a note on the adaptive estimation of a multiplicative separable regression function

Research ArticleA Note on the Adaptive Estimation of a Multiplicative Separable Regression Function Christophe Chesneau Laboratoire de Math´ematiques Nicolas Oresme, Universit´e de Caen

Research Article

A Note on the Adaptive Estimation of a Multiplicative Separable Regression Function

Christophe Chesneau

Laboratoire de Math´ematiques Nicolas Oresme, Universit´e de Caen Basse-Normandie, Campus II, Science 3, 14032 Caen, France

Correspondence should be addressed to Christophe Chesneau; christophe.chesneau@gmail.com

Received 18 January 2014; Accepted 25 February 2014; Published 20 March 2014

Academic Editors: F Ding, E Skubalska-Rafajlowicz, and H C So

Copyright © 2014 Christophe Chesneau This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

We investigate the estimation of a multiplicative separable regression function from a bidimensional nonparametric regression model with random design We present a general estimator for this problem and study its mean integrated squared error (MISE) properties A wavelet version of this estimator is developed In some situations, we prove that it attains the standard unidimensional rate of convergence under the MISE over Besov balls

1 Motivations

We consider the bidimensional nonparametric regression

model with random design described as follows Let

(𝑌𝑖, 𝑈𝑖, 𝑉𝑖)𝑖∈Zbe a stochastic process defined on a probability

space(Ω, A, P), where

𝑌𝑖= ℎ (𝑈𝑖, 𝑉𝑖) + 𝜉𝑖, 𝑖 ∈ Z, (1) (𝜉𝑖)𝑖∈Zis a strictly stationary stochastic process,(𝑈𝑖, 𝑉𝑖)𝑖∈Zis

a strictly stationary stochastic process with support in[0, 1]2,

and ℎ : [0, 1]2 → R is an unknown bivariate regression

function It is assumed thatE(𝜉1) = 0, E(𝜉2

1) exists, (𝑈𝑖, 𝑉𝑖)𝑖∈Z are independent,(𝜉𝑖)𝑖∈Zare independent, and, for any𝑖 ∈ Z,

(𝑈𝑖, 𝑉𝑖) and 𝜉𝑖 are independent In this study, we focus our

attention on the case where ℎ is a multiplicative separable

regression function: there exist two functions𝑓 : [0, 1] → R

and𝑔 : [0, 1] → R such that

ℎ (𝑥, 𝑦) = 𝑓 (𝑥) 𝑔 (𝑦) (2)

We aim to estimate ℎ from the 𝑛 random variables:

(𝑌1, 𝑈1, 𝑉1), , (𝑌𝑛, 𝑈𝑛, 𝑉𝑛) This problem is plausible in

many practical situations as in utility, production, and cost

function applications (see, e.g., Linton and Nielsen [1],

Yatchew and Bos [2], Pinske [3], Lewbel and Linton [4], and Jacho-Ch´avez [5])

In this note, we provide a theoretical contribution to the subject by introducing a new general estimation method for

ℎ A sharp upper bound for its mean integrated squared error (MISE) is proved Then we adapt our methodology to propose an efficient and adaptive procedure It is based on two wavelet thresholding estimators following the construction studied in Chaubey et al [6] It has the features to be adaptive for a wide class of unknown functions and enjoy nice MISE properties Further details on wavelet estimators can be found in, for example, Antoniadis [7], Vidakovic [8], and H¨ardle et al [9] Despite the so-called “curse of dimensionality” coming from the bidimensionality of (1),

we prove that our wavelet estimator attains the standard unidimensional rate of convergence under the MISE over Besov balls (for both the homogeneous and inhomogeneous zones) It completes asymptotic results proved by Linton and Nielsen [1] via nonadaptive kernel methods for the structured nonparametric regression model

The paper is organized as follows Assumptions on (1 and some notations are introduced inSection 2 Section 3 presents our general MISE result Section 4 is devoted to our wavelet estimator and its performances in terms of rate

http://dx.doi.org/10.1155/2014/271303

of convergence under the MISE over Besov balls Technical

proofs are collected inSection 5

2 Assumptions and Notations

For any𝑝 ≥ 1, we set

L𝑝([0, 1])

= {V : [0, 1] 󳨀→ R; ‖V‖𝑝= (∫1

0 |V (𝑥)|𝑝𝑑𝑥)

1/𝑝

< ∞}

(3)

We set

𝑒𝑜= ∫1

0 𝑓 (𝑥) 𝑑𝑥, 𝑒∗ = ∫1

0 𝑔 (𝑥) 𝑑𝑥, (4) provided that they exist

We formulate the following assumptions

(H1) There exists a known constant𝐶1> 0 such that

sup

𝑥∈[0,1]󵄨󵄨󵄨󵄨𝑓(𝑥)󵄨󵄨󵄨󵄨 ≤ 𝐶1 (5) (H2) There exists a known constant𝐶2> 0 such that

sup

𝑥∈[0,1]󵄨󵄨󵄨󵄨𝑔(𝑥)󵄨󵄨󵄨󵄨 ≤ 𝐶2 (6) (H3) The density of(𝑈1, 𝑉1), denoted by 𝑞, is known and

there exists a constant𝑐3> 0 such that

𝑐3≤ inf (𝑥,𝑦)∈[0,1] 2𝑞 (𝑥, 𝑦) (7) (H4) There exists a known constant𝜔 > 0 such that

󵄨󵄨󵄨󵄨𝑒𝑜𝑒∗󵄨󵄨󵄨󵄨 ≥ 𝜔 (8) The assumptions (H1) and (H2), involving the boundedness

of ℎ, are standard in nonparametric regression models

The knowledge of 𝑞 discussed in (H3) is restrictive but

plausible in some situations, the most common case being

(𝑈1, 𝑉1) ∼ U([0, 1]2) (the uniform distribution on [0, 1]2)

Finally, mention that (H4) is just a technical assumption more

realistic to the knowledge of𝑒𝑜and𝑒∗(depending on𝑓 and

𝑔, resp.)

3 MISE Result

Theorem 1presents an estimator forℎ and shows an upper

bound for its MISE

Theorem 1 One considers (1) under (H1)–(H4) One

intro-duces the following estimator forℎ (2):

̂ℎ (𝑥, 𝑦) = 𝑓 (𝑥) ̃𝑔 (𝑦)̃

̃𝑒 1{|̃𝑒|≥𝜔/2}, (9)

where ̃ 𝑓 denotes an arbitrary estimator for 𝑓𝑒∗inL2([0, 1]), ̃𝑔

denotes an arbitrary estimator for𝑔𝑒𝑜inL2([0, 1]), 1 denotes

the indicator function,

̃𝑒 = 1𝑛∑𝑛

𝑖=1

𝑌𝑖

𝑞 (𝑈𝑖, 𝑉𝑖), (10)

and 𝜔 refers to (H4).

Then there exists a constant 𝐶 > 0 such that

E (∬1

0(̂ℎ (𝑥, 𝑦) − ℎ (𝑥, 𝑦))2𝑑𝑥 𝑑𝑦)

≤ 𝐶 (E (󵄩󵄩󵄩󵄩̃𝑔 − 𝑔𝑒𝑜󵄩󵄩󵄩󵄩2

2) + E (󵄩󵄩󵄩󵄩󵄩 ̃𝑓− 𝑓𝑒∗󵄩󵄩󵄩󵄩󵄩2

2) + E (󵄩󵄩󵄩󵄩̃𝑔 − 𝑔𝑒𝑜󵄩󵄩󵄩󵄩2

2󵄩󵄩󵄩󵄩󵄩𝑓 − 𝑓𝑒̃ ∗󵄩󵄩󵄩󵄩󵄩2

2) +1𝑛)

(11)

The form of ̃ℎ (9) is derived to the multiplicative separable structure of ℎ (2) and a ratio-type normalization Other results about such ratio-type estimators in a general statistical context can be found in Vasiliev [10]

Based on Theorem 1, ̂ℎ is efficient for ℎ if and only if

̃

𝑓 is efficient for 𝑓𝑒∗ and ̃𝑔 is efficient for 𝑔𝑒𝑜 in terms of MISE Even if several methods are possible, we focus our attention on wavelet methods enjoying adaptivity for a wide class of unknown functions and having optimal properties under the MISE For details on the interests of wavelet methods in nonparametric statistics, we refer to Antoniadis [7], Vidakovic [8], and H¨ardle et al [9]

4 Adaptive Wavelet Estimation

Before introducing our wavelet estimators, let us present some basics on wavelets

4.1 Wavelet Basis on [0, 1] Let us briefly recall the

con-struction of wavelet basis on the interval[0, 1] introduced by Cohen et al [11] Let𝑁 be a positive integer, and let 𝜙 and 𝜓

be the initial wavelets of the Daubechies orthogonal wavelets 𝑑𝑏2𝑁 We set

𝜙𝑗,𝑘(𝑥) = 2𝑗/2𝜙 (2𝑗𝑥 − 𝑘) , 𝜓𝑗,𝑘(𝑥) = 2𝑗/2𝜓 (2𝑗𝑥 − 𝑘)

(12) With appropriate treatments at the boundaries, there exists

an integer𝜏 satisfying 2𝜏 ≥ 2𝑁 such that the collection S = {𝜙𝜏,𝑘(⋅), 𝑘 ∈ {0, , 2𝜏−1}; 𝜓𝑗,𝑘(⋅); 𝑗 ∈ N−{0, , 𝜏−1}, 𝑘 ∈ {0, , 2𝑗− 1}}, is an orthonormal basis of L2([0, 1]) AnyV ∈ L2([0, 1]) can be expanded on S as

V (𝑥) =2

𝜏 −1

∑

𝑘=0

𝛼𝜏,𝑘𝜙𝜏,𝑘(𝑥) +∑∞

𝑗=𝜏

2 𝑗 −1

∑

𝑘=0

𝛽𝑗,𝑘𝜓𝑗,𝑘(𝑥) , 𝑥 ∈ [0, 1] ,

(13) where𝛼𝑗,𝑘and𝛽𝑗,𝑘are the wavelet coefficients ofV defined by

𝛼𝑗,𝑘= ∫1

0 V (𝑥) 𝜙𝑗,𝑘(𝑥) 𝑑𝑥, 𝛽𝑗,𝑘= ∫1

0 V (𝑥) 𝜓𝑗,𝑘(𝑥) 𝑑𝑥

(14)

4.2 Besov Balls For the sake of simplicity, we consider the

sequential version of Besov balls defined as follows Let𝑀 >

0, 𝑠 ∈ (0, 𝑁), 𝑝 ≥ 1 and 𝑟 ≥ 1 A function V belongs to 𝐵𝑠

𝑝,𝑟(𝑀)

if and only if there exists a constant𝑀∗ > 0 (depending on

𝑀) such that the associated wavelet coefficients (14) satisfy

2𝜏(1/2−1/𝑝)(2

𝜏 −1

∑

𝑘=0

|𝛼𝜏,𝑘|𝑝)

1/𝑝

+ (∑∞

𝑗=𝜏

(2𝑗(𝑠+1/2−1/𝑝)(2

𝑗 −1

∑

𝑘=0󵄨󵄨󵄨󵄨󵄨𝛽𝑗,𝑘󵄨󵄨󵄨󵄨󵄨𝑝

)

1/𝑝

)

𝑟

)

1/𝑟

≤ 𝑀∗ (15)

In this expression,𝑠 is a smoothness parameter and 𝑝 and

𝑟 are norm parameters For a particular choice of 𝑠, 𝑝, and

𝑟, 𝐵𝑠

𝑝,𝑟(𝑀) contains the H¨older and Sobolev balls (see, e.g.,

DeVore and Popov [12], Meyer [13], and H¨ardle et al [9])

4.3 Hard Thresholding Estimators In the sequel, we consider

() under (H1)–(H4)

We consider hard thresholding wavelet estimators for ̃𝑓

and ̃𝑔 in (9) They are based on a term-by-term selection

of estimators of the wavelet coefficients of the unknown

function Those which are greater to a threshold are kept; the

others are removed This selection is the key to the adaptivity

and the good performances of the hard thresholding wavelet

estimators (see, e.g., Donoho et al [14], Delyon and Juditsky

[15], and H¨ardle et al [9])

To be more specific, we use the “double thresholding”

wavelet technique, introduced by Delyon and Juditsky [15]

then recently improved by Chaubey et al [6] The role of

the second thresholding (appearing in the definition of the

wavelet estimator for𝛽𝑗,𝑘) is to relax assumption on the model

(seeRemark 6)

Estimator ̃ 𝑓 for 𝑓𝑒∗ We define the hard thresholding wavelet

estimator ̃𝑓 by

̃

𝑓 (𝑥) =2

𝜏 −1

∑

𝑘=0̂𝛼𝜏,𝑘𝜙𝜏,𝑘(𝑥) +∑𝑗1

𝑗=𝜏

2 𝑗 −1

∑

𝑘=0

̂

𝛽𝑗,𝑘1{| ̂𝛽𝑗,𝑘|≥𝜅𝐶∗𝜆𝑛}𝜓𝑗,𝑘(𝑥) ,

(16) where

̂𝛼𝜏,𝑘= 𝑎1

𝑛

𝑎 𝑛

∑

𝑖=1

𝑌𝑖

𝑞 (𝑈𝑖, 𝑉𝑖)𝜙𝜏,𝑘(𝑈𝑖) , (17) where𝑎𝑛is the integer part of𝑛/2,

̂

𝛽𝑗,𝑘= 1

𝑎𝑛

𝑎 𝑛

∑

𝑖=1

𝑊𝑖,𝑗,𝑘1{|𝑊𝑖,𝑗,𝑘|≤𝐶∗/𝜆𝑛},

𝑊𝑖,𝑗,𝑘= 𝑌𝑖

𝑞 (𝑈𝑖, 𝑉𝑖)𝜓𝑗,𝑘(𝑈𝑖) ,

(18)

where𝑗1 is the integer satisfying(1/2)𝑎𝑛 < 2𝑗 1 ≤ 𝑎𝑛, 𝜅 =

2 + 8/3 + 2√4 + 16/9, 𝐶∗= √(2/𝑐3)(𝐶2𝐶2+ E(𝜉2)), and

𝜆𝑛= √ln𝑎𝑛

Estimator ̃𝑔 for 𝑔𝑒𝑜 We define the hard thresholding wavelet

estimator ̃𝑔 by

̃𝑔(𝑥) =2

𝜏 −1

∑

𝑘=0

̂𝜐𝜏,𝑘𝜙𝜏,𝑘(𝑥) +∑𝑗2

𝑗=𝜏

2 𝑗 −1

∑

𝑘=0

̂𝜃𝑗,𝑘1{|̂𝜃𝑗,𝑘|≥𝜅∗𝐶∗𝜂𝑛}𝜓𝑗,𝑘(𝑥) ,

(20) where

̂𝜐𝜏,𝑘= 𝑏1

𝑛

𝑏 𝑛

∑

𝑖=1

𝑌𝑎𝑛+𝑖

𝑞 (𝑈𝑎𝑛+𝑖, 𝑉𝑎𝑛+𝑖)𝜙𝜏,𝑘(𝑉𝑎𝑛 +𝑖) , (21) where𝑎𝑛is the integer part of𝑛/2, 𝑏𝑛= 𝑛 − 𝑎𝑛,

̂𝜃𝑗,𝑘= 1

𝑏𝑛

𝑏𝑛

∑

𝑖=1

𝑍𝑎𝑛+𝑖,𝑗,𝑘1{|𝑍𝑎𝑛+𝑖,𝑗,𝑘|≤𝐶∗/𝜂𝑛},

𝑍𝑎𝑛+𝑖,𝑗,𝑘= 𝑌𝑎𝑛 +𝑖

𝑞 (𝑈𝑎𝑛+𝑖, 𝑉𝑎𝑛+𝑖)𝜓𝑗,𝑘(𝑉𝑎𝑛 +𝑖) ,

(22)

Where𝑗2is the integer satisfying(1/2)𝑏𝑛 < 2𝑗2 ≤ 𝑏𝑛,𝜅∗ =

2 + 8/3 + 2√4 + 16/9, 𝐶∗= √(2/𝑐3)(𝐶2

1𝐶2

2+ E(𝜉2

1)), and

𝜂𝑛 = √ln𝑏𝑛

Estimator for ℎ From ̃𝑓 (16) and ̃𝑔 (20), we consider the following estimator forℎ (2):

̂ℎ (𝑥, 𝑦) = 𝑓 (𝑥) ̃𝑔(𝑦)̃ ̃𝑒 1{|̃𝑒|≥𝜔/2}, (24) where

̃𝑒 = 1𝑛∑𝑛

𝑖=1

𝑌𝑖

𝑞 (𝑈𝑖, 𝑉𝑖) (25) and𝜔 refers to (H4)

Let us mention that ̃ℎ is adaptive in the sense that it does not depend on𝑓 or 𝑔 in its construction

Remark 2 Since 𝑓̃ is defined with (𝑌1, 𝑈1,

𝑉1), , (𝑌𝑎𝑛, 𝑈𝑎𝑛, 𝑉𝑎𝑛) and ̃𝑔 is defined with (𝑌𝑎𝑛+1,

𝑈𝑎𝑛+1, 𝑉𝑎𝑛+1), , (𝑌𝑛, 𝑈𝑛, 𝑉𝑛), thanks to the independence of (𝑌1, 𝑈1, 𝑉1), , (𝑌𝑛, 𝑈𝑛, 𝑉𝑛), ̃𝑓 and ̃𝑔 are independent

Remark 3 The calibration of the parameters in ̃𝑓 and ̃𝑔 is based on theoretical considerations; thus defined, ̃𝑓 and ̃𝑔can attain a fast rate of convergence under the MISE over Besov balls (see [6], Theorem 6.1]) Further details are given in the proof ofTheorem 4

4.4 Rate of Convergence Theorem 4investigates the rate of

convergence attains by ̂ℎ under the MISE over Besov balls

Theorem 4 We consider (1) under (H1)–(H4) Let ̂ ℎ be (24)

and let ℎ be (2) Suppose that

(i)𝑓 ∈ 𝐵𝑠 1

𝑝 1 ,𝑟 1(𝑀1) with 𝑀1 > 0, 𝑟1 ≥ 1, either {𝑝1 ≥ 2

and𝑠1∈ (0, 𝑁)} or {𝑝1∈ [1, 2) and 𝑠1∈ (1/𝑝1, 𝑁)},

(ii)𝑔 ∈ 𝐵𝑠 2

𝑝2,𝑟2(𝑀2) with 𝑀2 > 0, 𝑟2 ≥ 1, either {𝑝2 ≥ 2

and𝑠2∈ (0, 𝑁)} or {𝑝2∈ [1, 2) and 𝑠2∈ (1/𝑝2, 𝑁)}.

Then there exists a constant 𝐶 > 0 such that

E (∬1

0(̂ℎ (𝑥, 𝑦) − ℎ (𝑥, 𝑦))2𝑑𝑥 𝑑𝑦) ≤𝐶(ln𝑛

𝑛 )

2𝑠 ∗ /(2𝑠 ∗ +1)

, (26)

where𝑠∗ = min(𝑠1, 𝑠2).

The rate of convergence(ln 𝑛/𝑛)2𝑠 ∗ /(2𝑠 ∗ +1)is the near

opti-mal one in the minimax sense for the unidimensional

regres-sion model with random design under the MISE over Besov

balls𝐵𝑠 ∗

𝑝,𝑟(𝑀) (see, e.g., Tsybakov [16], and H¨ardle et al [9])

ThusTheorem 4proves that our estimator escapes to the

so-called “curse of dimensionality.” Such a result is not possible

with the standard bidimensional hard thresholding wavelet

estimator attaining the rate of convergence (ln 𝑛/𝑛)2𝑠/(2𝑠+𝑑)

with𝑑 = 2 under the MISE over bidimensional Besov balls

defined with 𝑠 as smoothness parameter (see Delyon and

Juditsky [15])

Theorem 4completes asymptotic results proved by

Lin-ton and Nielsen [1] investigating this problem for the

struc-tured nonparametric regression model via another

estima-tion method based on nonadaptive kernels

Remark 5 In Theorem 4, we take into account both the

homogeneous zone of Besov balls, that is,{𝑝1 ≥ 2 and 𝑠1 ∈

(0, 𝑁)}, and the inhomogeneous zone, that is, {𝑝1∈ [1, 2) and

𝑠1 ∈ (1/𝑝1, 𝑁)}, for the case 𝑓 ∈ 𝐵𝑠1

𝑝 1 ,𝑟 1(𝑀1) and the same for

𝑔 ∈ 𝐵𝑠 2

𝑝 2 ,𝑟 2(𝑀2) This has the advantage to cover a very rich

class of unknown regression functionsℎ

Remark 6 Note thatTheorem 4does not require the

knowl-edge of the distribution of𝜉1;{E(𝜉1) = 0 and the existence of

E(𝜉2

1)} is enough

Remark 7 Let us mention that the phenomenon of curse of

dimensionality has also been studied via wavelet methods

by Neumann [17] but for the multidimensional Gaussian

white noise model and with different approaches based on

anysotropic frameworks

Remark 8 Our study can be extended to the

multidimen-sional case considered by Yatchew and Bos [2], that is,𝑓 :

[0, 1]𝑞 1 → R and 𝑔 : [0, 1]𝑞 2 → R; 𝑞1 and𝑞2 denoting

two positive integers In this case, adapting our framework

to the multidimensional case (𝑞1 dimensional Besov balls,

𝑞1 dimensional (tensorial) wavelet basis, 𝑞1 dimensional

hard thresholding wavelet estimator, see, e.g, Delyon and

Juditsky [15]), one can prove that (9) attains the rate of convergence(ln 𝑛/𝑛)2𝑠∗ /(2𝑠∗+𝑞∗), where𝑠∗ = min(𝑠1, 𝑠2) and

𝑞∗= max(𝑞1, 𝑞2)

5 Proofs

In this section, for the sake of simplicity,𝐶 denotes a generic constant; its value may change from one term to another

Proof of Theorem 1 Observe that

̂ℎ (𝑥, 𝑦) − ℎ (𝑥, 𝑦) =𝑓 (𝑥) ̃𝑔 (𝑦)̃

̃𝑒 1{|̃𝑒|≥𝜔/2}− 𝑓 (𝑥) 𝑔 (𝑦)

=1

̃𝑒( ̃𝑓 (𝑥) ̃𝑔 (𝑦) − 𝑓 (𝑥) 𝑔 (𝑦) ̃𝑒) 1{|̃𝑒|≥𝜔/2}

− 𝑓 (𝑥) 𝑔 (𝑦) 1{|̃𝑒|<𝜔/2}

(27) Therefore, using the triangular inequality, the Markov ine-quality, (H1), (H2), (H4), {|̃𝑒| < 𝜔/2} ∩ {|𝑒∗𝑒𝑜| ≥ 𝜔} ⊆ {|̃𝑒 − 𝑒∗𝑒𝑜| ≥ 𝜔/2}, and again the Markov inequality, we get

󵄨󵄨󵄨󵄨

󵄨̂ℎ (𝑥, 𝑦) − ℎ (𝑥, 𝑦)󵄨󵄨󵄨󵄨󵄨

≤ 2

𝜔󵄨󵄨󵄨󵄨󵄨𝑓 (𝑥) ̃𝑔 (𝑦) − 𝑓 (𝑥) 𝑔 (𝑦) ̃𝑒̃ 󵄨󵄨󵄨󵄨󵄨 +󵄨󵄨󵄨󵄨𝑓(𝑥)󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨𝑔(𝑦)󵄨󵄨󵄨󵄨1{|̃𝑒|<𝜔/2}

≤ 𝐶 (󵄨󵄨󵄨󵄨󵄨 ̃𝑓(𝑥) ̃𝑔(𝑦) − 𝑓(𝑥)𝑔(𝑦) ̃𝑒󵄨󵄨󵄨󵄨󵄨 + 1{|̃𝑒−𝑒 ∗ 𝑒 𝑜 |≥𝜔/2})

≤ 𝐶 (󵄨󵄨󵄨󵄨󵄨 ̃𝑓(𝑥) ̃𝑔(𝑦) − 𝑓(𝑥)𝑔(𝑦) ̃𝑒󵄨󵄨󵄨󵄨󵄨 +󵄨󵄨󵄨󵄨̃𝑒− 𝑒∗𝑒𝑜󵄨󵄨󵄨󵄨)

(28)

On the other hand, we have the decomposition

̃

𝑓 (𝑥) ̃𝑔(𝑦) − 𝑓 (𝑥) 𝑔 (𝑦) ̃𝑒

= 𝑓 (𝑥) 𝑒∗( ̃𝑔 (𝑦) − 𝑔 (𝑦) 𝑒𝑜) + 𝑔 (𝑦) 𝑒𝑜( ̃𝑓 (𝑥) − 𝑓 (𝑥) 𝑒∗) + ( ̃𝑔 (𝑦) − 𝑔 (𝑦) 𝑒𝑜) ( ̃𝑓 (𝑥) − 𝑓 (𝑥) 𝑒∗)

+ 𝑓 (𝑥) 𝑔 (𝑦) (𝑒∗𝑒𝑜− ̃𝑒)

(29) Owing to the triangular inequality, (H1) and (H2), we have

󵄨󵄨󵄨󵄨

󵄨𝑓 (𝑥) ̃𝑔 (𝑦) − 𝑓 (𝑥) 𝑔 (𝑦) ̃𝑒̃ 󵄨󵄨󵄨󵄨󵄨

≤ 𝐶 (󵄨󵄨󵄨󵄨̃𝑔(𝑦) − 𝑔 (𝑦) 𝑒𝑜󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨󵄨 ̃𝑓(𝑥)−𝑓(𝑥)𝑒∗󵄨󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨̃𝑔(𝑦) − 𝑔 (𝑦) 𝑒𝑜󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 ̃𝑓(𝑥)−𝑓(𝑥)𝑒∗󵄨󵄨󵄨󵄨󵄨 +󵄨󵄨󵄨󵄨̃𝑒− 𝑒∗𝑒𝑜󵄨󵄨󵄨󵄨)

(30) Putting (28) and (30) together, we obtain

󵄨󵄨󵄨󵄨

󵄨̂ℎ (𝑥, 𝑦) − ℎ (𝑥, 𝑦)󵄨󵄨󵄨󵄨󵄨

≤ 𝐶 (󵄨󵄨󵄨󵄨̃𝑔(𝑦) − 𝑔 (𝑦) 𝑒𝑜󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨󵄨 ̃𝑓(𝑥)−𝑓(𝑥)𝑒∗󵄨󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨̃𝑔(𝑦) − 𝑔 (𝑦) 𝑒𝑜󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 ̃𝑓(𝑥)−𝑓(𝑥)𝑒∗󵄨󵄨󵄨󵄨󵄨 +󵄨󵄨󵄨󵄨̃𝑒− 𝑒∗𝑒𝑜󵄨󵄨󵄨󵄨)

(31)

Therefore, by the elementary inequality:(𝑎 + 𝑏 + 𝑐 + 𝑑)2 ≤

8 (𝑎2 + 𝑏2 + 𝑐2 + 𝑑2), (𝑎, 𝑏, 𝑐, 𝑑) ∈ R4, an integration over

[0, 1]2and taking the expectation, it comes

E (∬1

0(̂ℎ (𝑥, 𝑦) − ℎ (𝑥, 𝑦))2𝑑𝑥 𝑑𝑦)

≤ 𝐶 (E (󵄩󵄩󵄩󵄩̃𝑔 − 𝑔𝑒𝑜󵄩󵄩󵄩󵄩2

2) + E (󵄩󵄩󵄩󵄩󵄩 ̃𝑓− 𝑓𝑒∗󵄩󵄩󵄩󵄩󵄩2

2) + E (󵄩󵄩󵄩󵄩̃𝑔 − 𝑔𝑒𝑜󵄩󵄩󵄩󵄩2

2󵄩󵄩󵄩󵄩󵄩𝑓 − 𝑓𝑒̃ ∗󵄩󵄩󵄩󵄩󵄩2

2) + E ((̃𝑒− 𝑒∗𝑒𝑜)2))

(32) Now observe that, owing to the independence of(𝑈𝑖, 𝑉𝑖)𝑖∈Z,

the independence between(𝑈1, 𝑉1) and 𝜉1, andE(𝜉1) = 0, we

obtain

E (̃𝑒) = E (𝑞 (𝑈𝑌1

1, 𝑉1))

= E (ℎ (𝑈1, 𝑉1)

𝑞 (𝑈1, 𝑉1)) + E (𝜉1) E (

1

𝑞 (𝑈1, 𝑉1))

= ∬1

0

𝑓 (𝑥) 𝑔 (𝑦)

𝑞 (𝑥, 𝑦) 𝑞 (𝑥, 𝑦) 𝑑𝑥 𝑑𝑦

= (∫1

0 𝑓 (𝑥) 𝑑𝑥) (∫1

0 𝑔 (𝑦) 𝑑𝑦) = 𝑒∗𝑒𝑜

(33)

Then, using similar arguments to (33),(𝑎 + 𝑏)2≤2(𝑎2+ 𝑏2),

(𝑎, 𝑏) ∈ R2, (H1), (H2), (H3), andE(𝜉21) < ∞, we have

E ((̃𝑒− 𝑒∗𝑒𝑜)2) = V (̃𝑒)

= 1𝑛V (𝑞 (𝑈𝑌1

1, 𝑉1))

≤ 1

𝑛E ((

𝑌1

𝑞 (𝑈1, 𝑉1))

2

)

≤ 2

𝑛E (

(ℎ (𝑈1, 𝑉1))2+ 𝜉2

1

(𝑞 (𝑈1, 𝑉1))2 )

≤ 2

𝑐2(𝐶21𝐶22+ E (𝜉21))1𝑛 = 𝐶𝑛1

(34)

Equations (32) and (34) yield the desired inequality:

E (∬1

0(̂ℎ (𝑥, 𝑦) − ℎ (𝑥, 𝑦))2𝑑𝑥 𝑑𝑦)

≤ 𝐶 (E (󵄩󵄩󵄩󵄩̃𝑔 − 𝑔𝑒𝑜󵄩󵄩󵄩󵄩2

2) + E (󵄩󵄩󵄩󵄩󵄩 ̃𝑓− 𝑓𝑒∗󵄩󵄩󵄩󵄩󵄩2

2) + E (󵄩󵄩󵄩󵄩̃𝑔 − 𝑔𝑒𝑜󵄩󵄩󵄩󵄩2

2󵄩󵄩󵄩󵄩󵄩𝑓 − 𝑓𝑒̃ ∗󵄩󵄩󵄩󵄩󵄩2

2) +1

𝑛)

(35)

Proof of Theorem 4 We aim to applyTheorem 1by

investigat-ing the rate of convergence attained by ̃𝑓 and ̃𝑔 under the

MISE over Besov balls

First of all, remark that, for𝛾 ∈ {𝜙, 𝜓}, any integer 𝑗 ≥ 𝜏 and any𝑘 ∈ {0, , 2𝑗− 1}

(i) Using similar arguments to (33), we obtain

E (1

𝑎𝑛

𝑎 𝑛

∑

𝑖=1

𝑌𝑖

𝑞 (U𝑖, 𝑉𝑖)𝛾𝑗,𝑘(𝑈𝑖))

= E ( 𝑌1

𝑞 (𝑈1, 𝑉1)𝛾𝑗,𝑘(𝑈1))

= E (ℎ (𝑈1, 𝑉1)

𝑞 (𝑈1, 𝑉1)𝛾𝑗,𝑘(𝑈1)) + E (𝜉1) E (

𝛾𝑗,𝑘(𝑈1)

𝑞 (𝑈1, 𝑉1))

= ∬1

0

𝑓 (𝑥) 𝑔 (𝑦)

𝑞 (𝑥, 𝑦) 𝛾𝑗,𝑘(𝑥) 𝑞 (𝑥, 𝑦) 𝑑𝑥 𝑑𝑦

= (∫1

0 𝑓 (𝑥) 𝛾𝑗,𝑘(𝑥) 𝑑𝑥) (∫1

0 𝑔 (𝑦) 𝑑𝑦)

= ∫1

0 (𝑓 (𝑥) 𝑒∗) 𝛾𝑗,𝑘(𝑥) 𝑑𝑥

(36)

(ii) Using similar arguments to (34) and‖𝛾𝑗,𝑘‖22 = 1, we have

𝑎𝑛

∑

𝑖=1

E (( 𝑌𝑖

𝑞 (𝑈𝑖, 𝑉𝑖)𝛾𝑗,𝑘(𝑈𝑖))

2

)

= E (( 𝑌1

𝑞 (𝑈1, 𝑉1)𝛾𝑗,𝑘(𝑈1))

2

) 𝑎𝑛

≤ 2E ((ℎ (𝑈1, 𝑉1))

2+ 𝜉2 1

(𝑞 (𝑈1, 𝑉1))2 (𝛾𝑗,𝑘(𝑈1))

2

) 𝑎𝑛

≤ 𝑐2

3(𝐶2

1𝐶2

2+ E (𝜉2

1)) E ((𝛾𝑗,𝑘(𝑈1))

2

𝑞 (𝑈1, 𝑉1) ) 𝑎𝑛

= 2

𝑐3(𝐶12𝐶2

2+ E (𝜉2

1)) ∬1

0

(𝛾𝑗,𝑘(𝑥))2

𝑞 (𝑥, 𝑦) 𝑞 (𝑥, 𝑦) 𝑑𝑥 𝑑𝑦𝑎𝑛

= 2

𝑐3(𝐶12𝐶2

2+ E (𝜉2

1)) 󵄩󵄩󵄩󵄩󵄩𝛾𝑗,𝑘󵄩󵄩󵄩󵄩󵄩2

2𝑎𝑛= 𝐶2

∗𝑎𝑛,

(37) with𝐶2

∗= (2/𝑐3)(𝐶2

1𝐶2

2+ E(𝜉2

1))

Applying [6, Theorem 6.1] (see the Appendix) with𝑛 =

𝜇𝑛= 𝜐𝑛= 𝑎𝑛,𝛿 = 0, 𝜃𝛾= 𝐶∗,𝑊𝑖= (𝑌𝑖, 𝑈𝑖, 𝑉𝑖),

𝑞𝑖(𝛾, (𝑦, 𝑥, 𝑤)) = 𝑦

𝑞 (𝑥, 𝑤)𝛾 (𝑥) (38)

and𝑓 ∈ 𝐵𝑠 1

𝑝 1 ,𝑟 1(𝑀1) (so 𝑓𝑒∗ ∈ 𝐵𝑠 1

𝑝 1 ,𝑟 1(𝑀1𝑒∗)) with 𝑀1 > 0,

𝑟1 ≥ 1, either {𝑝1 ≥ 2 and 𝑠1 ∈ (0, 𝑁)} or {𝑝1 ∈ [1, 2) and

𝑠1 ∈ (1/𝑝1, 𝑁)}, we prove the existence of a constant 𝐶 > 0

such that

E (󵄩󵄩󵄩󵄩󵄩 ̃𝑓− 𝑓𝑒∗󵄩󵄩󵄩󵄩󵄩2

2) ≤ 𝐶 (ln𝑎𝑛

𝑎𝑛 )

2𝑠 1 /(2𝑠 1 +1)

≤𝐶(ln𝑛𝑛)2𝑠1/(2𝑠1+1),

(39)

when𝑛 is large enough

The MISE of ̃𝑔 can be investigated in a similar way: for

𝛾 ∈ {𝜙, 𝜓}, any integer 𝑗 ≥ 𝜏 and any 𝑘 ∈ {0, , 2𝑗− 1}

(i) We show that

E (𝑏1

𝑛

𝑏 𝑛

∑

𝑖=1

𝑌𝑎𝑛+𝑖

𝑞 (𝑈𝑎𝑛+𝑖, 𝑉𝑎𝑛+𝑖)𝛾𝑗,𝑘(𝑉𝑎𝑛 +𝑖))

= ∫1

0 (𝑔 (𝑥) 𝑒𝑜) 𝛾𝑗,𝑘(𝑥) 𝑑𝑥

(40)

(ii) We show that

𝑏 𝑛

∑

𝑖=1

E (( 𝑌𝑎𝑛 +𝑖

𝑞 (𝑈𝑎𝑛+𝑖, 𝑉𝑎𝑛+𝑖)𝛾𝑗,𝑘(𝑉𝑎𝑛 +𝑖))

2

) ≤ 𝐶2∗𝑏𝑛, (41) with always𝐶2∗ = (2/𝑐3)(𝐶21𝐶22+ E(𝜉21))

Applying again [6, Theorem 6.1] (see the Appendix) with

𝑛 = 𝜇𝑛 = 𝜐𝑛 = 𝑏𝑛,𝛿 = 0, 𝜃𝛾= 𝐶∗,𝑊𝑖= (𝑌𝑖, 𝑈𝑖, 𝑉𝑖),

𝑞𝑖(𝛾, (𝑦, 𝑥, 𝑤)) = 𝑦

𝑞 (𝑥, 𝑤)𝛾 (𝑤) (42) and𝑔 ∈ 𝐵𝑠2

𝑝2,𝑟2(𝑀2) with 𝑀2 > 0, 𝑟2 ≥ 1, either {𝑝2 ≥ 2 and

𝑠2 ∈ (0, 𝑁)} or {𝑝2 ∈ [1, 2) and 𝑠2 ∈ (1/𝑝2, 𝑁)}; we prove the

existence of a constant𝐶 > 0 such that

E (󵄩󵄩󵄩󵄩̃𝑔 − 𝑔𝑒𝑜󵄩󵄩󵄩󵄩2

2) ≤ 𝐶(ln𝑏𝑏𝑛

𝑛 )2𝑠2/(2𝑠2+1) ≤𝐶(ln𝑛𝑛)2𝑠2/(2𝑠2+1),

(43)

when𝑛 is large enough

Using the independence between ̃𝑓 and ̃𝑔 (seeRemark 2),

it follows from (39) and (43) that

E (󵄩󵄩󵄩󵄩̃𝑔 − 𝑔𝑒𝑜󵄩󵄩󵄩󵄩2

2󵄩󵄩󵄩󵄩󵄩𝑓 − 𝑓𝑒̃ ∗󵄩󵄩󵄩󵄩󵄩2

2) = E (󵄩󵄩󵄩󵄩̃𝑔 − 𝑔𝑒𝑜󵄩󵄩󵄩󵄩2

2) E (󵄩󵄩󵄩󵄩󵄩 ̃𝑓− 𝑓𝑒∗󵄩󵄩󵄩󵄩󵄩2

2)

≤𝐶(ln𝑛𝑛)4𝑠1𝑠2/(2𝑠1+1)(2𝑠2+1)

(44)

Owing toTheorem 1, (39), (43) and (44), we get

E (∬1

0(̂ℎ (𝑥, 𝑦) − ℎ (𝑥, 𝑦))2𝑑𝑥 𝑑𝑦)

≤ 𝐶 (E (󵄩󵄩󵄩󵄩̃𝑔 − 𝑔𝑒𝑜󵄩󵄩󵄩󵄩2

2) + E (󵄩󵄩󵄩󵄩󵄩 ̃𝑓− 𝑓𝑒∗󵄩󵄩󵄩󵄩󵄩2

2) + E (󵄩󵄩󵄩󵄩̃𝑔 − 𝑔𝑒𝑜󵄩󵄩󵄩󵄩2

2󵄩󵄩󵄩󵄩󵄩𝑓 − 𝑓𝑒̃ ∗󵄩󵄩󵄩󵄩󵄩2

2) +1𝑛)

≤ 𝐶 ((ln𝑛𝑛)2𝑠2/(2𝑠2+1)+ (ln𝑛𝑛)2𝑠1/(2𝑠1+1) +(ln𝑛𝑛)4𝑠1𝑠2/(2𝑠1+1)(2𝑠2+1)+1𝑛)

≤𝐶(ln𝑛

𝑛 )

2𝑠 ∗ /(2𝑠 ∗ +1)

,

(45)

with𝑠∗= min(𝑠1, 𝑠2)

Theorem 4is proved

Appendix

Let us now present in detail [6, Theorem 6.1] which is used two times in the proof ofTheorem 4

We consider a general form of the hard thresholding wavelet estimator denoted by ̂𝑓𝐻for estimating an unknown function𝑓 ∈ L2([0, 1]) from 𝑛 independent random variables

𝑊1, , 𝑊𝑛:

̂

𝑓𝐻(𝑥) =2

𝜏 −1

∑

𝑘=0

̂𝛼𝜏,𝑘𝜙𝜏,𝑘(𝑥) +

𝑗1

∑

𝑗=𝜏

2 𝑗 −1

∑

𝑘=0

̂

𝛽𝑗,𝑘1{| ̂𝛽

𝑗,𝑘 |≥𝜅𝜗𝑗}𝜓𝑗,𝑘(𝑥) ,

(A.1) where

̂𝛼𝑗,𝑘= 1

𝜐𝑛

𝑛

∑

𝑖=1

𝑞𝑖(𝜙𝑗,𝑘, 𝑊𝑖) ,

̂

𝛽𝑗,𝑘= 1

𝜐𝑛

𝑛

∑

𝑖=1

𝑞𝑖(𝜓𝑗,𝑘, 𝑊𝑖) 1{|𝑞𝑖(𝜓𝑗,𝑘,𝑊𝑖)|≤𝜍𝑗},

𝜍𝑗= 𝜃𝜓2𝛿𝑗 𝜐𝑛

√𝜇𝑛ln𝜇𝑛, 𝜗𝑗 = 𝜃𝜓2

𝛿𝑗√ln𝜇𝑛

𝜇𝑛 ,

(A.2)

𝜅 ≥ 2 + 8/3 + 2√4 + 16/9 and 𝑗1is the integer satisfying

1

2𝜇𝑛1/(2𝛿+1)< 2𝑗1≤ 𝜇𝑛1/(2𝛿+1) (A.3) Here, we suppose that there exist

(i)𝑛 functions 𝑞1, , 𝑞𝑛with𝑞𝑖 : L2([0, 1]) × 𝑊𝑖(Ω) →

C for any 𝑖 ∈ {1, , 𝑛}, (ii) two sequences of real numbers (𝜐𝑛)𝑛∈N and(𝜇𝑛)𝑛∈N satisfying lim𝑛 → ∞𝜐𝑛 = ∞ and lim𝑛 → ∞𝜇𝑛= ∞,

such that, for𝛾 ∈ {𝜙, 𝜓},

(A1) any integer𝑗 ≥ 𝜏 and any 𝑘 ∈ {0, , 2𝑗− 1},

E (1

𝜐𝑛

𝑛

∑

𝑖=1

𝑞𝑖(𝛾𝑗,𝑘, 𝑊𝑖)) = ∫1

0 𝑓 (𝑥) 𝛾𝑗,𝑘(𝑥) 𝑑𝑥 (A.4)

(A2) there exist two constants,𝜃𝛾> 0 and 𝛿 ≥ 0, such that,

for any integer𝑗 ≥ 𝜏 and any 𝑘 ∈ {0, , 2𝑗− 1},

𝑛

∑

𝑖=1E (󵄨󵄨󵄨󵄨󵄨𝑞𝑖(𝛾𝑗,𝑘, 𝑊𝑖)󵄨󵄨󵄨󵄨󵄨2) ≤ 𝜃2

𝛾22𝛿𝑗𝜐𝑛2

𝜇𝑛. (A.5)

Let ̂𝑓𝐻 be (A.1) under (A1) and (A2) Suppose that 𝑓 ∈

𝐵𝑠𝑝,𝑟(𝑀) with 𝑟 ≥ 1, {𝑝 ≥ 2 and 𝑠 ∈ (0, 𝑁)} or {𝑝 ∈ [1, 2)

and𝑠 ∈ ((2𝛿 + 1)/𝑝, 𝑁)} Then there exists a constant 𝐶 > 0

such that

E (‖ ̂𝑓𝐻− 𝑓‖22) ≤ 𝐶(ln𝜇𝜇𝑛

𝑛 )2𝑠/(2𝑠+2𝛿+1) (A.6)

Conflict of Interests

The author declares that there is no conflict of interests

regarding the publication of this paper

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