On the adaptive estimation of a multiplicative separable regression function

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On the adaptive estimation of a multiplicative separable regression function

Christophe Chesneau

To cite this version:

Christophe Chesneau. On the adaptive estimation of a multiplicative separable regression function.

2013. �hal-00853587�

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On the adaptive estimation of a multiplicative separable regression function

Christophe Chesneau

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Abstract We investigate the estimation of a multiplicative separable regression function from a bi-dimensional 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.

Keywords Nonparametric regression· Multiplicative separable regression function·Wavelet methods.

2000 Mathematics Subject Classification62G08, 62G20.

1 Motivations

We consider the bi-dimensional nonparametric regression model with random design described as follows. Let (Yi, Ui, Vi)_i∈_Z be a stochastic process defined on a probability space (Ω,A,P), where

Yi=h(Ui, Vi) +ξi, i∈Z, (1) (ξ_i)_i∈_Z is a strictly stationary stochastic process, (U_i, V_i)_i∈_Z is 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 that E(ξ₁) = 0, E(ξ₁²) exists, (U_i, V_i)_i∈_Zare independent, (ξ_i)_i∈_Z are independent and, for anyi∈Z, (U_i, V_i) and ξ_i are independent. In this study, we focus our attention on the case wherehis a multiplicative separable regression function: there exist two functionsf : [0,1]→Randg: [0,1]→Rsuch that

h(x, y) =f(x)g(y). (2)

Laboratoire de Math´ematiques Nicolas Oresme, Universit´e de Caen Basse-Normandie, Cam- pus II, Science 3, 14032 Caen, France. E-mail: chesneau@math.unicaen.fr

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We aim to estimatehfrom thenrandom 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 (1995), Yatchew and Bos (1997), Pinske (2000), Lewbel and Linton (2007) and Jacho-Ch´avez et al.(2010).

In this paper, we provide a theoretical contribution to the subject by introducing a new general estimation method forh. A sharp upper bound for its associated mean integrated squared error (MISE) is proved. Then we adapt our methodology to propose an efficient and adaptive wavelet procedure. It is based on two wavelet thresholding estimators having the features to be adaptive for a wide class of unknown functions and enjoying nice MISE properties.

Further details on such wavelet estimators can be found in, e.g., Antoniadis (1997), Vidakovic (1999) and H¨ardleet al.(1998). Despite the so-called ”curse of dimension“ coming from the bi-dimensionality 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 (1995) via non adaptive kernel methods for the structured nonparametric regression model.

The paper is organized as follows. Assumptions on (1) and some notations are introduced in Section 2. Section 3 presents our general MISE result. Section 4 is devoted to our wavelet estimator and its performances in terms of rate of convergence under the MISE over Besov balls. Technical proofs are collected in Section 5.

2 Assumptions and notations

For anyp≥1, we set L^p([0,1]) =

(

v: [0,1]→R; ||v||p= Z 1

0

|v(x)|^pdx ^1/p

<∞ )

. We set

e_o= Z 1

0

f(x)dx, e_∗= Z 1

0

g(x)dx, (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. (H2) There exists a known constantC2>0 such that

sup

x∈[0,1]

|g(x)| ≤C₂.

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(H3) The density of (U1, V1), denoted by q, is known and there exist two constants c3>0 andC3>0 such that

c₃≤ inf

(x,y)∈[0,1]²q(x, y), sup

(x,y)∈[0,1]²

q(x, y)≤C₃. (H4) There exists a known constantω >0 such that

|eoe_∗| ≥ω.

The assumptions (H1) and(H2), involving the boundedness ofh, 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 ofe_o ande_∗(depending on f andgrespectively).

3 MISE result

Theorem 1 presents an estimator forhand shows an upper bound for its MISE.

Theorem 1 We consider (1)under (H1)-(H4). We introduce the following estimator for h(2):

ˆh(x, y) =

f˜(x)˜g(y)

˜

e 1_{|˜_e|≥ω/2}, (3)

where f˜ denotes an arbitrary estimator for f e_∗ in L²([0,1]), g˜ denotes an arbitrary estimator forgeo inL²([0,1]),1denotes the indicator function,

˜ e= 1

n

X

i=1

Yi

q(Ui, Vi) andω refers to (H4).

Then there exists a constant C >0 such that E

Z 1 0

(ˆh(x, y)−h(x, y))²dxdy

≤C

E(||˜g−ge_o||²₂) +E(||f˜−f e_∗||²₂) +E(||˜g−ge_o||²₂||f˜−f e_∗||²₂) +1 n

. The form of ˜h(3) is derived to the multiplicative separable structure ofh(2) and a ratio-type normalization. Other results about such ratio-type estimators in a general statistical context can be found in Vasiliev (2012).

Based on Theorem 1, ˆhis efficient forhif and only if ˜f is efficient forf e_∗ and ˜g is efficient for geo in terms of MISE. This result motivates the investigation of wavelet methods enjoying adaptivity for a wide class of unknown functions and having optimal properties under the MISE. For details on the in- terests of wavelet methods in nonparametric statistics, we refer to Antoniadis (1997), Vidakovic (1999) and H¨ardleet al.(1998).

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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 construction of wavelet basis on the interval [0,1]

introduced by Cohen et al. (1993). Let N be a positive integer, φ and ψ be the initial wavelets of the Daubechies orthogonal waveletsdb2N. We set

φ_j,k(x) = 2^j/2φ(2^jx−k), ψ_j,k(x) = 2^j/2ψ(2^jx−k).

With appropriate treatments at the boundaries, there exists an integerτ satisfying 2^τ ≥ 2N such that the collection S = {φτ,k(.), k ∈ {0, . . . ,2^τ − 1}; ψj,k(.); j ∈ N− {0, . . . , τ−1}, k ∈ {0, . . . ,2^j−1}}, is an orthonormal basis ofL²([0,1]).

For any integer`≥τ, any v∈L²([0,1]) can be expanded onS as

v(x) =

2^`−1

X

k=0

α`,kφ`,k(x) +

∞

X

j=`

2^j−1

X

k=0

βj,kψj,k(x), x∈[0,1],

whereα_j,k andβ_j,k are the wavelet coefficients ofv defined by αj,k=

Z 1 0

v(x)φj,k(x)dx, βj,k= Z 1

0

v(x)ψj,k(x)dx. (4)

4.2 Besov balls

For the sake of simplicity, we consider the sequential version of Besov balls defined as follows. LetM >0,s >0,p≥1 andr≥1. A functionv belongs to B_p,r^s (M) if and only if there exists a constant M^∗ >0 (depending onM) such that the associated wavelet coefficients (4) satisfy

2^{τ(1/2−1/p)}

2^τ−1

X

k=0

|ατ,k|^p

!^1/p +







∞

X

j=τ





2j(s+1/2−1/p)





2^j−1

X

k=0

|βj,k|^p





1/p





r





1/r

≤M^∗.

In this expression,sis a smoothness parameter andpandrare norm parameters. For a particular choice ofs, pandr, B_p,r^s (M) contains the H¨older and Sobolev balls. See, e.g., Devore and Popov (1988), Meyer (1992) and H¨ardle et al.(1998).

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4.3 Hard thresholding estimators

In the sequel, we consider (1) under(H1)-(H4).

We focus our attention on wavelet hard thresholding estimators for ˜fand ˜g in (3). 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 other are removed. This selection is the key to the adaptivity and the good performances of the hard wavelet estimators. See, e.g., Donoho et al.(1996), Delyon and Juditsky (1996) and H¨ardleet al. (1998).

Estimatorf˜forf e_∗.We define the hard thresholding estimator ˜f by f˜(x) =

2^τ−1

X

k=0

ˆ

ατ,kφτ,k(x) +

j₁

X

j=τ 2^j−1

X

k=0

βˆj,k1{^|^β^ˆj,k|≥κC∗λn}ψj,k(x), (5)

where

ˆ

ατ,k= 1 an

a_n

X

i=1

Yi

q(Ui, Vi)φτ,k(Ui), an is the integer part ofn/2,

βˆj,k= 1 an

an

X

i=1

Wi,j,k1_{|W_i,j,k_|≤C_∗/λ_n}, Wi,j,k= Yi

q(Ui, Vi)ψj,k(Ui), j₁is the integer satisfying

1 2

an

lnan

<2^j¹ ≤ an

lnan

, κis a large enough constant,C_∗=p

2(C₃/c²₃)(C₁²C₂²+E(ξ²₁)) and λn=

rlnan

an

.

Estimatorg˜forgeo.We define the hard thresholding estimator ˜gby

˜ g(x) =

2^τ−1

X

k=0

ˆ

υτ,kφτ,k(x) +

j₂

X

j=τ 2^j−1

X

k=0

θˆj,k1{^|^θ^ˆj,k|≥κ∗C∗ηn}ψj,k(x), (6)

where

ˆ υ_τ,k= 1

bn bn

X

i=1

Y_a_n_+i

q(Ua_n+i, Va_n+i)φ_τ,k(V_a_n_+i), a_n is the integer part ofn/2,b_n=n−a_n,

θˆj,k= 1 b_n

b_n

X

i=1

Za_n+i,j,k1_{|Z_an+i,j,k_|≤C_∗_/η_n_}, Za_n+i,j,k= Ya_n+i

q(U_a_n_+i, V_a_n_+i)ψj,k(Va_n+i),

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j2is the integer satisfying 1 2

bn

lnbn

<2^j² ≤ bn

lnbn

, κ_∗ is a large enough constant,C_∗=p

2(C₃/c²₃)(C₁²C₂²+E(ξ²₁)) and ηn=

rlnbn

b_n .

Estimator forh: From ˜f (5) and ˜g(6), we consider the following estimator for h(2):

ˆh(x, y) =

f˜(x)˜g(y)

˜

e 1_{|˜_e|≥ω/2}, (7)

where

˜ e= 1

n

X

i=1

Yi

q(Ui, Vi) andω refers to(H4).

Let us mention that ˜his adaptive in the sense that it does not depend on f or gin its construction.

Remark 1 Since ˜f is defined with (Y1, U1, V1), . . . ,(Ya_n, Ua_n, Va_n) and ˜gis defined with (Ya_n+1, Ua_n+1, Va_n+1), . . . ,(Yn, Un, Vn), thanks to the independence of (Y1, U1, V1), . . . ,(Yn, Un, Vn), ˜f and ˜gare independent.

Remark 2 The calibration of the parameters in ˜f and ˜gis based on theoretical considerations; thus defined, ˜fand ˜gcan attain a fast rate of convergence under the MISE over Besov balls. See (Chaubeyet al. 2013, Theorem 6.1). Further details are given in the proof of Theorem 2.

4.4 Rate of convergence

Theorem 2 investigates the rate of convergence attains by ˆhunder the MISE over Besov balls.

Theorem 2 We consider (1) under (H1)-(H4). Let ˆhbe (7) andh be (2).

Suppose that

– f ∈ B_p^s¹₁_,r₁(M1) with M1 > 0, r1 ≥ 1, either {p1 ≥ 2 and s1 > 0} or {p1∈[1,2)ands1>1/p1},

– g ∈ B_p^s²₂_,r₂(M2) with M2 > 0, r2 ≥ 1, either {p2 ≥ 2 and s2 > 0} or {p2∈[1,2)ands2>1/p2}.

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Then there exists a constant C >0 such that E

Z 1 0

≤C lnn

n

^2s∗/(2s∗+1)

, wheres_∗= min(s1, s2).

The rate of convergence (lnn/n)^2s^∗^/(2s^∗⁺¹⁾is the near optimal one in the minimax sense for the unidimensional regression model with random design under the MISE over Besov ballsB_p,r^s^∗(M). See, e.g., Tsybakov (2004) and H¨ardleet al.(1998). In this sense, Theorem 2 proves that our estimator escapes to the so-called “curse of dimension”. Such a result is not possible with the standard bi-dimensional hard thresholding estimator attaining the rate of convergence (lnn/n)^2s/(2s+d) withd= 2 under the MISE over bi-dimensional Besov balls defined withsas smoothness parameter. See Delyon and Juditsky (1996).

Theorem 2 completes asymptotic results proved by Linton and Nielsen (1995) investigating this problem for the structured nonparametric regression model via another estimation method based on non adaptive kernels.

Remark 3 In Theorem 2, we take into account both the homogeneous zone of Besov balls, i.e., {p1 ≥2 and s1 >0}, and the inhomogeneous zone, i.e., {p1 ∈ [1,2) and s1 > 1/p1}, for the case f ∈ B_p^s¹₁_,r₁(M1), and the same for g∈B_p^s₂²_,r₂(M2). This has the advantage to cover a very rich class of unknown regression functionsh.

Remark 4 Note that Theorem 2 does not require the knowledge of the distribution ofξ₁;{E(ξ₁) = 0 and the existence ofE(ξ₁²)} is enough.

Remark 5 Our study can be extended to the multidimensional case considered by Yatchew and Bos (1997), i.e.,f : [0,1]^q¹ →Randg : [0,1]^q² →R,q1 and q2 denoting two positive integer. In this case, adapting our framework to the multidimensional case (q1 dimensional Besov balls,q1dimensional (tensorial) wavelet basis,q1 dimensional wavelet hard thresholding estimator,. . . see, for instance, Delyon and Juditsky (1996)), one can prove that (3) attains the rate of convergence (lnn/n)^2s^∗^/(2s^∗^+q^∗⁾, where s∗ = 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 ˆh(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}.

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

|ˆh(x, y)−h(x, y)| ≤ 2

ω|f˜(x)˜g(y)−f(x)g(y)˜e|+|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|

. (8)

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).˜

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

|f˜(x)˜g(y)−f(x)g(y)˜e|

≤C |˜g(y)−g(y)e_o|+|f˜(x)−f(x)e_∗|+|˜g(y)−g(y)e_o||f˜(x)−f(x)e_∗| +|˜e−e_∗e_o|

. (9)

Putting (8) and (9) together, we obtain

|ˆh(x, y)−h(x, y)|

≤C |˜g(y)−g(y)e_o|+|f˜(x)−f(x)e_∗|+|˜g(y)−g(y)e_o||f˜(x)−f(x)e_∗| +|˜e−e_∗e_o|

.

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 Z 1

0

Z 1 0

≤C E(||˜g−ge_o||²₂) +E(||f˜−f e_∗||²₂) +E(||˜g−ge_o||²₂||f˜−f e_∗||²₂) +E((˜e−e∗eo)²)

. (10)

Now observe that, owing to the independence of (U_i, V_i)_i∈_Z, the independence between (U1, V1) andξ1, andE(ξ1) = 0, we obtain

E(˜e) =E

Y1

q(U1, V1)

=E

h(U1, V1) q(U1, V1)

+E(ξ1)E 1

q(U1, V1)

= Z 1

0

Z 1 0

f(x)g(y)

q(x, y) q(x, y)dxdy= Z 1

0

f(x)dx Z 1

0

g(y)dy

=e_∗e_o. (11)

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Then, using similar arguments to (11), (a+b)²≤2(a²+b²), (a, b)∈R²,(H1), (H2), (H3)andE(ξ₁²)<∞, we have

E((˜e−e_∗e_o)²) =V(˜e) = 1 nV

Y1

q(U₁, V₁)

≤ 1 nE

Y1

q(U₁, V₁) ²!

≤ 2 nE

(h(U1, V1))²+ξ₁² (q(U₁, V₁))²

≤ 2

c²₃ C₁²C₂²+E(ξ₁²)1 n

=C1

n. (12)

Equations (10) and (12) yield the desired inequality:

E Z 1

0

Z 1 0

≤C

E(||˜g−geo||²₂) +E(||f˜−f e_∗||²₂) +E(||˜g−geo||²₂||f˜−f e_∗||²₂) +1 n

.

Proof of Theorem 2.We aim to apply Theorem 1 by investigating the rate of convergence attains by ˜f and ˜g under the MISE over Besov balls.

First of all, remark that, for γ ∈ {φ, ψ}, any integer j ≥ τ and any k ∈ {0, . . . ,2^j−1},

– using similar arguments to (11), we obtain

E 1 an

a_n

X

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

γ_j,k(U₁) q(U1, V1)

= Z 1

0

Z 1 0

f(x)g(y)

q(x, y) γ_j,k(x)q(x, y)dxdy

= Z 1

0

f(x)γj,k(x)dx Z 1

0

g(y)dy

= Z 1

0

(f(x)e_∗)γj,k(x)dx.

(11)

– using similar arguments to (12) and||γj,k||²₂= 1, we have

an

X

i=1

E

Y_i

q(Ui, Vi)γ_j,k(U_i) 2!

=E

Y1

q(U1, V1)γj,k(U1) 2!

an

≤2E

(h(U1, V1))²+ξ²₁

(q(U1, V1))² (γj,k(U1))²

an

≤ 2

c²₃ C₁²C₂²+E(ξ₁²)

E (γj,k(U1))² an

= 2

c²₃ C₁²C₂²+E(ξ₁²) Z 1

0

(γj,k(x))² Z 1

0

q(x, y)dy

dxan

≤2C3

c²₃ C₁²C₂²+E(ξ₁²)

||γj,k||²₂an =C_∗²an, withC_∗²= 2(C₃/c²₃)(C₁²C₂²+E(ξ₁²)).

Applying (Chaubeyet al.2013, Theorem 6.1) (see Appendix) with ”n=µ_n= υ_n =a_n“, ”δ = 0“, ”θ_γ =C_∗“, and f ∈B_p^s¹

1,r₁(M₁) (so f e_∗ ∈B_p^s¹

1,r₁(M₁e_∗)) withM1>0,r1≥1, either{p1≥2 ands1>0}or{p1∈[1,2) ands1>1/p1}, we prove the existence of a constantC >0 such that

E(||f˜−f e_∗||²₂)≤C lnan

a_n

^2s1/(2s₁+1)

≤C lnn

n

^2s1/(2s₁+1)

, (13) fornlarge enough.

The MISE of ˜g can be investigated in a similar way: for γ ∈ {φ, ψ}, any integerj ≥τ and anyk∈ {0, . . . ,2^j−1},

– we show that E

1 bn

bn

X

i=1

Y_a_n_+i

q(Ua_n+i, Va_n+i)γ_j,k(V_a_n_+i)

!

= Z 1

0

(g(x)e_o)γ_j,k(x)dx.

– we show that

b_n

X

i=1

E

Ya_n+i

q(U_a_n_+i, V_a_n_+i)γj,k(Va_n+i) ²!

≤C_∗²bn, with alwaysC_∗²= 2(C3/c²₃)(C₁²C₂²+E(ξ₁²)).

Applying again (Chaubeyet al.2013, Theorem 6.1) (see Appendix) with ”n= µ_n=υ_n =b_n“, ”δ= 0“, ”θ_γ =C_∗“ andg∈B_p^s²

2,r₂(M₂) withM₂>0,r₂≥1, either {p2 ≥ 2 and s₂ > 0} or {p2 ∈ [1,2) and s₂ > 1/p₂}, we prove the existence of a constantC >0 such that

E(||˜g−geo||²₂)≤C lnbn

b_n

^2s2/(2s₂+1)

≤C lnn

n

^2s2/(2s₂+1)

, (14)

(12)

fornlarge enough.

Using the independence between ˜f and ˜g (see Remark 1), it follows from (13) and (14) that

E(||˜g−geo||²₂||f˜−f e_∗||²₂) =E(||˜g−geo||²₂)E(||f˜−f e_∗||²₂)

≤C lnn

n

4s1s2/(2s1+1)(2s2+1)

. (15)

Owing to Theorem 1, (13), (14) and (15), we get E

Z 1 0

≤C

E(||˜g−ge_o||²₂) +E(||f˜−f e_∗||²₂) +E(||˜g−ge_o||²₂||f˜−f e_∗||²₂) +1 n

≤C

lnn n

2s2/(2s2+1)

+ lnn

n

2s1/(2s1+1)

+ lnn

n

4s1s2/(2s1+1)(2s2+1)

+1 n

!

≤C lnn

n

^2s∗/(2s∗+1)

, withs_∗= min(s1, s2).

Theorem 2 is proved.

Appendix

Let us now present in details (Chaubey et al. 2013, Theorem 6.1) used two times in the proof of Theorem 2.

We consider a general form of the hard thresholding estimator denoted by fˆH for estimating an unknown function f ∈ L²([0,1]) from n independent random variablesW1, . . . , Wn:

fbH(x) =

2^τ−1

X

k=0

αbτ,kφτ,k(x) +

j1

X

j=τ 2^j−1

X

k=0

βbj,k1{|bβ_j,k|≥κϑj}ψj,k(x), (16) where

αb_j,k= 1 υn

n

X

i=1

q_i(φ_j,k, W_i),

βbj,k= 1 υ_n

n

X

i=1

qi(ψj,k, Wi)1_{|q_i_(ψ_j,k_,W_i_)|≤ς_j_},

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ςj=θψ2^δj υn

√µnlnµn

, ϑj =θψ2^δj s

lnµn

µn

, κ≥2 + 8/3 + 2p

4 + 16/9andj1 is the integer satisfying 1

2µ^1/(2δ+1)_n <2^j¹ ≤µ^1/(2δ+1)_n . Here, we suppose that there exist

– n functionsq1, . . . , qn with qi:L²([0,1])×R→R for anyi∈ {1, . . . , n}, – two sequences of real numbers(υ_n)_n∈_Nand(µ_n)_n∈_Nsatisfyinglim_n→∞υ_n =

∞ andlim_n→∞µ_n=∞ such that, forγ∈ {φ, ψ},

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

υ_n

n

X

i=1

qi(γj,k, Wi)

!

= Z 1

0

f(x)γj,k(x)dx.

(A2). there exist two constants,θγ>0 andδ≥0, such that, for any integer j ≥τ and any k∈ {0, . . . ,2^j−1},

n

X

i=1

E

(q_i(γ_j,k, W_i))²

≤θ_γ²2^2δjυ²_n µn

.

Let fˆ_H be (16)under (A1) and (A2). Suppose that f ∈B^s_p,r(M) withr ≥1, {p ≥2 ands ∈ (0, N)} or {p ∈[1,2) and s∈ ((2δ+ 1)/p, N)}. Then there exists a constantC >0 such that

E

kfbH−fk²₂

≤C lnµn

µ_n

2s/(2s+2δ+1)

.

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