Nonparametric regression estimation for random fields in a fixed-design

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Nonparametric regression estimation for random fields in a fixed-design

Mohamed El Machkouri

To cite this version:

Mohamed El Machkouri. Nonparametric regression estimation for random fields in a fixed-design.

Statistical Inference for Stochastic Processes, Springer Verlag, 2007. �hal-00004096�

ccsd-00004096, version 1 - 4 Feb 2005

Nonparametric regression estimation for random fields in a fixed-design

Mohamed EL MACHKOURI 4th February 2005

Abstract

We investigate the nonparametric estimation for regression in a fixed-design setting when the errors are given by a field of dependent random variables. Sufficient conditions for kernel estimators to con- verge uniformly are obtained. These estimators can attain the optimal rates of uniform convergence and the results apply to a large class of random fields which contains martingale-difference random fields and mixing random fields.

AMS Subject Classifications (2000): 60G60, 62G08

Key words and phrases: nonparametric regression estimation, kernel estimators, strong consistency, fixed-design, exponential inequalities, martingale difference random fields, mixing, Orlicz spaces.

Short title: Nonparametric regression in a fixed design.

1 Introduction

Over the last few years nonparametric estimation for random fields (or spatial processes) was given increasing attention stimulated by a growing demand from applied research areas (see Guyon [18]). In fact, spatial data arise in various areas of research including econometrics, image analysis, meterology, geostatistics... Our aim in this paper is to investigate uniform strong con- vergence rates of a regression estimator in a fixed design setting when the errors are given by a stationary field of dependent random variables which show spatial interaction. We are most interested in conditions which ensure convergence rates to be identical to those in the case of independent errors (see Stone [33]). Currently the author is working on extensions of the present results to the random design framework. Let Z^d, d ≥ 1 denote the integer lattice points in the d-dimensional Euclidean space. By a stationary real random field we mean any family (εk)_k∈Z^d of real-valued random variables defined on a probability space (Ω,F,P) such that for any (k, n)∈Z^d×N^∗ and any (i1, ..., in) ∈ (Z^d)ⁿ, the random vectors (εi1, ..., εin) and (εi1+k, ..., εin+k) have the same law. The regression model which we are interested in is

Yi =g(i/n) +εi, i∈Λn={1, ..., n}^d (1) wheregis an unknown smooth function and (εi)_i∈Z^dis a zero mean stationary real random field. Note that this model was considered also by Bosq [8] and Hall et Hart [19] for time series (d= 1). LetK be a probability kernel defined on R^d and (hn)n≥1 a sequence of positive numbers which converges to zero and which satisfies (nhn)n≥1 goes to infinity. We estimate the function g by the kernel-type estimator g_n defined for anyx in [0,1]^d by

gn(x) = P

i∈ΛnY_iK

x−i/n hn

P

i∈ΛnK

x−i/n hn

. (2) Note that Assumption A1) in section 2 ensures thatgnis well defined. Until now, most of existing theoretical nonparametric results of dependent random variables pertain to time series (see Bosq [9]) and relatively few generaliza- tions to the spatial domain are available. Key references on this topic are Biau [5], Carbon et al. [10], Carbon et al. [11], Hallin et al. [20], [21], Tran [34], Tran and Yakowitz [35] and Yao [36] who have investigated nonpara- metric density estimation for random fields and Altman [2], Biau and Cadre [6], Hallin et al. [22] and Lu and Chen [25], [26] who have studied spatial prediction and spatial regression estimation. The classical asymptotic theory in statistics is built upon central limit theorems, law of large numbers and

large deviations inequalities for the sequences of random variables. These classical limit theorems have been extended to the setting of spatial pro- cesses. In particular, some key results on the central limit theorem and its functional versions are Alexander and Pyke [1], Bass [3], Basu and Dorea [4], Bolthausen [7] and more recently Dedecker [12], [13], El Machkouri [16]

and El Machkouri and Voln´y [17]. For a survey on limit theorems for spa- tial processes and some applications in statistical physics, one can refer to Nahapetian [28]. Note also that the main results (section 3) of this work are obtained via exponential inequalities for random fields discovered by El Machkouri [16].

The paper is organized as follows. The next section sets up the notations and the assumptions which will be considered in the sequel. In section 3, we present our main results on both weak and strong consistencies rates of the estimator gn. The last section is devoted to the proofs.

2 Notations and Assumptions

In the sequel we denotekxk= max1≤k≤d|xk| for anyx= (x1, ..., xd)∈[0,1]^d. With a view to obtain optimal convergence rates for the estimator gn defined by (2), we have to make the following assumptions on the regression function g and the probability kernel K:

A1) The probability kernel K is symmetric, nonnegative, supported by [−1,1]^d and satisfies a Lipschitz condition |K(x)−K(y)| ≤ ηkx−yk for anyx, y ∈[−1,1]^dand someη >0. In addition there exists c, C >0 such that c≤K(x)≤C for anyx∈[−1,1]^d.

A2) There exists a constant B >0 such that |g(x)−g(y)| ≤Bkx−yk for any x, y ∈[0,1]^d, that is g isB-Lipschitz.

A Young function ψ is a real convex nondecreasing function defined on R⁺ which satisfies limt→∞ψ(t) = +∞and ψ(0) = 0. We define the Orlicz space Lψ as the space of real random variables Z defined on the probability space (Ω,F,P) such that E[ψ(|Z|/c)] < +∞ for some c > 0. The Orlicz space Lψ equipped with the so-called Luxemburg norm k.k^ψ defined for any real random variable Z by

kZk^ψ = inf{c >0 ; E[ψ(|Z|/c)]≤1}

is a Banach space. For more about Young functions and Orlicz spaces one can refer to Krasnosel’skii and Rutickii [24]. Let β > 0. We denote by ψβ

the Young function defined for any x∈R⁺ by

ψ_β(x) = exp((x+ξ_β)^β)−exp(ξ_β^β) where ξ_β = ((1−β)/β)^1/β11{0<β<1}. On the latticeZ^dwe define the lexicographic order as follows: ifi= (i1, ..., id) and j = (j1, ..., jd) are distinct elements of Z^d, the notation i <lex j means that either i1 < j1 or for some p in {2,3, ..., d}, ip < jp and iq = jq for 1≤q < p. Let the sets {V_i^k; i∈Z^d, k ∈N^∗} be defined as follows:

V_i¹ ={j ∈Z^d; j <lex i}, and for k ≥2

V_i^k=V_i¹∩ {j ∈Z^d;|i−j| ≥k} where |i−j|= max

1≤l≤d|il−jl|. For any subset Γ of Z^d defineF^Γ =σ(εi; i∈Γ) and set

E|k|(εi) = E(εi|F_V^|k|

i ), k ∈V_i¹.

Denoteβ(q) = 2q/(2−q) for 0< q <2 and consider the following conditions:

C1) ε0 ∈L^∞ and

X

k∈V₀¹

kεkE|k|(ε0)k^∞<∞.

C2) There exists 0< q <2 such that ε₀ ∈L_ψβ(q) and X

k∈V₀¹

q

|εkE|k|(ε0)|

2 ψβ(q)

<∞.

C3) There existsp >2 such thatε₀ ∈L^p and X

k∈V₀¹

kεkE|k|(ε0)k^p₂ <∞.

C4) ε0 ∈L² and P

k∈Z^d|E(ε0εk)|<∞.

Remark 1 Note that Dedecker [12] established the central limit theorem for any stationary square-integrable random field (ε_k)_k∈Z^d which satisfies the condition P

k∈V₀¹kεkE|k|(ε0)k¹<∞.

In classical statistical physics, there exists spatial processes which satisfy conditions C1),...,C4). For example, Nahapetian and Petrosian [29] gave sufficient conditions for a Gibbs field (εk)_k∈Z^d to possess the following mar- tingale difference property: for any i in Z^d, E(εi|FV_i¹) = 0 a.s. Another examples of random fields which satisfy conditions C1),...,C4) can be found also among the class of mixing random fields. More precisely, given two sub-σ-algebras U and V of F, different measures of their dependence have been considered in the literature. We are interested by two of them. The α-mixing and φ-mixing coefficients had been introduced by Rosenblatt [31]

and Ibragimov [23] respectively and can be defined by

α(U,V) = sup{|P(U∩V)−P(U)P(V)|, U ∈ U, V ∈ V}

φ(U,V) = sup{kP(V|U)−P(V)k^∞, V ∈ V}.

We have 2α(U,V) ≤ φ(U,V) and these coefficients equal zero if and only if the σ-algebrasU andV are independent. Denote by♯Γ the cardinality of any subset Γ of Z^d. In the sequel, we shall use the following non-uniform mixing coefficients defined for any (k, l, n) in (N^∗∪ {∞})²×Nby

α_k,l(n) = sup{α(FΓ1,FΓ2), ♯Γ₁ ≤k, ♯Γ₂ ≤l, ρ(Γ₁,Γ₂)≥n}, φk,l(n) = sup{φ(F^Γ1,F^Γ2), ♯Γ1 ≤k, ♯Γ2 ≤l, ρ(Γ1,Γ2)≥n},

where the distance ρ is defined by ρ(Γ1,Γ2) = min{|i−j|, i ∈ Γ1, j ∈ Γ2}. We say that the random field (ε_k)_k∈Z^d isα-mixing orφ-mixing if there exists a pair (k, l) in (N^∗∪ {∞})²such that limn→∞αk,l(n) = 0 or limn→∞φk,l(n) = 0 respectively. For more about mixing coefficients one can refer to Doukhan [15]. We consider the following mixing conditions:

C^′1)ε0 ∈L^∞ and

X

k∈Z^d

φ∞,1(|k|)<∞. C^′2) There exists 0< q <2 such that ε0 ∈Lψ_β(q) and

X

k∈Z^d

q

φ∞,1(|k|)<∞

or X

k∈Z^d

c²_k(β(q))<∞ where for any β >0

ck(β) = inf (

c >0

Z α1,∞(|k|) 0

ψβ

Qε0(u) c

du≤1 )

. (3)

C^′3) There exists p >2 such that ε0 ∈L^p and X

k∈Z^d

Z α1,∞(|k|) 0

Q^p_ε0(u)du

!2/p

<∞ (4)

where Qε0 is the inverse cadlag of the tail function t → P(|ε0|> t) (i.e. for any u≥0, Qε0(u) = inf{t >0|P(|ε0|> t)≤u}).

Remark 2 Let us note that if p= 2 +δ for some δ >0 then the condition

∞

X

m=1

m^d−1α

δ 2+δ−ε

1,∞ (m)<∞ for some ε >0

is more restrictive than condition (4) and is known to be sufficient for the ran- dom field (ε_k)_k∈Z^d to satisfy a functional central limit theorem (cf. Dedecker [13]).

In statistical physics, using the Dobrushin’s uniqueness condition (cf. [14]), one can construct Gibbs fields satisfying a uniform exponential mixing con- dition which is more restrictive than conditions C^′1), C^′2) and C^′3) (see Guyon [18], theorem 2.1.3, p. 52).

3 Main results

Let (Zn)n≥1 be a sequence of real random variables and (vn)n≥1be a sequence of positive numbers. We say that

Zn=Oa.s.[vn] if there exists λ >0 such that

lim sup

n→∞

|Zn|

v_n ≤λ a.s.

Our main result is the following.

Theorem 1 Assume that the assumptionA1) holds.

1) If C1) holds then

sup

x∈[0,1]^d|gn(x)−Egn(x)|=Oa.s.

(logn)^1/2 (nhn)^d/2

. (5)

2) If C2) holds for some 0< q <2 then sup

x∈[0,1]^d|gn(x)−Egn(x)| =Oa.s.

(logn)^1/q (nhn)^d/2

. (6)

3) Assume that C3)holds for some p >2andhn =n^−θ2(logn)^θ1 for some θ1, θ2 >0. Leta, b≥0 be fixed and denote

vn = n^a(logn)^b

(nhn)^d/2 and θ = 2a(d+p)−d²−2 d(3d+ 2) . If θ ≥θ2 and d(3d+ 2)θ1+ 2(d+p)b > 2 then

sup

x∈[0,1]^d|g_n(x)−Eg_n(x)|=O_a.s.[v_n]. (7) Remark 3 Theorem 1 shows that the optimal uniform convergence rate is obtained for bounded errors (cf. estimation (5)) and that it is “almost”

optimal if one considers errors with only finite exponential moments (cf.

estimation (6)).

Theorem 2 Assume that the assumption A1) holds.

1) Assume that C3) holds for somep > 2. Let a >0 be fixed and denote v_n = n^a

(nhn)^d/2 and θ = 2a(d+p)−d² d(3d+ 2) . If θ > 0 and h_n≥n^−θ then

sup

x∈[0,1]^d|g_n(x)−Eg_n(x)| p

=O[v_n]. (8)

2) If C4) holds then sup

x∈[0,1]^dkgn(x)−Egn(x)k2 =O

(nhn)^−d/2

. (9)

In the sequel, we denote by Lip(B) the set of B-Lipschitz functions. The following proposition gives the convergence of Egn(x) to g(x).

Proposition 1 Assume that the assumption A2) holds then sup

x∈[0,1]^d

sup

g∈Lip(B)|Egn(x)−g(x)|=O[hn].

From Proposition 1 and Theorem 1 we derive the following corollary.

Corollary 1 Assume thatA1)andA2)hold and lethn = n^−dlogn1/(2+d)

.

1) If C1) holds then

sup

x∈[0,1]^d

sup

g∈Lip(B)|gn(x)−g(x)|=Oa.s.

"

logn n^d

_2+d¹ #

. (10)

2) If C2) holds for some 0< q <2 then sup

x∈[0,1]^d

sup

g∈Lip(B)|gn(x)−g(x)|=Oa.s.

"

u(n)

logn n^d

_2+d¹ #

(11) where u(n) = (logn)^(2−q)/2q.

3) Let ε >0 be fixed. If C3) holds for some p >2 satisfying p≥ 4d³+ (4−2ε)d²+ (2−4ε)d+ 4

2ε(2 +d) (12)

then sup

x∈[0,1]^d

sup

g∈Lip(B)|gn(x)−g(x)|=Oa.s.

"

u(n)

logn n^d

_2+d¹ #

(13) where u(n) =n^ε.

Remark 4 Note that the consistency rate (n^−dlogn)^1/(2+d) is known to be the optimal one (see Stone [33]).

From Proposition 1 and Theorem 2 we derive the following corollary.

Corollary 2 Assume that A1) and A2) hold and let hn=n^−d/(2+d). 1) Let ε >0 be fixed. If C3) holds for some p >2 satisfying

p≥ 4d³+ (4−2ε)d²−4εd

2ε(2 +d) (14)

then

sup

x∈[0,1]^d

sup

g∈Lip(B)|gn(x)−g(x)| p

=Oh

n^{−2+dd +ε}i

. (15)

2) If C4) holds then

sup

x∈[0,1]^d

sup

g∈Lip(B)|gn(x)−g(x)| 2

=Oh

n^−2+dd i

. (16)

Finally the rates of convergence obtained above are valid when the errors are given by a mixing random field. More precisely, we have the following corollary.

Corollary 3 Theorems 1 and 2 and Corollaries 1 and 2 still hold if one replace conditions C1), C2) and C3) by conditions C^′1), C^′2) and C^′3) respectively.

4 Proofs

For any x in [0,1]^d and any integer n ≥1 we define Bn(x) = Egn(x)−g(x) and Vn(x) =gn(x)−Egn(x). More precisely

Bn(x) = P

i∈Λnai(x)g(i/n) P

i∈Λna_i(x) −g(x) V_n(x) =

P

i∈Λnai(x)εi

P

i∈Λnai(x) whereai(x) =K

x−i/n hn

. In the sequel, we denote alsoSn(x) =P

i∈Λnai(x)εi

for any x∈[0,1]^d. We start with the following lemma.

Lemma 1 There exists constants c, C >0 such that for any x∈[0,1]^d and any n∈N^∗,

c

d

Y

k=1

[n(xk+hn)]≤ X

i∈Λn

ai(x)≤C

d

Y

k=1

[n(xk+hn)] (17) where [.] denote the integer part function.

Proof of Lemma 1. Since the kernel K is supported by [−1,1]^d, we have X

i∈Λn

ai(x) =

[n(x1+hn)]

X

i1=1

. . .

[n(xd+hn)]

X

id=1

ai(x).

By assumption, there exists constants c, C > 0 such that c≤K(y)≤C for any y∈[−1,1]^d. The proof of Lemma 1 is complete.

4.1 Proof of Theorem 1

Let (υn)n≥1 be a sequence of positive numbers going to zero. Following Carbon and al. [11] the compact set [0,1]^d can be covered by rn cubes Ik

having sides of length l_n = υ_nh^2d+1_n and center at c_k. Clearly there exists c > 0 such that rn≤c/l^d_n. Define

A1,n(g) = max

1≤k≤rn

sup

x∈Ik

|gn(x)−gn(ck)| A2,n(g) = max

1≤k≤rn

sup

x∈Ik

|Egn(x)−Egn(ck)| A3,n = max

1≤k≤rn|gn(ck)−Egn(ck)| then

sup

x∈[0,1]^d|g_n(x)−Eg_n(x)| ≤ sup

g∈Lip(B)

[A_1,n(g) +A_2,n(g)] +A_3,n. (18)

Lemma 2 Fori= 1,2 we have sup

g∈Lip(B)

A_i,n(g) =O_a.s.[v_n].

Proof of Lemma 2. Since g ∈ Lip(B), we can assume without loss of gener- ality that g is bounded by B on the set [0,1]^d. For any x∈Ik, we have

gn(x)−gn(ck) =σ1+σ2

where

σ1 = P

i∈ΛnYi(ai(x)−ai(ck)) P

i∈Λnai(x) and

σ2 = P

i∈Λn(ai(ck)−ai(x)) P

i∈Λnai(x)×P

i∈Λnai(ck) X

i∈Λn

Yiai(ck).

Now, by Lemma 1 and AssumptionA1), we derive that there exists constants c, η > 0 such that for any n sufficiently large

|σ1| ≤ 2^dηln/hn

c(nhn)^d X

i∈Λn

|Yi| ≤ ηvnh^d_n

c B+ 1

n^d X

i∈Λn

|εi|

!

and

|σ2| ≤ 4^dηn^dln/hn

c²(nh_n)^2d X

i∈Λn

|Yi| ≤ ηvn

c² B+ 1 n^d

X

i∈Λn

|εi|

!

Since (εi) is a stationary ergodic random field the lemma easily follows from the last inequalities and the Birkhoff ergodic theorem. The proof of Lemma 2 is complete.

Lemma 3 Assume that either C1) holds and vn = (logn)^1/2/(nhn)^d/2 or C2) holds for some 0< q <2 and vn= (logn)^1/q/(nhn)^d/2 then

A3,n =Oa.s.[vn]

Proof of Lemma 3. Let 0 < q ≤ 2 be fixed. We consider the exponential Young function define for any x ∈ R⁺ by ψq(x) = exp((x+ξq)^q)−exp(ξ_q^q) where ξq= ((1−q)/q)^1/q11{0<q<1}. Let λ >0 and x∈[0,1]^d be fixed

P(|Vn(x)|> λvn) =P Sn(x)

> λvn

X

i∈Λn

ai(x)

!

≤(1 +e^ξqq) exp

−

λvnP

i∈Λnai(x)

||P

i∈Λnai(x)εi||^ψq +ξ_q q

.

For any i∈Λn and any 0< q <2 denote bi,q(a(x)ε) =

ai(x)εi

2

ψ_β(q) + X

k∈V_i¹

q

ak(x)εkE|k−i|(ai(x)εi)

2 ψ_β(q)

(19) and

b_i,2(a(x)ε) =

a_i(x)ε_i

2

∞+ X

k∈V_i¹

ka_k(x)ε_kE_|k−i|(a_i(x)ε_i)k∞ (20) where V_i¹ = {j ∈ Z^d; j <_lex i}. Using Kahane-Khintchine inequalities (cf.

El Machkouri [16], Theorem 1) we derive that if Condition C2) holds for some 0< q <2 then

P(|Vn(x)|> λvn)≤(1 +e^ξqq) exp

−

λ vnP

i∈Λnai(x) M(P

i∈Λnbi,q(a(x)ε))^1/2 +ξq

q (21) where M is a positive constant depending only on q and on the probability kernel K. Now using the definition (19) and Lemma 1 there exist constants c, M >0 such that

sup

x∈[0,1]^d

P(|V_n(x)|> λv_n)≤(1 +e^ξqq) exp

− λ vn(P

i∈Λnai(x))^1/2

M +ξ_q

!q

≤(1 +e^ξqq) exp

−c^qλ^qv_n^q([nh_n])^dq/2 M^q

So if vn = (logn)^1/q/(nhn)^d/2 and n is sufficiently large then sup

x∈[0,1]^d

P(|Vn(x)|> λvn)≤(1 +e^ξqq) exp

− c^qλ^q logn 2^dq/2M^q

. (22)

If ConditionC1) holds then (21) still hold withq = 2 (cf. El Machkouri [16], Theorem 1). So if vn= (logn)^1/2/(nhn)^d/2 and n is large it follows that

sup

x∈[0,1]^d

P(|Vn(x)|> λvn)≤2 exp

− c²λ² logn 2^dM²

. (23)

Since

P(|A3,n|> λvn)≤ rn sup

x∈[0,1]^d

P(|Vn(x)|> λvn),

using (22) and (23), choosingλsufficiently large and applying Borel-Cantelli’s lemma, we derive

P

lim sup

n→∞ {|A3,n|> λvn}

= 0 and

P

lim sup

n→∞

|A3,n| vn ≤λ

= 1.

The proof of points 1) and 2) of Theorem 1 are completed by combining Inequality (18) with Lemmas 2 and 3.

Lemma 4 Assume that C3) holds for some p > 2 and hn = n^−θ2(logn)^θ1 for some θ1, θ2 >0. Let a, b≥0 be fixed and denote

vn= n^a(logn)^b

(nhn)^d/2 and θ = 2a(d+p)−d²−2 d(3d+ 2) . If θ≥θ2 and d(3d+ 2)θ1+ 2(d+p)b >2 then

n→+∞lim

|A_3,n| vn

= 0 a.s.

Proof of Lemma 4. Let p >2 be fixed. For any λ >0 P(|V_n(x)|> λv_n) =P |S_n(x)|> λv_nX

i∈Λn

a_i(x)

!

≤ λ^−pE|Sn(x)|^p v^pn(P

i∈Λnai(x))^p

≤ λ^−p v^pn(P

i∈Λna_i(x))^p 2pX

i∈Λn

ci(x)

!p/2

where ci(x) = ai(x)²kεik²p +ai(x)P

k∈V_i¹ak(x)kεkE|k−i|(εi)k^p₂. The last es- timate follows from a Marcinkiewicz-Zygmund type inequality by Dedecker (see [13]) for real random fields. Noting that there exists γ > 0 such that ci(x) ≤ γai(x), x ∈ [0,1]^d and using Lemma 1, we derive that there exists γ^′ >0 such that

P(|A3,n|> λvn)≤rn sup

x∈[0,1]^d

P(|Vn(x)|> λvn)≤ γ^′ τnλ^p

where τn =l^d_nv_n^p([nhn])^dp/2. Since vn =n^a(logn)^b/(nhn)^d/2 and ln =vnh^2d+1_n it follows

1 τn

= (nhn)^d(d+p)/2

h^d(2d+1)n n^a(d+p)(logn)^b(d+p)([nhn])^dp/2. If n is sufficiently large, we derive

1

τn ≤ 2^dp/2(nhn)^d(d+p)/2

h^d(2d+1)n n^a(d+p)(logn)^b(d+p)(nhn)^dp/2

= 2^dp/2

h^d(3d+2)/2n n^{a(d+p)−d2/2}(logn)^b(d+p)

≤ 2^dp/2

n(logn)^{b(d+p)+θ1d(3d+2)/2} since θ ≥θ2. Now b(d+p) +θ1d(3d+ 2)/2 >1 implies P

n≥1τ_n⁻¹ <∞. Applying Borel- Cantelli’s lemma, it follows that for any λ >0

P

lim sup

n→∞ {|A3,n|> λvn}

= 0, that is for any λ >0

P

lim sup

n→∞

|A3,n| vn ≤λ

= 1.

The proof of Lemma 4 is complete and the point3) of Theorem 1 is obtained by combining Inequality (18) with Lemmas 2 and 4. The proof of Theorem 1 is complete.

4.2 Proof of Theorem 2

We follow the first part of the proof of Theorem 1 and we consider the estimation (18).

Lemma 5 Assume that C3) holds for some p >2. Let a > 0 be fixed and denote

v_n= n^a

(nhn)^d/2 and θ = 2a(d+p)−d² d(3d+ 2) . If θ >0 and hn ≥n^−θ then

kA_3,nk_p =O[v_n].

Proof of Lemma5. Letp >2 andx∈[0,1]^dbe fixed. Using the Marcinkiewicz- Zygmund type inequality by Dedecker (see [13]) as in the proof of Lemma 4 there exist γ^′′, c >0 such that

kVn(x)k^p = E|Sn(x)|^p P

i∈Λnai(x)p

!1/p

≤γ^′′ X

i∈Λn

ai(x)

!−1/2

≤ γ^′′

√c([nhn])^−d/2 by Lemma 1.

It follows that

r_n^1/p sup

x∈[0,1]^dkVn(x)k^p =O vn

τn

whereτn=l^d/pn vn([nhn])^d/2. Ifnis sufficiently large thenτn ≥2^−d/2ln^d/pvn(nhn)^d/2, hence using hn ≥n^−θ we obtainτn≥2^−d/2. Finally, we derive

kA3,nk^p =k max

1≤k≤rn|Vn(xk)|k^p ≤r_n^1/p sup

x∈[0,1]^dkVn(x)k^p =O[vn].

The proof of Lemma 5 is complete. The point 1) of Theorem 2 is obtained by combining inequality (18) and lemmas 2 and 5.

Now, we are going to prove the point 2) of Theorem 2. We have E(Sn(x)²) = X

k,l∈Λn

ak(x)al(x)E(εkεl)

= X

k∈Λn

ak(x)²E(ε²_k) +X

k6=l

ak(x)al(x)E(εkεl)

=E(ε²₀) X

k∈Λn

ak(x)²+ X

k∈Λn

ak(x) X

l∈Λn\{k}

al(x)E(εkεl)

≤X

l∈Z^d

|E(ε0εl)| × X

k∈Λn

ak(x).

If Condition C4) holds then using Lemma 1 there exists γ >0 such that for any x ∈ [0,1]^d we have E(S_n(x)²) ≤ γQd

k=1[n(x_k+h_n)]. Let x ∈ [0,1]^d be fixed, using Lemma 1, there exists c >0 such that

kVn(x)k² = kSn(x)k² P

i∈Λnai(x)

≤

√γ c

d

Y

k=1

[n(x_k+h_n)]

!^−1/2

≤

√γ c([nh_n])^d/2

≤ 2^d/2√γ

c(nh_n)^d/2 for n sufficiently large.

The proof of Theorem 2 is complete.

4.3 Proof of Proposition 1

Since g ∈ Lip(B), it follows that

|Bn(x)|=

P

i∈Λn(g(i/n)−g(x))ai(x) P

i∈Λnai(x)

≤Bh_n P

i∈Λnk(i/n−x)/hnkai(x) P

i∈Λnai(x)

≤Bhn.

The proof of Proposition 1 is complete.

4.4 Proof of Corollary 1

Let h_n= (n^−dlogn)^1/(2+d) then Proposition 1 gives sup

x∈[0,1]^d

sup

g∈Lip(B)|Eg_n(x)−g(x)|=O

"

logn n^d

_2+d¹ #

. (24)

Assume that C1) holds. Noting that (logn)^1/2

(nhn)^d/2 =

logn n^d

_2+d¹

and using (5) we obtain sup

x∈[0,1]^d|gn(x)−Egn(x)|=Oa.s.

"

logn n^d

_2+d¹ #

. (25)

Combining (24) and (25) we derive (10).

Assume that C2) holds for some 0< q <2. Noting that (logn)^1/q

(nhn)^d/2 =

logn n^d

_2+d¹

×(logn)^(2−q)/2q and using (6) we obtain

sup

x∈[0,1]^d|gn(x)−Egn(x)|=Oa.s.

"

logn n^d

_2+d¹

×(logn)^(2−q)/2q

#

. (26) Combining (24) and (26) we derive (11).

Letε >0 be fixed and assume that C3) holds for some p >2 which satisfies condition (12). Applying the point 3) of Theorem 1 withθ1 = 1/(2 +d) and θ₂ =d/(2 +d) and noting that

v_n= n^a(logn)^b (nhn)^d/2 =n^ε

logn n^d

_2+d¹

⇐⇒

a=ε and b= 1 2

it follows

sup

x∈[0,1]^d|gn(x)−Egn(x)|=Oa.s.

"

n^ε

logn n^d

_2+d¹ #

. (27)

Combining (24) and (27) we derive (13). The proof of Corollary 1 is complete.

4.5 Proof of Corollary 2

Let h_n=n^−d/(2+d) then Proposition 1 gives sup

x∈[0,1]^d

sup

g∈Lip(B)|Egn(x)−g(x)|=Oh

n^−2+dd i

. (28)

Letε >0 be fixed and assume that C3) holds for some p >2 which satisfies condition (14). Applying the point 1) of Theorem 2 and noting that

vn= n^a

(nhn)^d/2 =n^{−2+dd +ε} ⇐⇒a=ε

it follows that

sup

x∈[0,1]^d|gn(x)−Egn(x)| p

=Oh

n^{−2+dd +ε}i

. (29)

Combining (28) and (29) we derive (15).

Since h_n =n^−d/(2+d) then (nh_n)^−d/2 =h_n. So, if C4) holds then combining (28) and (9) we derive (16). The proof of Corollary 2 is complete.

4.6 Proof of Corollary 3

Let p > 2 be fixed. Using Rio’s inequality [30] (see also Dedecker [13]) we obtain the bound

kεkE|k|(ε0)k^p₂ ≤4

Z α1,∞(|k|) 0

Q^p_ε0(u)du

!2/p

(30) hence condition C^′3) is more restrictive than condition C3).

By Serfling’s inequality (see McLeish [27] or Serfling [32]) we know that kε_kE_|k|(ε₀)k∞≤2kε₀k²∞φ_∞,1(|k|)

so condition C^′1) is more restrictive than condition C1).

Now for 0 < q < 2 there exists C(q) > 0 (cf. Inequality (17) in [16]) such that

q

|εkE_|k|(ε0)|

2

ψ_β(q) ≤C(q)q

φ∞,1(|k|). (31) In [16] we used the following lemma which can be obtain by the expansion of the exponential function.

Lemma 6 Let β be a positive real number and Z be a real random variable.

There exist positive universal constants Aβ andBβ depending only onβ such that

Aβ sup

p>2

kZk^p

p^1/β ≤ kZk^ψβ ≤Bβ sup

p>2

kZk^p p^1/β . Consider the coefficient ck(β) given by (3) and denote

dk(p) =

Z α1,∞(|k|) 0

Q^p_ε0(u)du

!1/p

then the following version of lemma 6 holds.

Lemma 7 Let β be a positive real number. There exist positive universal constants A_β and B_β depending only on β such that for any k ∈Z^d

Aβsup

p>2

d_k(p)

p^1/β ≤ck(β)≤Bβsup

p>2

d_k(p) p^1/β .

Now combining lemmas 6 and 7 and inequality (30) there exists C^′(q) > 0 such that

q

|εkE|k|(ε0)|

2

ψβ(q) ≤C^′(q)c²_k(β(q)). (32) Finally condition C^′2) is more restrictive than conditionC2) and the proof of Corollary 3 is complete.

Aknowledgements. I would like to express my thanks to the anonymous referee for his/her careful reading of the manuscript and valuable sugges- tions. I am indebted for ´E. Youndje for many stimulating conversations on nonparametric estimation.

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Mohamed EL MACHKOURI

Laboratoire de Math´ematiques Rapha¨el Salem UMR 6085, Universit´e de Rouen

Site Colbert,

F76821 Mont-Saint-Aignan Cedex

email : [email protected]