On adaptive wavelet estimation of the regression function and its derivatives in an errors-in-variables model

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function and its derivatives in an errors-in-variables model

Christophe Chesneau

To cite this version:

Christophe Chesneau. On adaptive wavelet estimation of the regression function and its derivatives

in an errors-in-variables model. 2010. �hal-00466830v2�

in an errors-in-variables model

Christophe Chesneau

University of Caen (LMNO), Caen, France. Email: chesneau@math.unicaen.fr

Abstract

We consider a regression model with errors-in-variables:(Y, X), whereY =f(Z) + ξ and X = Z +W. Our goal is to estimate the unknown regression functionf and its derivatives under mild assumptions onξ(only finite moments of order2are required). To reach this goal, we develop a new adaptive wavelet estimator based on a hard thresholding rule. Taking the minimax approach under the mean integrated squared error over Besov balls, we prove that it attains a sharp rate of convergence.

Keywords: Errors-in-variables model, Derivatives function estimation, Minimax ap- proach, Wavelets, Hard thresholding.

2000 Mathematics Subject Classification:62G07, 62G20.

1 Motivations

We observe nindependent pairs of random variables (Y1, X1), . . . ,(Yn, Xn)in the following errors-in-variables model: for anyv∈ {1, . . . , n},

( Yv =f(Zv) +ξv,

X_v=Z_v+W_v, (1.1)

wheref is an unknown function, X1, . . . , Xn are n i.i.d. random variables having the uniform distribution on[0,1],W₁, . . . , W_nareni.i.d. unobserved random variables and ξ1, . . . , ξn areni.i.d. unobserved zero mean random variables. We assume that all these random variables are independent, the density function ofW₁, denotedg, is known and ξ1 admits finite moments of order 2. We aim to estimate f and its derivatives from (Y₁, X₁), . . . ,(Y_n, X_n).

The estimation of f from the model (1.1) has received a lot of attention. See e.g.

[6, 11–13, 15, 18, 19]. In this paper, we focus on a more general problem: the adaptive estimation of thed-th derivative off: f^(d). This is of interest to detect possible bumps, concavity or convexity properties of f. The estimation off^(d) has been investigated in

This work is supported by ANR grant NatImages, ANR-08-EMER-009

several papers for various models (but never (1.1)) starting with [1]. For references using wavelet methods, see e.g. [3–5, 23].

Another feature of the study concernsξ1, . . . , ξn: to estimatef^(d)(includingf =f⁽⁰⁾), we only assume thatξ₁has finite moments of order2. And thus we do not need to know the distribution ofξ1. Moreover, this relaxes the assumption onξ1in [6] where finite moments of order>6are required.

Under this general framework, considering the ordinary smooth case ong(see (2.2)), we estimatef^(d) by a new wavelet estimator based on a hard thresholding rule. It has the originality to combine a singular value decomposition (SVD) approach similar to the one of [17] and some technical tools introduced in wavelet estimation theory by [7].

We evaluate its performances by taking the minimax approach under the mean inte- grated squared error (MISE) over a wide class of functions: the Besov balls B_p,r^s (M) (to be defined in Section 3). We prove that our estimator attains the rate of convergence vn = (lnn/n)2s/(2s+2δ+2d+1)

, whereδ is a factor related to the ordinary smooth case.

This rate of convergence is sharp in the sense that it is the one attains by the best non- realistic linear wavelet estimator up to a logarithmic term.

The paper is organized as follows. Assumptions on the model and some notations are introduced in Section 2. Section 3 briefly describes the periodized wavelet basis on[0,1]

and the Besov balls. The estimators are presented in Section 4. The results are set in Section 5. The proofs are gathered in Section 6.

2 Assumptions and notations

We assume in the sequel thatf^(d)andgbelong toL²per([0,1]), the space of periodic functions of period one that are square-integrable on[0,1]:

L²per([0,1]) = (

h; his1-periodic andkhk2= Z 1

0

h²(x)dx ^1/2

<∞ )

. We assume that there exists a known constantC_∗>0such that

kfk∞= sup

x∈[0,1]

|f(x)| ≤C_∗<∞. (2.1) Any functionh∈L²per([0,1])can be represented by its Fourier series

h(t) =X

`∈Z

F(h)(`)e<sup>2iπ`t</sup>, t∈[0,1],

where the equality is intended in mean-square convergence sense, andF(h)(`)denotes the Fourier coefficient given by

F(h)(`) = Z 1

0

h(x)e<sup>−2iπ`x</sup>dx, `∈Z,

whenever this integral exists. The notation · will be used for the complex conjugate.

We consider the ordinary smooth case ong: there exist three constants,c_g >0,C_g>0 andδ >1, such that, for any`∈Z, the Fourier coefficient ofg, i.e.F(g)(`), satisfies

cg

(1 +`<sup>2</sup>)<sup>δ/2</sup> ≤ | F(g)(`)| ≤ Cg

(1 +`²)^δ/2. (2.2)

This assumption controls the decay of the Fourier coefficients ofg, and thus the smoothness ofg. It is a standard hypothesis usually adopted in the field of nonparametric estimation for deconvolution problems. See e.g. [14, 17, 21].

3 Wavelets and Besov balls

3.1 Periodized Meyer Wavelets

We consider an orthonormal wavelet basis generated by dilations and translations of a

”father” Meyer-type waveletφand a ”mother” Meyer-type waveletψ. The main features of such wavelets are:

1. they are bandlimited, i.e. the Fourier transforms ofφandψhave compact supports respectively included in[−4π/3,4π/3]and[−8π/3,−2π/3]∪[2π/3,8π/3].

2. for any frequency in[−2π,−π]∪[π,2π], there exists a constantc >0such that the magnitude of the Fourier transform ofψis lower bounded byc.

3. the functions(φ, ψ)areC^∞as their Fourier transforms have a compact support, and ψhas an infinite number of vanishing moments as its Fourier transform vanishes in a neighborhood of the origin, i.e. for anyu∈N,R∞

−∞x^uψ(x)dx= 0.

For the purpose of this paper, we use the periodized Meyer wavelet bases on the unit interval. For anyx∈[0,1], any integerjand anyk∈ {0, . . . ,2^j−1}, let

φ_j,k(x) = 2^j/2φ(2^jx−k), ψ_j,k(x) = 2^j/2ψ(2^jx−k) be the elements of the wavelet basis, and

φ^per_j,k(x) =X

l∈Z

φj,k(x−l), ψ^per_j,k(x) =X

l∈Z

ψj,k(x−l),

their periodized versions. There exists an integer j_∗ such that the collection B = {φ^per_j

∗,k, k ∈ {0, . . . ,2^j∗−1}; ψ_j,k^per, j ∈ N− {0, . . . , j_∗−1}, k ∈ {0, . . . ,2^j −1}}

forms an orthonormal basis ofL²per([0,1]). In what follows, the superscript ”per” will be dropped to lighten the notation.

Letjc be an integer such thatjc ≥j_∗. A functionh∈ L²per([0,1])can be expanded into a wavelet series as

h(x) =

2^jc−1

X

k=0

α_jc,kφ_jc,k(x) +

∞

X

j=j_c 2^j−1

X

k=0

β_j,kψ_j,k(x), x∈[0,1], where

αj,k= Z 1

0

h(x)φ_j,k(x)dx, βj,k= Z 1

0

h(x)ψ_j,k(x)dx. (3.1) See [20, Vol. 1 Chapter III.11] for a detailed account on periodized orthonormal wavelet bases.

3.2 Besov balls

LetM >0,s >0,p≥1andr≥1. Setβj∗−1,k =αj∗,k. A functionhbelongs to the Besov ballsB_p,r^s (M)if and only if there exists a constantM^∗ >0such that the wavelet coefficients (3.1) satisfy







∞

X

j=j∗−1





2j(s+1/2−1/p)





2^j−1

X

k=0

|βj,k|^p





1/p





r





1/r

≤M^∗.

For a particular choice of parameterss,pandr, these sets contain the H¨older and Sobolev balls. See [20].

4 Estimators

Wavelet coefficient estimators. The first step to estimatef^(d)consists in expandingf^(d) onBand estimating its unknown wavelet coefficients.

For any integer j ≥ j_∗ and any k ∈ {0, . . . ,2^j − 1}, we estimate αj,k = R1

0 f^(d)(x)φ_j,k(x)dxby αb_j,k= 1

n

n

X

v=1

X

`∈Cj

(2iπ`)<sup>d</sup>F(φj,k)(`)

F(g)(`) Y<sub>v</sub>e<sup>−2iπ`Xv</sup>, (4.1) Cj= supp (F(φj,0)) = supp (F(φj,k)), andβj,k=R1

0 f^(d)(x)ψ_j,k(x)dxby βbj,k= 1

n

n

X

v=1

Gv1_{|Gv|≤ηj}, (4.2)

where

Gv= X

`∈D_j

(2iπ`)<sup>d</sup>F(ψj,k)(`)

F(g)(`) Yve<sup>−2iπ`Xv</sup>,

Dj = supp (F(ψj,0)) = supp (F(ψj,k)), for any random eventA,1_Ais the indicator function onA, and the thresholdη_jis defined by

η_j =θ2^(δ+d)j r n

lnn, (4.3)

θ = p

C_∗∗(C_∗²+E(ξ₁²)),C_∗ is (2.1) andC_∗∗ = 2^δ−1(2(2π)^2d/c²_g)(8π/3)^2(δ+d). The estimators αbj,k andβbj,k are constructed via a SVD approach similar to the one of [17].

The idea of the thresholding in (4.2) is to operate a selection on the observations: when, forv ∈ {1, . . . , n},f is ”too noisy” byξv, the observation(Yv, Xv)is neglected. Such a technique has been introduced by [7] in wavelet estimation. Statistical properties ofαb_j,k andβbj,kare given in Propositions 6.1, 6.2 and 6.3.

We consider two wavelets estimators forf^(d): a linear estimator and a hard threshold- ing estimator.

Linear estimator. Assuming that f^(d) ∈ B^s_p,r(M) withp ≥ 2, we define the linear estimatorfb_d^Lby

fb_d^L(x) =

2^j0−1

X

k=0

αbj₀,kφj₀,k(x), (4.4) whereαbj,kis defined by (4.1) andj0is the integer satisfying

2⁻¹n1/(2s+2δ+2d+1)<2^j0≤n1/(2s+2δ+2d+1). It is not adaptive since it depends ons, the smoothness parameter off^(d). Hard thresholding estimator.We define the hard thresholding estimatorfb_d^Hby

fb_d^H(x) =

2^j∗−1

X

k=0

αb_j∗,kφ_j∗,k(x) +

j₁

X

j=j∗

2^j−1

X

k=0

βb_j,k1{^|bβj,k|≥κλj}ψ_j,k(x), (4.5) whereαbj,kandβbj,kare defined by (4.1) and (4.2),j1is the integer satisfying

2⁻¹n1/(2δ+2d+1)<2^j1≤n1/(2δ+2d+1), κ≥8/3 + 2 + 2p

16/9 + 4andλjis the threshold λj =θ2^(δ+d)j

rlnn

n . (4.6)

The definitions ofηjandλjare chosen to minimize the MISE offb^H and to make it adap- tive. Further statistical results on the hard thresholding estimator for the standard regression model can be found in [8–10].

5 Results

Theorem 5.1. Consider(1.1)under the assumptions of Section2. Suppose thatf^(d) ∈ B_p,r^s (M)withs > 0,p ≥ 2 andr ≥ 1. Letfb_d^L be(4.4). Then there exists a constant C >0such that

E Z 1

0

fb_d^L(x)−f^(d)(x)² dx

≤Cn−2s/(2s+2δ+2d+1).

The proof of Theorem 5.1 uses a moment inequality on (4.1) and a suitable decompo- sition of the MISE.

Since the distribution ofξ₁is unknown, we can not use the likelihood function related to the model and the optimal lower bound seems difficult to determine (see e.g. [16, 25]).

For this reason, our benchmark will be the rate of convergence attains by the ”optimal non-realistic”fb_d^L, i.e. vn =n−2s/(2s+2δ+2d+1). Note that, under the standard Gaussian assumption onξ₁, one can prove thatv_nis optimal in the minimax sense.

Theorem 5.2. Consider(1.1)under the assumptions of Section2. Letfb_d^Hbe(4.5). Suppose thatf^(d)∈B^s_p,r(M)withr≥1,{p≥2ands >0}or{p∈[1,2)ands >(2δ+ 2d+ 1)/p}. Then there exists a constantC >0such that

E Z 1

0

fb_d^H(x)−f^(d)(x)2

dx

≤C lnn

n

2s/(2s+2δ+2d+1)

.

The proof of Theorem 5.2 is based on several probability results (moment inequalities, concentration inequality,. . . ) and a suitable decomposition of the MISE.

Theorem 5.2 proves that fb_d^H attains vn up to the logarithmic term (lnn)2s/(2s+2δ+2d+1).

Conclusion and perspectives. We have constructed a new adaptive estimatorfb_d^H for f^(d)under mild assumption onξ1. It is based on wavelet and thresholding. It has ”near- optimal” minimax properties for a wide class of functionsf^(d). Possible perspectives of this work are

- to potentially improve the estimation off^(d)by considering other kinds of thresh- olding rules as the block thresholding one introduced by [2],

- to investigate the random design case where the distribution ofX1is unknown.

6 Proofs

In this section, C represents a positive constant which may differ from one term to another.

6.1 Auxiliary results

Proposition 6.1. For any integerj ≥ j_∗ and any k ∈ {0, . . . ,2^j −1}, letα_j,kbe the wavelet coefficient(3.1)off^(d)andαbj,kbe(4.1). Then there exists a constantC >0such that

E

(αbj,k−αj,k)²

≤C2^2(δ+d)j1 n. Proof of Proposition 6.1.For anyv∈ {1, . . . , n}, let us set

Hv= X

`∈C_j

(2iπ`)<sup>d</sup>F(φj,k)(`)

F(g)(`) Yve<sup>−2iπ`Xv</sup>.

SinceX1,W1andξ1are independent, using the convolution product betweenfandg, i.e.

(f ? g)(x) =R1

0 f(x−y)g(y)dy, we have E Y1e^−2iπ`X1

= E f(X1−W1)e^−2iπ`X1

+E(ξ1)E e^−2iπ`X1

= E f(X1−W1)e^−2iπ`X1

= Z 1

0

Z 1 0

f(x−y)g(y)e^−2iπ`xdxdy= Z 1

0

(f ? g)(x)e^−2iπ`xdx

= F(f ? g)(`) =F(f)(`)F(g)(`).

Moreover, sincef is1-periodic, for anyu∈ {0, . . . , d},f^(u)is1-periodic andf^(u)(0) = f^(u)(1). Bydintegrations by parts, for any`∈Z, we have

(2iπ`)<sup>d</sup>F(f)(`) =F f^(d)

(`).

The Parseval-Plancherel theorem gives E(H1) = X

`∈Cj

(2iπ`)<sup>d</sup>F(φ<sub>j,k</sub>)(`)

F(g)(`) E Y1e<sup>−2iπ`X1</sup>

= X

`∈Cj

(2iπ`)<sup>d</sup>F(φj,k)(`)

F(g)(`) F(f)(`)F(g)(`)

= X

`∈Cj

F(φj,k)(`)(2iπ`)^dF(f)(`) =X

`∈Cj

F(φj,k)(`)F f^(d)

(`)

= Z 1

0

φ_j,k(x)f^(d)(x)dx=αj,k. (6.1)

HenceE(αbj,k) =αj,k. So E

(αb_j,k−α_j,k)²

= V(αb_j,k) =V 1 n

n

X

v=1

H_v

!

= 1 n²

n

X

v=1

V(H_v)

= 1

nV(H1)≤ 1 nE H₁²

. (6.2)

SinceX1,W1andξ1are independent withE(ξ1) = 0and|f(X1−W1)| ≤ kfk_∞≤C_∗<

∞, we have

E H₁²

= E





f²(X1−W1)



 X

`∈C_j

(2iπ`)<sup>d</sup>F(φj,k)(`)

F(g)(`) e<sup>−2iπ`X1</sup>





2





+ 2E(ξ1)E





f(X1−W1)



 X

`∈Cj

(2iπ`)<sup>d</sup>F(φj,k)(`)

F(g)(`) e<sup>−2iπ`X1</sup>





2





+ E ξ₁² E









 X

`∈Cj

(2iπ`)<sup>d</sup>F(φ<sub>j,k</sub>)(`)

F(g)(`) e<sup>−2iπ`X1</sup>





2





= E





f²(X₁−W₁)



 X

`∈Cj

(2iπ`)<sup>d</sup>F(φj,k)(`)

F(g)(`) e<sup>−2iπ`X1</sup>





2





+ E ξ₁² E









 X

`∈C_j

(2iπ`)<sup>d</sup>F(φj,k)(`)

F(g)(`) e<sup>−2iπ`X1</sup>





2





≤ C_∗²+E ξ₁² E









 X

`∈Cj

(2iπ`)<sup>d</sup>F(φj,k)(`)

F(g)(`) e<sup>−2iπ`X1</sup>





2



. (6.3) The assumption (2.2) implies

sup

`∈Cj

(2π`)^2d

|F(g)(`)|² ≤ (2π)^2d c²_g sup

`∈Cj

`<sup>2d</sup>(1 +`²)^δ≤2^δ−1(2π)^2d c²_g sup

`∈Cj

`<sup>2d</sup>(1 +`^2δ)

≤ 2^δ−12(2π)^2d c²_g

8π 3

^2(δ+d)

2^2(δ+d)j =C_∗∗2^2(δ+d)j. (6.4) Using the fact that (e<sup>−2iπ`x</sup>)<sub>`∈Z</sub> is an orthonormal basis of L²per([0,1]), (6.4) and the Parseval-Plancherel theorem, we obtain

E









 X

`∈Cj

(2iπ`)<sup>d</sup>F(φj,k)(`)

F(g)(`) e<sup>−2iπ`X1</sup>





2





= Z 1

0



 X

`∈C_j

(2iπ`)<sup>d</sup>F(φj,k)(`) F(g)(`) e<sup>−2iπ`x</sup>





2

dx=X

`∈C_j

(2π`)<sup>2d</sup>|F(φj,k) (`)|²

|F(g)(`)|²

≤ C_∗∗2^2(δ+d)j X

`∈Cj

|F(φ_j,k) (`)|²=C_∗∗2^2(δ+d)j Z 1

0

|φj,k(x)|²dx

= C_∗∗2^2(δ+d)j. (6.5)

Putting (6.3) and (6.5) together, we obtain E H₁²

≤θ²2^2(δ+d)j. (6.6)

It follows from (6.2) and (6.6) that E

(αbj,k−αj,k)²

≤C2^2(δ+d)j1 n.

Proposition 6.2. For any integerj ≥ j_∗ and anyk ∈ {0, . . . ,2^j −1}, letβ_j,k be the wavelet coefficient(3.1)off^(d)andβbj,kbe(4.2). Then there exists a constantC >0such that

E

βbj,k−βj,k

⁴

≤C2^4(δ+d)j(lnn)² n² .

Proof of Proposition 6.2.Proceeding as in (6.1) (withψinstead ofφ), we have βj,k=

Z 1 0

f^(d)(x)ψj,k(x)dx=E(Gv) =E(Gv1_{|Gv|≤ηj}) +E(G11_{|G1|>ηj}). (6.7) We have

E

βb_j,k−β_j,k4

= E



 1 n

n

X

v=1

Gv1_{|Gv|≤ηj}−E Gv1_{|Gv|≤ηj}

−E G11_{|G1|>ηj}

!4



≤ 8(A+B), (6.8)

where

A=E



 1 n

n

X

v=1

G_v1_{|Gv|≤ηj}−E(G_v1_{|Gv|≤ηj})

!⁴



and

B = E(|G1|1_{|G1|>ηj})⁴ .

Let us boundAandB, in turn. To boundA, we need the Rosenthal inequality presented in lemma below (see [24]).

Lemma 6.1(Rosenthal’s inequality). Letp ≥ 2,n ∈ N^∗ and(Uv)v∈{1,...,n} benzero mean i.i.d. random variables such thatE(|U1|^p)<∞. Then there exists a constantC >0 such that

E

n

X

v=1

Uv

p!

≤Cmax

nE(|U1|^p), nE U₁^2p/2 .

Applying the Rosenthal inequality withp= 4and, for anyv∈ {1, . . . , n}, Uv=Gv1_{|Gv|≤ηj}−E Gv1_{|Gv|≤ηj}

, we obtain

A= 1 n⁴E





n

X

v=1

U_v

!4

≤C 1 n⁴max

nE U₁⁴

, nE U₁²² .

Using (6.6) (withψinstead ofφ), we have, for anya∈ {2,4}, E(U₁^a)≤2^aE G^a₁1_{|G1|≤ηj}

≤2^aη^a−2_j E G²₁

≤2^aθ²η^a−2_j 2^2(δ+d)j. Hence

A ≤ C 1 n⁴max

η²_jθ²2^2(δ+d)jn,

θ²2^2(δ+d)jn2

= C 1 n⁴max

2^4(δ+d)j n²

lnn,2^4(δ+d)jn²

=C2^4(δ+d)j 1

n². (6.9) Let us now boundB. Using again (6.6) (withψinstead ofφ), we obtain

E |G1|1_{|G1|>ηj}

≤ E G²₁

η_j ≤ 1

θ2^(δ+d)j rlnn

n 2^2(δ+d)jθ²

= θ2^(δ+d)j rlnn

n . (6.10)

Hence

B≤C2^4(δ+d)j(lnn)²

n² . (6.11)

Combining (6.8), (6.9) and (6.11), we have E

βb_j,k−β_j,k4

≤C

2^4(δ+d)j 1

n² + 2^4(δ+d)j(lnn)² n²

≤C2^4(δ+d)j(lnn)² n² .

Proposition 6.3. For any integerj ≥ j_∗ and anyk ∈ {0, . . . ,2^j −1}, letβj,k be the wavelet coefficient(3.1)off^(d)andβbj,kbe(4.2). Then, for anyκ≥8/3+2+2p

16/9 + 4, P

|βb_j,k−β_j,k| ≥κλ_j/2

≤2n⁻². Proof of Proposition 6.3.Using (6.7), we have

|βbj,k−βj,k|

= 1 n

n

X

v=1

G_v1_{|Gv|≤ηj}−E G_v1_{|Gv|≤ηj}

−E G₁1_{|G1|>ηj}

≤ 1 n

n

X

v=1

G_v1_{|Gv|≤ηj}−E G_v1_{|Gv|≤ηj}

+E |G1|1_{|G1|>ηj} .

Using (6.10), we obtain

E |G₁|1_{|G1|>ηj}

≤θ2^(δ+d)j rlnn

n =λ_j. Hence

S = P

|bβj,k−βj,k| ≥κλj/2

≤ P 1 n

n

X

v=1

G_v1_{|Gv|≤ηj}−E G_v1_{|Gv|≤ηj}

≥(κ/2−1)λ_j

! .

Now we need the Bernstein inequality presented in the lemma below (see [22]).

Lemma 6.2 (Bernstein’s inequality). Let n ∈ N^∗ and (U_v)v∈{1,...,n} be n zero mean i.i.d. random variables such that there exists a constant M > 0 satisfying, for any v∈ {1, . . . , n},|Uv| ≤M <∞. Then, for anyλ >0, holds

P

n

X

v=1

Uv

≥λ

!

≤2 exp − λ²

2 nE(U₁²) +^λM₃

! .

Let us set, for anyv∈ {1, . . . , n},

Uv=Gv1_{|Gv|≤ηj}−E Gv1_{|Gv|≤ηj} . ThenE(U1) = 0,

|U_v| ≤

G_v1_{|Gv|≤ηj}

+E |G_v|1_{|Gv|≤ηj}

≤2η_j and, using again (6.6) (withψinstead ofφ),

E U₁²

=V G11_{|G1|≤ηj}

≤E G²₁

≤θ²2^2(δ+d)j. It follows from the Bernstein inequality that

S≤2 exp



− n²(κ/2−1)²λ²_j 2

θ²n2^2(δ+d)j+^{2n(κ/2−1)λ}₃ ^jηj



.

Since

λjηj=θ2^(δ+d)j rlnn

n θ2^(δ+d)j r n

lnn =θ²2^2(δ+d)j, λ²_j =θ²2^2(δ+d)jlnn n , we have, for anyκ≥8/3 + 2 + 2p

16/9 + 4,

S≤2 exp



− (κ/2−1)²lnn 2

1 +^2(κ/2−1)₃



≤2n⁻².

6.2 Proofs of the main results

Proof of Theorem 5.1.We expand the functionf^(d)as

f^(d)(x) =

2^j0−1

X

k=0

αj₀,kφj₀,k(x) +

∞

X

j=j₀ 2^j−1

X

k=0

βj,kψj,k(x),

where

αj0,k= Z 1

0

f^(d)(x)φ_j0,k(x)dx, βj,k= Z 1

0

f^(d)(x)ψ_j,k(x)dx.

We have

fb_d^L(x)−f^(d)(x) =

2^j0−1

X

k=0

(αbj₀,k−αj₀,k)φj₀,k(x)−

∞

X

j=j0

2^j−1

X

k=0

βj,kψj,k(x).

Hence

E Z 1

0

fb_d^L(x)−f^(d)(x)² dx

=A+B, where

A=

2^j0−1

X

k=0

E

(αbj₀,k−αj₀,k)²

, B=

∞

X

j=j₀ 2^j−1

X

k=0

β_j,k² .

Using Proposition 6.1, we obtain

A≤C2^j0(1+2δ+2d)1

n ≤Cn−2s/(2s+2δ+2d+1). Sincep≥2, we haveB_p,r^s (M)⊆B^s_2,∞(M). Hence

B≤C2^−2j0s≤Cn−2s/(2s+2δ+2d+1). So

E Z 1

0

fb_d^L(x)−f^(d)(x)2

dx

≤Cn−2s/(2s+2δ+2d+1). The proof of Theorem 5.1 is complete.

Proof of Theorem 5.2.We expand the functionf^(d)as

f^(d)(x) =

2^j∗−1

X

k=0

α_j∗,kφ_j∗,k(x) +

∞

X

j=j∗

2^j−1

X

k=0

β_j,kψ_j,k(x),

where

αj∗,k= Z 1

0

f^(d)(x)φ_j∗,k(x)dx, βj,k= Z 1

0

f^(d)(x)ψ_j,k(x)dx.

We have

fb_d^H(x)−f^(d)(x)

=

2^j∗−1

X

k=0

(αb_j∗,k−α_j∗,k)φ_j∗,k(x) +

j1

X

j=j∗

2^j−1

X

k=0

βb_j,k1{^|bβj,k|≥κλj} −β_j,k ψ_j,k(x)

−

∞

X

j=j1+1 2^j−1

X

k=0

βj,kψj,k(x).

Hence

E Z 1

0

fb_d^H(x)−f^(d)(x)2

dx

=R+S+T, (6.12)

where R=

2^j∗−1

X

k=0

E

(αbj∗,k−αj∗,k)²

, S=

j₁

X

j=j_∗ 2^j−1

X

k=0

E

βbj,k1{^|bβj,k|≥κλ_j} −βj,k

²

and

T =

∞

X

j=j₁+1 2^j−1

X

k=0

β_j,k² . Let us boundR,TandS, in turn.

Using Proposition 6.1, we have R≤C2^{j∗(1+2δ+2d)}1

n ≤C1 n ≤C

lnn n

2s/(2s+2δ+2d+1)

. (6.13)

Forr≥1andp≥2, we haveB_p,r^s (M)⊆B^s_2,∞(M). So T ≤C

∞

X

j=j1+1

2^−2js≤C2^−2j1s≤Cn−2s/(2δ+2d+1)≤C lnn

n

2s/(2s+2δ+2d+1)

.

Forr≥1andp∈[1,2), we haveB_p,r^s (M)⊆B_2,∞^s+1/2−1/p(M). Sinces >(2δ+2d+1)/p, we have(s+ 1/2−1/p)/(2δ+ 2d+ 1)> s/(2s+ 2δ+ 2d+ 1). So

T ≤ C

∞

X

j=j₁+1

2−2j(s+1/2−1/p)≤C2^−2j1(s+1/2−1/p)

≤ Cn−2(s+1/2−1/p)/(2δ+2d+1)≤C lnn

n

2s/(2s+2δ+2d+1)

.

Hence, forr≥1,{p≥2ands >0}or{p∈[1,2)ands >(2δ+ 2d+ 1)/p}, we have T ≤C

lnn n

2s/(2s+2δ+2d+1)

. (6.14)

The termScan be decomposed as

S=e₁+e₂+e₃+e₄, (6.15)

where

e1=

j1

X

j=j∗

2^j−1

X

k=0

E

βbj,k−βj,k

2

1{^|bβj,k|≥κλj}1_{{|βj,k|<κλj/2}}

,

e2=

j₁

X

j=j_∗ 2^j−1

X

k=0

E

βbj,k−βj,k

²

1{^|bβj,k|≥κλ_j}1_{{|βj,k|≥κλj/2}}

,

e₃=

j₁

X

j=j∗

2^j−1

X

k=0

E

β_j,k² 1{^|bβj,k|<κλj}1_{{|βj,k|≥2κλj}} and

e4=

j₁

X

j=j_∗ 2^j−1

X

k=0

E

β_j,k² 1{^|bβj,k|<κλj}1_{{|βj,k|<2κλj}} . Let us analyze each terme1,e2,e3ande4in turn.

Upper bounds fore₁ande₃.We have n|βb_j,k|< κλ_j, |βj,k| ≥2κλ_jo

⊆n

|βb_j,k−β_j,k|> κλ_j/2o ,

n|bβ_j,k| ≥κλ_j, |β_j,k|< κλ_j/2o

⊆n

|βb_j,k−β_j,k|> κλ_j/2o and

n|βbj,k|< κλj, |βj,k| ≥2κλj

o⊆n

|βj,k| ≤2|βbj,k−βj,k|o . So

max(e₁, e₃)≤C

j₁

X

j=j∗

2^j−1

X

k=0

E

βb_j,k−β_j,k2

1{^|bβj,k−βj,k|>κλj/2}

.

It follows from the Cauchy-Schwarz inequality and Propositions 6.2 and 6.3 that E

βb_j,k−β_j,k²

1{^|bβj,k−βj,k|>κλj/2}

≤

E

βb_j,k−β_j,k⁴1/2

P

|βb_j,k−β_j,k|> κλ_j/2^1/2

≤ C2^2(δ+d)jlnn n² .

Hence

max(e1, e3) ≤ Clnn n²

j1

X

j=j∗

2^j(1+2δ+2d)≤Clnn

n² 2^j1(1+2δ+2d)

≤ Clnn n ≤C

lnn n

2s/(2s+2δ+2d+1)

. (6.16)

Upper bound for the terme2. Using Proposition 6.2 and the Cauchy-Schwarz inequality, we obtain

E

βbj,k−βj,k

²

≤

E

βbj,k−βj,k

⁴1/2

≤C2^2(δ+d)jlnn n . Hence

e₂≤Clnn n

j1

X

j=j∗

2^2(δ+d)j

2^j−1

X

k=0

1_{{|βj,k|>κλj/2}}.

Letj2be the integer defined by 2⁻¹ n

lnn

1/(2s+2δ+2d+1)

<2^j2≤ n lnn

1/(2s+2δ+2d+1)

. (6.17)

We have

e₂≤e_2,1+e_2,2, where

e2,1=Clnn n

j2

X

j=j∗

2^2(δ+d)j

2^j−1

X

k=0

1_{{|βj,k|>κλj/2}}

and

e_2,2=Clnn n

j₁

X

j=j₂+1

2^2(δ+d)j

2^j−1

X

k=0

1_{{|βj,k|>κλj/2}}. We have

e2,1≤Clnn n

j2

X

j=j∗

2^j(1+2δ+2d)≤Clnn

n 2^j2(1+2δ+2d)≤C lnn

n

2s/(2s+2δ+2d+1)

.

Forr≥1andp≥2, sinceB_p,r^s (M)⊆B_2,∞^s (M),

e_2,2 ≤ Clnn n

j1

X

j=j₂+1

2^2(δ+d)j 1 λ²_j

2^j−1

X

k=0

β_j,k² =C

∞

X

j=j₂+1 2^j−1

X

k=0

β_j,k² ≤C2^−2j2s

≤ C lnn

n

2s/(2s+2δ+2d+1)

.

Forr≥1,p∈[1,2)ands >(2δ+ 2d+ 1)/p, sinceB^s_p,r(M)⊆B_2,∞^s+1/2−1/p(M)and (2s+ 2δ+ 2d+ 1)(2−p)/2 + (s+ 1/2−1/p+δ+d−2(δ+d)/p)p= 2s, we have

e2,2 ≤ Clnn n

j1

X

j=j2+1

2^2(δ+d)j 1 λ^p_j

2^j−1

X

k=0

|βj,k|^p

≤ C lnn

n

^{(2−p)/2 ∞} X

j=j2+1

2j(δ+d)(2−p)2−j(s+1/2−1/p)p

≤ C lnn

n

^(2−p)/2

2^−j2(s+1/2−1/p+δ+d−2(δ+d)/p)p

≤ C lnn

n

2s/(2s+2δ+2d+1)

.

So, forr≥1,{p≥2ands >0}or{p∈[1,2)ands >(2δ+ 2d+ 1)/p}, e2≤C

lnn n

2s/(2s+2δ+2d+1)

. (6.18)

Upper bound for the terme4.We have e4≤

j1

X

j=j∗

2^j−1

X

k=0

β_j,k² 1_{{|βj,k|<2κλj}}.

Letj2be the integer (6.17). We have

e₄≤e_4,1+e_4,2, where

e4,1=

j2

X

j=j∗

2^j−1

X

k=0

β²_j,k1_{{|βj,k|<2κλj}}, e4,2=

j1

X

j=j2+1 2^j−1

X

k=0

β_j,k² 1_{{|βj,k|<2κλj}}.

We have

e_4,1 ≤ C

j₂

X

j=j∗

2^jλ²_j =Clnn n

j₂

X

j=j∗

2^j(1+2δ+2d)≤Clnn

n 2^j2(1+2δ+2d)

≤ C lnn

n

2s/(2s+2δ+2d+1)

.

Forr≥1andp≥2, sinceB_p,r^s (M)⊆B_2,∞^s (M), we have

e_4,2≤

∞

X

j=j2+1 2^j−1

X

k=0

β_j,k² ≤C2^−2j2s≤C lnn

n

2s/(2s+2δ+2d+1)

.

Forr≥1,p∈[1,2)ands >(2δ+ 2d+ 1)/p, sinceB^s_p,r(M)⊆B_2,∞^s+1/2−1/p(M)and (2−p)(2s+ 2δ+ 2d+ 1)/2 + (s+ 1/2−1/p+δ+d−2(δ+d)/p)p= 2s, we have

e4,2 ≤ C

j1

X

j=j2+1

λ^2−p_j

2^j−1

X

k=0

|βj,k|^p=C lnn

n

(2−p)/2 j1

X

j=j2+1

2^jδ(2−p)

2^j−1

X

k=0

|βj,k|^p

≤ C lnn

n

(2−p)/2 ∞

X

j=j2+1

2j(δ+d)(2−p)2−j(s+1/2−1/p)p

≤ C lnn

n

(2−p)/2

2^−j2(s+1/2−1/p+δ+d−2(δ+d)/p)p≤C lnn

n

2s/(2s+2δ+2d+1)

. So, forr≥1,{p≥2ands >0}or{p∈[1,2)ands >(2δ+ 2d+ 1)/p},

e₄≤C lnn

n

2s/(2s+2δ+2d+1)

. (6.19)

It follows from (6.15), (6.16), (6.18) and (6.19) that S ≤C

lnn n

2s/(2s+2δ+2d+1)

. (6.20)

Combining (6.12), (6.13), (6.14) and (6.20), we have, forr≥1,{p≥2ands >0}or {p∈[1,2)ands >(2δ+ 2d+ 1)/p},

E Z 1

0

fb_d^H(x)−f^(d)(x)² dx

≤C lnn

n

2s/(2s+2δ+2d+1)

. The proof of Theorem 5.2 is complete.

This work is supported by ANR grant NatImages, ANR-08-EMER-009.

References

[1] P.K. Bhattacharya, Estimation of a probability density function and its derivatives, Sankhya Ser. A.29 (1967), 373–382.

[2] T. Cai, Adaptive Wavelet Estimation: A Block Thresholding And Oracle Inequality Approach,The Annals of Statistics27 (1999), 898–924.

[3] Y.P. Chaubey and H. Doosti, Wavelet based estimation of the derivatives of a density for m-dependent random variables, Journal of the Iranian Statistical Society 4 (2) (2005), 97–105.

[4] Y.P. Chaubey, H. Doosti and B.L.S. Prakasa Rao, Wavelet based estimation of the derivatives of a density with associated variables,International Journal of Pure and Applied Mathematics27 (1) (2006), 97–106.

[5] C. Chesneau, Wavelet estimation of the derivatives of an unknown function from a convolution model, Preprint (2009).

[6] F. Comte and M.-L. Taupin, Nonparametric estimation of the regression function in an errors-in-variables model,Statistica Sinica173 (2007), 1065-1090.

[7] B. Delyon and A. Juditsky, On minimax wavelet estimators,Applied Computational Harmonic Analysis3(1996), 215–228.

[8] D.L. Donoho and I.M. Johnstone, Ideal spatial adaptation by wavelet shrinkage, Biometrika81(1994), 425-455.

[9] D. L. Donoho and I.M. Johnstone, Adapting to unknown smoothness via wavelet shrinkage,Journal of the American Statistical Association90432 (1995), 1200-1224.

[10] D.L. Donoho and I.M. Johnstone, Minimax estimation via wavelet shrinkage, The Annals of Statistics26(1998), 879-921.

[11] J. Fan and M. Masry, Multivariate regression estimation with errors-in-variables:

asymptotic normality for mixing process,J. Multivariate Anal.43(1992), 237-272.

[12] J. Fan and Y.K. Truong, Nonparametric regression with errors-in-variables,The An- nals of Statistics214 (1993), 1900-1925.

[13] J. Fan, Y.K. Truong and Y. Wang, Nonparametric function estimation involving errors-in-variables,Nonparametric Functional Estimation and Related Topic(1991), 613-627.

[14] J. Fan and J.Y. Koo, Wavelet deconvolution,IEEE transactions on information theory 48 (2002), 734–747.

[15] D.A. Ioannides and P.D. Alevizos, Nonparametric regression with errors in variables and applications,Statist. Probab. Lett.32(1997), 35-43.

[16] W. H¨ardle, G. Kerkyacharian, D. Picard and A. Tsybakov,Wavelet, Approximation and Statistical Applications, Lectures Notes in Statistics New York, 129 Springer Verlag, 1998.

[17] I.M. Johnstone, G. Kerkyacharian, D. Picard and M. Raimondo, Wavelet deconvolu- tion in a periodic setting,Journal of the Royal Statistical Society Serie B663 (2004), 547–573.

[18] J.Y. Koo and K.W. Lee, B-splines estimations of regression functions with errors in variables,Statist. Probab. Lett.40(1998), 57-66.

[19] E. Masry, Multivariate regression estimation with errors-in-variables for stationary processes,Nonparametric Statistics3(1993), 13-36.

[20] Y. Meyer,Ondelettes et Op´erateurs, Hermann, Paris, 1990.

[21] M. Pensky and B. Vidakovic, Adaptive wavelet estimator for nonparametric density deconvolution,The Annals of Statistics27 (1999), 2033–2053.

[22] V. V. Petrov,Limit Theorems of Probability Theory: Sequences of Independent Ran- dom Variables, Oxford: Clarendon Press, 1995.

[23] B.L.S. Prakasa Rao, Nonparametric estimation of the derivatives of a density by the method of wavelets,Bull. Inform. Cyb.28 (1996), 91–100.

[24] H. P. Rosenthal, On the subspaces ofL^p(p≥2) spanned by sequences of independent random variables,Israel Journal of Mathematics8(1970), 273-303.

[25] A.B. Tsybakov,Introduction a l’estimation non-parametrique, Springer, 2004.