n}(\gamma)$ or $\mathbb{V}_{n}(\gamma)$ cannot tend to zero). Then

[TABLE]

with the intermediate case

[TABLE]

Indeed, if $\gamma\beta<1$, then

[TABLE]

On the other hand,

[TABLE]

As a consequence, we have the following result:

Corollary 2.6**.**

Under Assumptions 2.1 and 2.4, choosing

[TABLE]

yields the rate

[TABLE]

where $c=c(\beta,L,K)$ is a positive constant depending on $L,\beta$ and the kernel $K$, but not on $n$.

Clearly, this is the optimal rate in the minimax sense, see Goldenshluger and Lepski [4] and the references therein. The bounds are uniform on the set of densities belonging to the ball ${\mathcal{H}}(\beta,L)$.

Proof.

Consider

[TABLE]

Then,

[TABLE]

Moreover, $\partial_{\gamma}\varphi_{n}(\gamma)=0$ if and only if,

[TABLE]

Therefore, $\gamma=1/(2\beta+1)$ makes the upper bound on the risk minimal. ∎

3. Goldenshluger-Lepski’s method for Wolverton-Wagner’s estimator

This section provides an extension of the well-known Goldenshluger-Lepski’s bandwidth selection method for Parzen-Rosenblatt’s estimator to Wolverton-Wagner’s estimator.

Throughout this section, assume that

[TABLE]

where $\gamma\in[0,1]$ and the maps $h_{1}(.),\dots,h_{n}(.)$ from $[0,1]$ into $(0,\infty)$ fulfill the following assumption.

Assumption 3.1**.**

For every $\gamma^{\prime}\in[0,1]$,

[TABLE]

Moreover, $h_{n}(.)$ is decreasing and one to one from $[0,1]$ into $(0,1]$.

For instance, one can take as above $h_{k}(\gamma^{\prime}):=k^{-\gamma^{\prime}}$ for every $k\in\{1,\dots,n\}$ and $\gamma^{\prime}\in[0,1]$.

Consider

[TABLE]

and the set $\Gamma_{n}:=\{\gamma_{1},\dots,\gamma_{N(n)}\}\subset[0,1]$, where $N(n)\in\{1,\dots,n\}$ and

[TABLE]

Consider also

[TABLE]

where $\gamma^{\prime}\in[0,1]$.

A way to extend the Goldenshluger-Lepski bandwidth selection method to Wolverton-Wagner’s estimator is to solve the minimization problem

[TABLE]

where

[TABLE]

with $\upsilon>0$ not depending on $n$ and

[TABLE]

In the sequel, the map $\mathfrak{h}_{n}(.)$ fulfills the following assumption.

Assumption 3.2**.**

For every $c>0$ and $r\in\{1/2,1\}$,

[TABLE]

Example. Consider

[TABLE]

and

[TABLE]

where $[x]$ denotes the interger part of $x$. For every $\gamma^{\prime}\in\Gamma_{n}$,

[TABLE]

Then, for any $c>0$ and $r\in\{1/2,1\}$,

[TABLE]

Proposition 3.3**.**

Under Assumptions 3.1 and 3.2, if $f$ is bounded and $\widehat{\gamma}_{n}$ is a solution of the minimization problem (5), then there exists a constant $\upsilon_{0}$ such that, for $\upsilon\geqslant\upsilon_{0}$,

[TABLE]

*where $c_{0}$ is a numerical constant ($c_{0}=18$ suits) and $c_{1}$ is a constant depending on $K$ and $\|f\|_{\infty}$ (but not on $n$).

If in addition Assumptions 2.1 and 2.4 hold, then*

[TABLE]

where ${\mathbb{B}}_{n}(\gamma)$ and ${\mathbb{V}}_{n}(\gamma)$ are defined in (2), (3) and (4).

Note that the proof leads to the value $\kappa_{0}=24$, which would be too large in practice.

Remark. By Corollary 2.6, the infimum in bound (7) has the order of the optimal rate, and is reached automatically by the data driven estimator. This result is more precise than the heuristics associated with cross-validation.

We mentioned previously that the optimal theoretical choice for $\gamma$ under Assumptions 2.1 and 2.4 is $\gamma=1/(2\beta+1)$. Here, the selected $\gamma$ should be at nearest of this value, e.g. if $\Gamma_{n}$ is as in (6), distant from less than $1/\sqrt{\log(n)}$ of the good choice. We may therefore consider that $\widehat{\gamma}_{n}$ provides an estimate of $1/(2\beta+1)$ and thus an estimate of the regularity $\beta$ of $f$ (at least for huge values of $n$).

4. The Lacour-Massart-Rivoirard (LMR) estimator

4.1. Estimator and main result

The Goldenshluger-Lepski method has been acknowledged as being difficult to implement, due to the square grid in $\gamma,\gamma^{\prime}$ required to compute intermediate versions of the criterion and to the lack of intuition in the choice of the constant $\upsilon$ which should be calibrated from preliminary simulation experiments. This is the reason why Lacour et al. [8] investigated and proposed a simplified criterion relying on deviation inequalities for $U$-statistics due to Houdré and Reynaud-Bouret [6]. This inequality applies in our more complicated context and Lacour-Massart-Rivoirard’s result can be extended here as follows.

Let us recall that $K_{\varepsilon}(\cdot):=(1/\varepsilon)K(\cdot/\varepsilon)$ for every $\varepsilon>0$ and set

[TABLE]

Let $\gamma_{\max}$ be the maximal proposal in $\Gamma_{n}$ and consider

[TABLE]

with

[TABLE]

Then, we define

[TABLE]

In the sequel, $K$, $f$ and $\mathbf{h}_{n}$ fulfill the following assumption.

Assumption 4.1**.**

The kernel $K$ is symmetric, $K(0)>0$,

[TABLE]

and $\|f\|_{\infty}<\infty$.

Proposition 4.2**.**

Consider $\lambda\in[1,\infty[$ and $\varepsilon\in(0,1)$. Under Assumption 4.1, there exists three deterministic constants $c_{1},c_{2},c_{3}>0$, not depending on $n$, $\lambda$ and $\gamma$, such that with probability larger than $1-c_{1}|\Gamma_{n}|e^{-\lambda}$,

[TABLE]

Remark. The term $\|f_{n,\gamma_{{\rm max}}}-f\|_{2}^{2}$ is negligible because it is a pure bias term for smallest bandwidth (e.g., under Assumption 2.4, it has order $n^{-2\beta\gamma_{{\rm max}}}$, see (3), and thus $o(1/n)$ if $\gamma_{{\rm max}}$ is near of 1 and $\beta>1/2$). The terms following are of order $O(1/n)$ and are always negligible compared to nonparametric rates in our setting. Therefore, the bound given in Proposition 4.2 says that the MISE of the adaptive estimator has the order of the best estimator of the collection, up to a multplicative factor larger than 1. This is the method we implement in the next section: it is faster than GL method and with no constant to calibrate in the penalty.

4.2. Simulation experiments

We consider basic densities with different types and orders of regularity:

• •

$X\rightsquigarrow\mathcal{N}(0,1)$, density $f_{1}$,

• •

a mixed gaussian $X\rightsquigarrow 0.5\mathcal{N}(-2,1)+0.5\mathcal{N}(2,1)$, density $f_{m,1}$,

• •

$X\rightsquigarrow\beta(3,3)$, density $f_{2}$,

• •

a mixed beta $X\rightsquigarrow 0.5(\beta(3,3)-1)+0.5\beta(3,3)$, density $f_{m,2}$,

• •

$X\rightsquigarrow\gamma(5,5)/10$, density $f_{3}$,

• •

a mixed gamma $X\rightsquigarrow 0.4.\gamma(2,1/3)+0.6\gamma(7,6)/10$, density $f_{m,3}$,

• •

$X\rightsquigarrow f_{4}$ with $f_{4}(x)=e^{-|x|}$, a Laplace density.

The densities $f_{1}$ and $f_{m,1}$ have infinite regularity, $f_{2}$ and $f_{m,2}$ should rather have regularity of order less than 2, $f_{3}$ and $f_{m,3}$ less than 4, and $f_{4}$ less than 1. This choice should allow to study the influence of the order of the kernel.

Denoting by $n_{j}(x)$ the density of a centered Gaussian random variable with variance equal to $j$, we consider the following kernels:

• •

a Gaussian kernel, $K_{1}(x)=e^{-x^{2}/2}/\sqrt{2\pi}$ which is of order 1,

• •

a Gaussian-type kernel of order 3, $K_{3}(x)=2n_{1}(x)-n_{2}(x)$,

• •

a Gaussian-type kernel of order 5, $K_{5}(x)=3n_{1}(x)-3n_{2}(x)+n_{3}(x)$,

• •

a Gaussian-type kernel of order 7, $K_{7}(x)=4n_{1}(x)-6n_{2}(x)+4n_{3}(x)-n_{4}(x)$.

With all these kernels, the penalty terms are computed analytically and without approximation. Indeed, for $n_{i,h}(x)=(1/h)n_{i}(x/h)$, it holds that

[TABLE]

We compute the variable bandwidth estimator as described in Section 4 and select $\widetilde{\gamma}_{n}$ in a collection of $M=40$ equispaced values between 0 and 0.5 while the bandwidth associated with observation $i$ is $h_{i}(\gamma)=i^{-\gamma}$. We also compute the original estimator of Lacour et al. [8] with bandwidth $h$ which does not depend on the observation and is selected among $M=40$ values in the set \{k/M\textrm{ ; }k=1,\dots,M\}.

For comparison, we give the performance of the Matlab density estimator obtained from ${\tt ksdensity}$ function (denoted by ks in Table 1), which entails a different bandwidth selection method and relies on a gaussian kernel.

We compute the integrated ${\mathbb{L}}^{2}$-risk associated with all the final estimators, evaluated at $P=100$ equispaced points in the range $[a,b]$ of the observations, averaged over $R=200$ repetitions:

[TABLE]

where $\widehat{f}^{(j)}_{\widehat{h}^{(j)}}$ is the estimator computed for path $j$.

Results are gathered in Table 1 and deserve some comments. As expected, when increasing $n$ from 250 to 1000, the resulting MSEs decrease and seem to be more improved in LMR methods of both types than for ks estimator. Increasing the order of the kernel systematically improves the results, except for the lowest regularity density $f_{4}$, which is at best with $K_{3}$, but it is interesting to note that taking higher order kernel is always a good strategy: if a loss occurs, it is negligible while the improvement, when it happens, is in all cases significant. Estimator ks fails for all mixed densities $f_{m,1}$, $f_{m,2}$ and $f_{m,3}$ and provides rather bad results in these cases, for both sample sizes. For the other densities ($f_{1}$, $f_{2}$, $f_{3}$, $f_{4}$), the results obtained with kernel $K_{7}$ and LMR method are better than with ks for $f_{1}$ (Gaussian case), and of comparable order in all other cases. Now if we compare the LMR and WW-LMR results both with kernel $K_{7}$, we conclude that the WW-LMR method wins in 10 cases out of 14, but not significantly.

The first line of Figure 1 illustrates in the left picture the way Matlab estimator fails for mixed densities (here the mixed Gaussian $f_{m,1}$) by probably selecting a too large bandwidth, here $n=1000$. The two LMR estimators are almost confounded. The middle and right pictures present the $M=40$ estimators among which the LMR procedure makes the selection, for the same path: we observe that the collection of proposals are rather different. The second line of Figure 1 presents the same type of results for density $f_{2}$, and sample size $n=250$. Figure 2 shows beams of 30 final estimators for sample size $n=250$, for the three estimators LMR-WW with $K_{7}$, LMR with $K_{7}$ and ks, showing very similar behaviours.

A last remark corresponding to numerical results we do not report in detail is the following. For most densities, the value of $\gamma$ selected by the LMR strategy decreases, and the value of $h$ increases, when the order of the kernel increases. Exceptions are densities with lower regularity (the beta $f_{2}$, mixed beta $f_{2,m}$ and Laplace $f_{4}$ densities) for which the last value of selected $h$ with $K_{7}$ is less than the one selected with $K_{5}$. This illustrates the fact that, asymptotically, if $\beta$ is the regularity index of the density and $\ell$ the order of the kernel, the optimal choice is for $h$ of order $n^{-1/(2\min(\beta,\ell)+1)}$ and for $\gamma$, $1/(2\min(\beta,\ell)+1)$. This point is further investigated hereafter.

4.3. Back to recursivity

However, one may wonder how to keep these ideas compatible with recursive procedures and online updating of the kernel estimator.

A first possibility is to consider that the adaptive bandwidth, whatever its type, can be selected on a preliminary sample and then, "frozen" to this selected value. The estimator may then be recursively updated with this frozen value, and the procedure would exploit all the observations.

We compute the mean over 200 repetitions of the selected values $\widetilde{\gamma}_{n}$ for $n$ observations, with increasing values of $n$, for the functions $f_{1},\dots,f_{4}$ defined in Section 4.2. We can see in Table 2 that, if there is a convergence towards a value, it is very slow. Indeed, keeping in mind the value $1/(2\min(\beta,\ell)+1)$ for $\gamma$, we may expect $\widetilde{\gamma}_{n}$ to tend to $1/15=0.67$ for $f_{1}$, $1/5=0.2$ for $f_{2}$, $1/9=0.11$ for $f_{3}$ and a quantity less that $1/5=0.2$ for $f_{4}$.

We provide in Table 3 the results obtained for densities $f_{1},\dots,f_{4}$ and sample size 1000 splitted in two parts: $n_{0}=500$ observations used for the selection of $\widetilde{\gamma}_{n_{0}}$ and $n_{1}=500$ updates of the resulting estimator with the "frozen" value $\widetilde{\gamma}_{n_{0}}$. We compute the MISE for the first step and final estimator, which relies on $n=n_{0}+n_{1}$ observations. The results are given in Table 3. We can see an improvement when going from $n_{0}$ to $n$ observations, but the results are deteriorated compared with what is obtained with $n=n_{0}+n_{1}=1000$ in Table 1.

This is why our idea is to exploit the recursive formula to select the $\gamma$-parameter at each step. The price to pay is to store a matrix instead of storing a vector, but the matrix size is fixed. Thus, for one given sample, the procedure is fast and the storage size under control. Precisely, adding an observation leads to update the matrix ${\mathbf{F}}_{n}=\left(\widehat{f}_{n,{\mathbf{h}}_{n}(\gamma^{(j)})}(x_{k})\right)_{1\leqslant j\leqslant M,1\leqslant k\leqslant K}$ for $\Gamma_{M}=\{\gamma^{(1)},\dots,\gamma^{(M)}\}$ the collection of proposed values of $\/gamma$ and $x_{k},k=1,\dots,K$ the set points at which the function is estimated. This is the collection of values used to select $\widetilde{\gamma}_{n}$. Then we have

[TABLE]

and we can select $\widetilde{\gamma}_{n+1}$. The resulting estimator is already computed since it is one of the collection. At each stage, the $M\times K$ matrix is stored to be updated, and the selection procedure has entries the matrix, the value of $n$ and the domain of observations, and gives as output the selected value of $\gamma\in\Gamma_{M}$ and the corresponding functional estimator, which corresponds to a line of the matrix. We show in Figure 3 the 950 selected values of $\widetilde{\gamma}_{n}$ for $n=50$ to $n=1000$, for one path of a random sample with density $f_{1}$, …, $f_{4}$ on the left and $f_{m,1},f_{m,2},f_{m,3}$ on the right plot. We consider $M=50$ and $K=100$. We observe on this example a stabilization of the selected value when $n_{1}+n_{0}$ gets near of 1000 observations.

5. Concluding remarks

Our study illustrates that bandwidth selection is an important step for kernel functional estimation, and recent methods are really powerful whatever the type of density to recover.

Our simulations show also that, even if it implies non necessarily nonnegative kernels and thus density estimators, increasing the order of the kernel improves the estimation both in the theory and in practice. Also, we proved that variable bandwidth for WW-type estimators can reach excellent rates, again both in theory and in practice, provided that adaptive choice of this variable bandwidth is performed. The orders of practical MISEs show that this WW-strategy provides results of the same order as the more classical bandwidth methods.

Lastly, we illustrate that recursivity formula can be used for fast online updating of the whole collection of estimators and the selection of the best one in the sense of our criterion.

6. Proofs

6.1. Proof of Proposition 2.3

First, by the bias-variance decomposition,

[TABLE]

where

[TABLE]

Let us find controls for $b_{n}(f,x)$ and $\textrm{var}(\widehat{f}_{n,\mathbf{h}_{n}}(x))$.

On the one hand,

[TABLE]

where $c_{1}\geqslant\|f\|_{\infty}\int_{-\infty}^{\infty}K(z)^{2}dz.$ Note that, from Theorem 1.1 in Tsybakov [13], there exists a constant $c_{\max}$ such that

[TABLE]

where $c_{\max}$ depends on $\beta,L$ only. Thus, we can set $c_{1}:=c_{\max}\|K\|^{2}_{2}$. On the other hand,

[TABLE]

For every $k\in\{1,\dots,n\}$ and $z\in\mathbb{R}$, by Taylor-Lagrange’s formula there exists $\tau\in[0,1]$ such that

[TABLE]

Then, by Assumption 2.1,

[TABLE]

Therefore, by Assumption 2.2,

[TABLE]

where $c_{2}:=L\int_{-\infty}^{\infty}|z|^{\beta}|K(z)|dz.$ In conclusion, by Equation (8), setting $c:=c_{1}\vee c_{2}^{2}$, we get

[TABLE]

6.2. Proof of Proposition 2.5

In order to prove Proposition 2.5, the two following inequalities are required.

Lemma 6.1**.**

For any Borel function $\varphi:\mathbb{R}^{2}\rightarrow\mathbb{R}$, if $y\mapsto\varphi(y,z)$ is integrable and

[TABLE]

is a Borel function, then

1. (1)

$\displaystyle{\int_{-\infty}^{\infty}\left(\int_{-\infty}^{\infty}\varphi(y,z)dy\right)^{2}dz\leqslant\left(\int_{-\infty}^{\infty}\left(\int_{-\infty}^{\infty}\varphi(y,z)^{2}dz\right)^{1/2}dy\right)^{2}}$. 2. (2)

$\displaystyle{\int_{-\infty}^{\infty}\left(\sum_{k=1}^{n}\varphi(k,z)\right)^{2}dz\leqslant\left(\sum_{k=1}^{n}\left(\int_{-\infty}^{\infty}\varphi(k,z)^{2}dz\right)^{1/2}\right)^{2}}$.

Proof.

Result (1) is proved in Tsybakov [13], see Lemma A.1 for a proof. Result (2) follows from triangular inequality applied to the $\mathbb{L}^{2}$-norm of the sum of $n$ functions. ∎

It has been established in the proof of Proposition 2.3 that

[TABLE]

and

[TABLE]

On the one hand, by Inequality (9),

[TABLE]

where $c_{1}:=\|K\|^{2}_{2}$. On the other hand, by Taylor’s formula with integral remainder,

[TABLE]

Then, by Assumption 2.1,

[TABLE]

By Lemma 6.1.(2),

[TABLE]

where

[TABLE]

By Lemma 6.1.(1), for every $k\in\{1,\dots,l\}$,

[TABLE]

Therefore, by Assumption 2.4,

[TABLE]

where

[TABLE]

since $\left(\int_{0}^{1}(1-\tau)^{l-1}\tau^{\beta-l}d\tau\right)^{2}\leqslant 1$. In conclusion, by Equation (8), setting $c:=c_{1}\vee c_{2}$, we get

[TABLE]

6.3. Proof of Proposition 3.3

For any $\gamma$ in $\Gamma_{n}$, we have

[TABLE]

By the definition of $A_{n}$,

[TABLE]

and

[TABLE]

Thus we get that, for any $\gamma\in\Gamma_{n}$,

[TABLE]

Since

[TABLE]

it follows that, for any $\gamma\in\Gamma_{n}$,

[TABLE]

Now, let us find a suitable control for $\mathbb{E}(A_{n}(\gamma))$. For that, consider

[TABLE]

with obviously $f_{n,\gamma}(x)={\mathbb{E}}(\widehat{f}_{n,\mathbf{h}_{n}(\gamma)}(x))$ and $f_{n,\gamma,\gamma^{\prime}}(x)={\mathbb{E}}(\widehat{f}_{n,\gamma,\gamma^{\prime}}(x))$. Now, for any $\gamma,\gamma^{\prime}\in\Gamma_{n}$,

[TABLE]

Thus, using the definition of $A_{n}(\gamma)$, we get that for any $\gamma\in\Gamma_{n}$,

[TABLE]

Let us control each terms of the right-hand side of Inequality (6.3). On the one hand, by Lemma 6.1.(2),

[TABLE]

On the other hand, let $\mathcal{C}$ be a countable and dense subset of the unit sphere of $\mathbb{L}^{2}(\mathbb{R},dx)$. Then,

[TABLE]

where, for every $\psi\in\mathcal{C}$,

[TABLE]

and

[TABLE]

In order to apply Talagrand’s inequality (see Klein and Rio [7]):

1. (1)

For every $\psi\in\mathcal{C}$, $h\in(1/n,1)$ and $y\in\mathbb{R}$,

[TABLE]

Then,

[TABLE] 2. (2)

For every $\psi\in\mathcal{C}$,

[TABLE]

Then, as established in the proof of Proposition 2.5,

[TABLE]

as $\|K\|_{1}\geqslant 1=|\int K|$. 3. (3)

For every $\psi\in\mathcal{C}$ and $k\in\{1,\dots,n\}$,

[TABLE]

Then,

[TABLE]

By applying Talagrand’s inequality to $(v_{\psi})_{\psi\in\mathcal{C}}$ and to the independent random variables $(h_{1}(\gamma^{\prime}),X_{1}),\dots,(h_{n}(\gamma^{\prime}),X_{n})$, there exist two numerical constants $c_{1},c_{2}>0$, and two constant $c_{3},c_{4}>0$ depending only on $f$, $K$ and $\upsilon$, such that

[TABLE]

Then, by Assumption 3.2, with $\upsilon\geqslant 24$, there exists a constant $c_{5}>0$, not depending on $n$, such that

[TABLE]

The same ideas give that there exists a constant $c_{6}>0$, not depending on $n$, such that

[TABLE]

By Inequalities (6.3)-(15), we get that, for all $\gamma\in\Gamma_{n}$,

[TABLE]

Plugging this in Inequality (11), using that

[TABLE]

we get

[TABLE]

where $c_{7}=3\max(2+\frac{1}{\upsilon},1+6\|K\|_{1}^{2})$ and $c_{8}>0$ is not depending on $n$. Note that, as $\upsilon\geqslant 24$ and $\|K\|_{1}^{2}\geqslant 1$, $c_{7}=18\|K\|_{1}^{2}$. We obtain the first inequality of Proposition 3.3.

To get (7), we write

[TABLE]

and, if Assumptions 2.1 and 2.4 hold, as established in the proof of Proposition 2.5, there exists a constant $c_{9}>0$ which does not depend on $n$ such that

[TABLE]

Plugging these last two bounds in (16) yields (7) and ends the proof of Proposition 3.3.

6.4. Proof of Proposition 4.2

For this proof, we use the tools and follow the lines given in the proof of Theorem 2 in Lacour et al. [8].

Throughout this section, for every $h>0$, we consider $f_{h}:=f\ast K_{h}$, where $\ast$ is the convolution product and we recall that $K_{h}=1/hK(\cdot/h)$. Note that for every $h>0$ and $k\in\{1,\dots,n\}$,

[TABLE]

We also consider $\lambda\in[1,\infty[$, $\varepsilon,\theta\in(0,1)$ and $\gamma\in\Gamma_{n}$.

In order to prove Proposition 4.2, let us first establish the three following lemmas providing suitable bounds for key quantities involved in the decomposition of

[TABLE]

Throughout this section, the positive constants $\kappa_{i}$ ; $i\in\mathbb{N}^{*}$ are not depending on $n$, $\lambda$, $\theta$, $\varepsilon$ and $\gamma$.

6.4.1. Steps of the proof

The proof relies on three Lemmas, which are stated first. Lemma 6.2 follows from an exponential inequality for $U$-statistics, which is applied here in a non identically distributed context.

Lemma 6.2**.**

Consider the $U$-statistic

[TABLE]

There exists a universal constant $\mathfrak{c}>0$ such that with probability larger than $1-5.54|\Gamma_{n}|e^{-\lambda}$,

[TABLE]

Lemmas 6.3 and 6.4 rely on Bernstein’s inequality for non identically distributed variables.

Lemma 6.3**.**

There exists a deterministic constant $c>0$, not depending on $n$, $\lambda$, $\theta$ and $\gamma$, such that for every $\gamma^{\prime}\in\Gamma_{n}$, with probability larger than $1-2e^{-\lambda}$,

[TABLE]

satisfies

[TABLE]

Lemma 6.4**.**

Under Assumption 4.1, there exists two deterministic constants $c_{1},c_{2}>0$, not depending on $n$, $\lambda$, $\varepsilon$ and $\gamma$, such that with probability larger than $1-c_{1}|\Gamma_{n}|e^{-\lambda}$,

[TABLE]

The proof of Proposition 4.2 is dissected in three steps.

Step 1. In this step, a suitable decomposition of

[TABLE]

is provided. On the one hand,

[TABLE]

Since $\widetilde{\gamma}_{n}\in\arg\min_{\gamma\in\Gamma_{n}}\textrm{Crit}(\gamma)$, for any $\gamma\in\Gamma_{n}$,

[TABLE]

with

[TABLE]

On the other hand,

[TABLE]

where

[TABLE]

Step 2. Some bounds for $\psi_{n,1}(\gamma)$, $\psi_{n,2}(\gamma)$ and $\psi_{n,3}(\gamma)$ are provided in this step.

1. (1)

Consider

[TABLE]

By Lemma 6.2, with probability larger than $1-5.54|\Gamma_{n}|e^{-\lambda}$,

[TABLE] 2. (2)

On the one hand, for any $\gamma^{\prime}\in\Gamma_{n}$,

[TABLE]

On the other hand,

[TABLE]

Therefore,

[TABLE] 3. (3)

By applying Lemma 6.3 to $V_{n}(\gamma,\gamma_{\max})$ and $V_{n}(\gamma_{\max},\gamma)$, with probability larger than $1-2e^{-\lambda}$,

[TABLE]

Step 3. Consider

[TABLE]

By Step 2 and Lemma 6.4, with probability larger than $1-\kappa_{2}|\Gamma_{n}|e^{-\lambda}$,

[TABLE]

Therefore, choosing $\theta$ as in Lemma 6.4 ($1/(1-\theta)=1+\varepsilon$), by Inequality (17), with probability larger than $1-\kappa_{5}|\Gamma_{n}|e^{-\lambda}$,

[TABLE]

This concludes the proof. $\Box$

6.4.2. Proof of Lemma 6.2

Consider

[TABLE]

The $U$-statistic satisfies

[TABLE]

where

[TABLE]

for every $(k,l)\in\Delta_{n}$, $a,b\in\{\gamma,\gamma_{\max}\}$ and $(\alpha,\beta)\in\mathbb{R}^{2}$.

By Houdré and Reynaud-Bourret [6], Theorem 3.4, there exists a universal constant $\mathfrak{c}>0$ such that

[TABLE]

where the constants $A$, $B$, $C$ and $D$ will be defined and controlled in the sequel.

• •

The constant $A$. Consider

[TABLE]

with

[TABLE]

For any $(k,l)\in\Delta_{n}$ and $(\alpha,\beta)\in\mathbb{R}^{2}$,

[TABLE]

Therefore,

[TABLE]

• •

The constant $B$. Consider

[TABLE]

For any $(k,l)\in\Delta_{n}$, $a,b\in\{\gamma,\gamma_{\max}\}$ and $(\alpha,\beta)\in\mathbb{R}^{2}$,

[TABLE]

Then,

[TABLE]

Therefore,

[TABLE]

• •

The constant $C$. Consider

[TABLE]

For any $(k,l)\in\Delta_{n}$ and $a,b\in\{\gamma,\gamma_{\max}\}$,

[TABLE]

Moreover,

[TABLE]

Then,

[TABLE]

Therefore, since $\lambda\in[1,\infty[$,

[TABLE]

• •

The constant $D$. Consider

[TABLE]

where

[TABLE]

For any $(a,b)\in\mathcal{S}$,

[TABLE]

with

[TABLE]

and

[TABLE]

Then,

[TABLE]

Therefore,

[TABLE]

Plugging the bounds obtained for $A,B,C,D$ in Inequality (18) gives the announced result and ends the proof. $\Box$

6.4.3. Proof of Lemma 6.3

For any $\gamma^{\prime}\in\Gamma_{n}$,

[TABLE]

where, for any $k\in\{1,\dots,n\}$,

[TABLE]

Indeed,

[TABLE]

In order to apply Bernstein’s inequality to $g_{\gamma^{\prime}}(h_{1}(\gamma),X_{1}),\dots,g_{\gamma^{\prime}}(h_{n}(\gamma),X_{n})$, let us find suitable controls of

[TABLE]

On the one hand,

[TABLE]

as $\|K\|_{1}\geqslant 1$. On the other hand,

[TABLE]

Then, by Bernstein’s inequality, with probability larger than $1-2e^{-\lambda}$,

[TABLE]

with $c=7/6\|f\|_{\infty}\|K\|_{1}^{2}$. This is the announced inequality. $\Box$

6.4.4. Proof of Lemma 6.4

First of all,

[TABLE]

Then, by Lemma 6.3, with probability larger than $1-2e^{-\lambda}$,

[TABLE]

where

[TABLE]

and

[TABLE]

with, for any $k\in\{1,\dots,n\}$,

[TABLE]

and

[TABLE]

Since

[TABLE]

and

[TABLE]

by Bernstein’s inequality, with probability larger than $1-2e^{-\lambda}$,

[TABLE]

Moreover, by Jensen’s inequality,

[TABLE]

Then, by Lemma 6.2, with probability larger than $1-\kappa_{3}e^{-\lambda}$,

[TABLE]

Therefore, by Inequality (19), with probability larger than $1-\kappa_{5}e^{-\lambda}$,

[TABLE]

This is the announced result if we set $1+\varepsilon=1/(1-\theta)$, which gives $1/[\theta(1-\theta)]=(1+\varepsilon)^{2}/\varepsilon$. $\Box$