S Plancade

ESTIMATION OF THE DENSITY OF REGRESSION ERRORS BY POINTWISE MODEL SELECTION

Keywords: MSC 2000 Subject Classifications. 62G07-62G08 Keywords and phrases. density, regression error, pointwise model selection, adaptivity

This paper presents two results: a density estimator and an estimator of regression error density. We first propose a density estimator constructed by model selection, which is adaptive for the quadratic risk at a given point. Then we apply this result to estimate the error density in an homoscedastic regression framework Yi = b(Xi) + ǫi, from which we observe a sample (Xi, Yi). Given an adaptive estimator b b of the regression function, we apply the density estimation procedure to the residuals b ǫi = Yib b(Xi). We get an estimator of the density of ǫi whose rate of convergence for the quadratic pointwise risk is the maximum of two rates: the minimax rate we would get if the errors were directly observed and the minimax rate of convergence of b b for the quadratic integrated risk.

Introduction

Consider a sample (X i , Y i ) from the homoscedastic regression framework:

(1)

Y i = b(X i ) + ǫ i
where the (ǫ i ) are unobserved independent identically distributed (i.i.d.) data with common density f , with zero mean and independent of the (X i ). The main goal of this paper is to propose an estimator for the density of ǫ i , and to provide an upper bound for the quadratic risk of this estimator at a fixed point x 0 . The main issue in regression problems is to predict Y i by measuring only X i . The first step in such study is the estimation of the regression function b(x) = E[Y |X = x]. This question has already been studied at length. The second step consists in studying the variations of Y i around its conditional mean, which are characterized by the density of the errors (ǫ i ).

The knowledge of an estimator of the error density has many applications: for example, it allows model validation and, combined with an estimator of the regression function, it provides confidence intervals for future observations Y . The reader is referred to [START_REF] Efromovich | Estimation of the density of regression errors[END_REF] for practical applications. Many papers are devoted to density estimation but the difficulty in our problem is to estimate the density from a sample (ǫ i ) which is not observed. The natural approach consists in computing proxies of the (ǫ i ), i.e. quantities based on the data which estimate the true (ǫ i ), and applying to them a density estimation procedure as if they were the true error sample. Observing that ǫ i = Y ib(X i ), we naturally estimate the errors by the residuals ( ǫ i = Y ib(X i )), where b is an estimator of the regression function. Efromovich applies this strategy with a thresholding density estimation procedure (see for example [START_REF] Efromovich | Estimation of the density of regression errors[END_REF]). He gets an estimator of the density of the (ǫ i ) whose L 2 -risk reaches the same minimax rate of convergence we would obtain if the (ǫ i ) were observed. Nevertheless, this result requires strong conditions of regularity on the regression function b, and on the density of the (X i ) and (ǫ i ). Another estimator is built in [START_REF] Plancade | Non parametric estimation of the density of the regression noise[END_REF] by model selection. Its L 2 -risk has a rate equal to the maximum of the minimax rates of estimation of b and f if the sample (ǫ i ) was observed. Let us also mention the papers [START_REF] Akritas | Non parametric estimation of the residuals distribution[END_REF] and [START_REF] Kiwitt | Empirical likelihood for the error distribution in nonparametric regression models[END_REF] which propose estimators of the regression errors distribution functions. But to the author's knowledge, no paper studies pointwise estimation of the error density by any method.

The estimators presented in this paper are based on a pointwise model selection procedure. Model selection theory has been initiated by Birgé and Massart (see for example [START_REF] Birgé | Minimum contrast estimators on sieves: exponential bounds and rates of convergence[END_REF]), and adapted to regression function estimation in [START_REF] Baraud | Model selection for regression on a random design[END_REF] in the study of integrated quadratic risks. We will use here the estimator b of b proposed in [START_REF] Baraud | Model selection for regression on a random design[END_REF], constructed by a model selection procedure based on least square estimators. Although the principle of pointwise model selection is the same, the techniques to carry it out are different. In particular, the key tool to prove the adaptivity of classical model selection estimators is the Talagrand inequality, whereas the adaptivity of pointwise model selection estimators comes out of a simpler Bernstein inequality. The techniques developed in this paper are based on [START_REF] Laurent | Adaptive estimation of linear functionals by model selection[END_REF], in which they develop these methods in a different framework.

This paper presents two results. On the one hand, we build a density estimator which proves to be adaptive for the pointwise risk over some classical classes of regularity. Such estimators have been constructed using kernel methods in [START_REF] Butucea | Exact adaptive pointwise estimation on sobolev classes of densities[END_REF], with the same adaptivity properties, along with minimax results over Sobolev classes. Nevertheless, our estimator is completely data driven, whereas the estimation procedure in [START_REF] Butucea | Exact adaptive pointwise estimation on sobolev classes of densities[END_REF] brings into play upper bounds on unknown quantities. The second result proceeds from the application of the above density estimation procedure to residuals from the framework (1). We get an estimator of the error density, whose pointwise rate of convergence is the maximum of these two rates: the pointwise minimax rate of estimation of f we would get if the errors (ǫ i ) where observed and the L 2 -minimax rate of estimation of b.

The paper is organized as follows. Section 2 presents the estimator of the error density and the main result. The theoretical tools used to obtain this result are described in Sections 3 and 4. More precisely, Section 3 is devoted to the construction of a density estimator by pointwise model selection, and the study of its convergence properties. In Section 4, we present an estimator of the regression function and apply the density estimation procedure described in Section 3 to the residuals. Section 5 is dedicated to numerical results. Most of the proofs are gathered in Section 6.

Main result

2.1. Notations. Let t be a function defined on an interval I of R and µ be a density on I. We consider different norms of t:

t ∞ := sup x∈I |t(x)|, t := I t 2 (x)dx 1/2 , t µ := I t 2 (x)µ(x)dx 1/2
.

Besides, we consider the following spaces of functions over I:

L 2 (I) := {t : I → R, t < +∞}, L ∞ (I) := {t : I → R, t ∞ < +∞}.
If t is a function k times differentiable, we denote by t (k) its k-th derivative.

For every set A, we denote by 1I A the indicator function of A, that is 1I A (x) = 1 if x ∈ A and 1I A (x) = 0 otherwise. For every function t : R → R, we denote by t * the Fourier transform of t:

t * (u) = x∈R t(x)e -iux dx, ∀u ∈ R
For every linear space S m we denote by t m the L 2 -orthogonal projection of t onto S m . We consider the following Sobolev classes, for every α, L > 0:

W (α, L) = {F ∈ L 2 (R), 1 2π R |F * (u)| 2 u 2α du ≤ L 2 }.
The Hölder classes are defined as follows. For every β, L > 0 , and r the largest integer less than β, let:

H(β, L) = {F ∈ L 2 (R), |F (r) (x) -F (r) (y)| ≤ L|x -y| β-r , ∀x, y ∈ R}
Finally, for every x ∈ R, we denote by E(x) its integer part, that is E(x) ∈ Z and:

E(x) ≤ x < E(x) + 1.
All throughout the paper, C i denotes a universal numerical constant, and C, C ′ , C" denote numerical constants which only depends on the given constants of the problem and may change from one line to another.

2.2. Assumptions. We consider a 3n-sample (X i , Y i ) i∈{-n,...,-1}∪{1,...,2n} from the regression framework (1), where the (X i ) are i.i.d, the (ǫ i ) are i.i.d, independent of the (X i ) and E(ǫ 1 ) = 0. We suppose also that the following assumption holds.

H 0 (f ) : The density f is upper bounded by ν := f ∞ and is supported on I = R or on a known compact set I, that we will suppose equal to [-1, 1].

We define two collections of functions, one on R and one on [-1, 1] We consider collections of functions on R constructed from the sine-cardinal function:

φ(x) := sin(πx) πx For every m > 0, k ∈ Z, we consider φ m,k (x) := √ mφ(mx -k) for every x ∈ R, and A m
is the following model:

(2)

A m = vect{φ m,k , k ∈ Z}
The collection of models incorporates the models A m for m belonging to a grid of step 1/B, B being a fixed positive integer:

A n := {A m , m ∈ 1 B N, m ≤ M n } and M n ≤ n.
We consider also collections of functions on [-1, 1] constructed from the compact wavelet decomposition. We only recall here the definition of wavelet bases, the reader is referred to [START_REF] Meyer | Ondelettes et operateurs[END_REF] for more details. Let ψ be a function, called mother wavelet, supported on a compact set [-B, B] of regularity r, which satisfies the following conditions:

1) ψ, . . . , ψ (r) are bounded on [-B, B] 2) For every 0 ≤ k ≤ r, and ℓ ≥ 1 there exists a constant C ℓ such that:

|ψ (k) (x)| ≤ C ℓ (1 + |x|) -ℓ , ∀x ∈ [-B, B]
3)

B -B x k ψ(x)dx = 0 , ∀ 0 ≤ k ≤ r, 4) The set of functions {ψ j,k : x → 2 j/2 ψ(2 j/2 x -k), (j, k) ∈ Z 2 } is an orthonormal basis of L 2 (R)
. Consider a function ϕ called the father wavelet of regularity r and supported on [-B, B] which satisfies Assumptions 1) et 2) above, and the following assumptions: [START_REF] Meyer | Ondelettes et operateurs[END_REF] for examples of such functions ψ and ϕ. The set

3') B -B ϕ(x)dx = 1 4') The set of functions {ϕ k : x → ϕ(x -k), k ∈ Z} ∪ {ψ j,k , j ∈ N, k ∈ Z} is an orthonormal basis of L 2 (R). See
{ψ j,k , j ≥ 0, k ∈ Z} ∪ {ϕ k , k ∈ Z} is an orthonormal basis of L 2 [-1, 1]. As ψ is supported on [-B, B], the restriction of ψ j,k to [-1, 1] is identically equal to zero for all j ∈ N and k / ∈ [-2 j -B, 2 j +B]. Let us denote Γ(j) := Z ∩ [-2 j -B, 2 j + B]. Similarly, ϕ k is identically equal to zero for all k / ∈ [-B -1, B + 1] = Γ(0)
. Finally, we consider the following models:

B m := vect({ψ j,k , j = 0, . . . , m -1, k ∈ Γ(j)} ∪ {ϕ k , k ∈ Γ(0)})
and the collection of models:

B n := {B m , m ∈ N * , 2 m ≤ M n } with M n ≤ n.
Proposition 2.1. 1) For every m > 0:

(3)

k∈Z φ 2 m,k ∞ ≤ √ m.
2) There exists a constant K(B) such that, for every m ∈ N * , (4)

m-1 j=0 k∈Γ(j) ψ 2 j,k + k∈Γ(0) ϕ 2 k ∞ ≤ K 2 (B) √ 2 m .
From now on, we use common notations for these two collections of models. The collection M n is A n if f is supported on R, and B n if f is supported on [-1, 1]. We denote by S m the model A m or B m and M n = {S m , m ∈ J n }. Moreover, we denote by :

S m = vect(χ λ , λ ∈ I m }
where the functions χ λ denote the ψ j,k and the ϕ k , or the φ m,k . Thus according to Inequalities (3) and (4), we have:

λ∈Im χ 2 λ ∞ ≤ K 2 D m
with D m = m and K = 1 for the sine-cardinal models, and D m = 2 m and K = K(B) for the wavelets models. We make different assumptions, for the cases of f supported on R or on a compact set : H 1 (β) : Take I = R, we consider the collection of model M n = A n . We assume that there exist β > 0 and C 0 > 0 such that for every model

A m ∈ A n , the L 2 -orthogonal projection f m of f onto A m satisfies: f -f m ∞ ≤ C 0 D -β m
Moreover, we suppose that f is Lipschitz, i.e. there exists a constant L > 0 such that for every x, y ∈ I, |f (x)f (y)| ≤ L|x -y|. The following Proposition gives conditions ensuring that Assumption H 1 holds. The proof is given in Section 7.

Proposition 2.2. If f ∈ W (α, L) with α > 1/2, then f -f m ∞ ≤ D α-1/2 m . Moreover, if α > 3/2 then f is Lipschitz.
H 2 (β) : Take I = [-1, 1], we consider the collection of models M n = B n with regularity r, and we suppose that f ∈ H(β, L) for some 1 ≤ β ≤ r and L > 0.

2.3. Construction of the estimator and main result. In this subsection, we give the definition of the estimator of f , the heuristical motivation concerning its construction being developed in the following sections. Let x 0 be a fixed point in I. We split the sample (X i , Y i ) i∈{-n,...,-1}∪{1,...,n} into three independent samples:

(5) Z -:= (X i , Y i ) ∈{-n,...,-1} , Z + 0 := (X i , Y i ) i∈{1,...,2n} , Z + 1 := (X i , Y i ) i∈{n+1,...,2n} Let b be any estimator of b built out of the first sample Z -. An example of such an estimator is given in Section 4. Consider the residuals from the second sample:

ǫ i := Y i -b(X i ) , i ∈ {1, . . . , 2n}
Given Z -, the ( ǫ i ) are i.i.d. with common density denoted by f -.

Let ν - n be an estimator of ν -:= f -∞ built from the sample Z + 1 such that the probability P [{ν -/2 ≤ ν - n < 2ν -} c ] decreases exponentially in n. We give an explicit construction of ν - n in Section 3.5. For every model S m = vect{χ λ , λ ∈ I m }, we consider the projection density estimator associated to the sample ( ǫ i ):

(6) f - m = λ∈Im 1 n n i=1 χ λ ( ǫ i ) χ λ .
The selected model is:

(7) m = arg min m∈Jn sup j∈Jm,j≥m {( f - j -f - m ) 2 (x 0 ) -Ax j,m ν - n D m + D j n } + + AK 2 x m ν - n D m n
where A is a positive constant, and (x m ) and (x j,m ) are weights of order ln(D m ) and ln(D j + D m ) more precisely described in (19) and (20).

Finally, we define the following numerical constants, depending on the collection of models:

(8)

α 1 = [ D 1 1 + D 1 + ln(1 + D 1 )] -1
where D 1 = min{D m , m ∈ M n } and:

(9)

α 2 = max x>0 x 1/(1+x)
and we consider a positive number α 3 such that:

α 3 ≥ α 1/3 1 α 2 .
We have α 2 < 1.4, and D 1 ≤ 1 so that α 1 ≤ (1/2 + ln 2) -1 . This implies that α 3 = 1.4 works.

We can prove the following result for our estimator:

Theorem 2.1. We suppose that Assumption H 0 (f ) holds. Moreover, we suppose that either

H 1 (β) or H 2 (β) hold for some β ≥ 1, with M n = E(α 3 n 1/3 ) + 1. Then (10) E[( f - b m -f ) 2 (x 0 )] ≤ κ( n ln n ) -2β 2β+1 + κ ′ E[ b -b 2 µ ]
for some constant κ and κ ′ depending on the parameters of the problem but not on n.

Comments: Suppose that f ∈ W (β, L) for some β > 3/2, then (10) holds.

On the one hand, [START_REF] Butucea | Exact adaptive pointwise estimation on sobolev classes of densities[END_REF] proves that the minimax rate of estimation of a density over Sobolev class W (β, L) is n

-2β-1
2β . She also proves that the adaptive minimax rate of convergence (which is the best rate of convergence for adaptive estimators over all

classes of convergence W (β, L)) is (n/ ln n) -2β-1 2β .
On the other hand, we present in Section 4 an adaptive estimator b of b which reaches the minimax rate over Besov balls, from [START_REF] Baraud | Model selection for regression on a random design[END_REF].

Thus, the rate of convergence of our estimator is the maximum of the two following rates:

• the minimax rate of estimation of b over Besov balls.

• the minimax rate of estimation we would obtain for f if the (ǫ i ) were directly observed. An analogous comment holds if Assumption H 2 (β) holds.

Density estimation by pointwise model selection

In this section, we present a density estimation procedure which products adaptive estimators for the pointwise risk. This procedure is the one which is applied to the pseudo observations ǫ i of ǫ i .

The results of this section require weaker assumptions on regularity than the error density estimation, and are staten for a more general collection of models. The assumptions considered here are satisfied in particular by the collections defined in Section 2. We consider a collection of model M n = {S m , m ∈ J n } which satisfies:

H dens (β) : For every m ∈ J n and {χ λ , λ ∈ I m } an orthonormal basis of S m , there exists an positive number D m such that:

(11) f -f m ∞ ≤ C 0 D -β m (12) λ∈Im χ 2 λ ∞ ≤ K D m
for some K > 0. We denote by M n = max m∈Jn D m and we suppose that M n ≤ n.

Moreover, we suppose that the collection M n si rich enough. More precisely, we assume that there exists a constant M ≥ 1 such that for every n, for every α ∈]0, 1[ such that n α M ≤ M n , there exists a model m and ( 13)

n α ≤ D m ≤ M n α
Remark: The Property ( 13) is satisfied by the collections described in Section 2, and by most of the clasical collections.

3.1.

A preliminary risk bound. Let (V 1 , . . . , V 2n ) be a i.i.d. sample drawn from a density g, split into two samples:

(14) Z 0 := (V i ) i∈{1,...,n} , Z 1 := (V i ) i∈{n+1...,2n}
Let x 0 be a fixed point in I. For every model m ∈ M n , let g m be the projection estimator of g on S m from the sample Z 0 :

(15)

g m = λ∈Im ( 1 n n i=1 χ λ (V i ))χ λ Let g m be the L 2 -projection of g onto S m .
Observing that E[ g m (x 0 )] = g m (x 0 ), we get the following bias-variance decomposition for every model m:

E[( g m -g) 2 (x 0 )] = E[( g m -g m ) 2 (x 0 )] + (g m -g) 2 (x 0 )
On the one hand, the variance term is replaced by a bound obtained thanks to ( 13) in H dens (β). Indeed:

E[( g m (x 0 ) -g m (x 0 )) 2 ] = V ar[ λ∈Im ( 1 n n i=1 χ λ (V i ))χ λ (x 0 )] = V ar[ 1 n n i=1 ( λ∈Im χ λ (V i )χ λ (x 0 ))]
As the (V i ) are i.i.d. we get:

E[( g m (x 0 ) -g m (x 0 )) 2 ] = 1 n V ar[ λ∈Im χ λ (V 1 )χ λ (x 0 )] ≤ 1 n E[( λ∈Im χ λ (V 1 )χ λ (x 0 )) 2 ] = 1 n x∈R ( λ∈Im χ λ (x)χ λ (x 0 )) 2 g(x)dx ≤ g ∞ n x∈R ( λ∈Im χ λ (x)χ λ (x 0 )) 2 dx.
We develop the square in the integral:

E[( g m (x 0 ) -g m (x 0 )) 2 ] ≤ g ∞ n λ,λ ′ ∈Im [ x∈R χ λ (x)χ λ ′ (x)dx]χ λ (x 0 )χ λ ′ (x 0 )
Using that the functions (χ λ ) are orthonormal and (13) leads to:

E[( g m (x 0 ) -g m (x 0 )) 2 ] ≤ g ∞ n λ∈Im χ 2 λ (x 0 ) ≤ K 2 g ∞ D m n = K 2 ν D m n
This bound is standard for a variance term. Finally, for every model m ∈ J n we have the following non adaptive bound for g m :

(16) E[( g m -g) 2 (x 0 )] ≤ (g -g m ) 2 (x 0 ) + K 2 ν D m n 3.2.
Construction of the adaptive estimator. The model selection procedure developed by Birgé and Massart relies on this idea: the best model among the collection M n is the one which minimizes the squared bias-variance sum above, thus the natural idea consists in building an estimator of the right hand side in ( 16) and selecting the model m which minimizes it.

The term K 2 νD m /n is estimated by K 2 ν n D m /n where ν n is an estimator of ν defined in Section 3.3.

Let us consider the bias term (gg m ) 2 (x 0 ). Contrary to the L 2 -bias term gg m 2 in classical model selection procedure, the pointwise bias term (g mg) 2 (x 0 ) is not easy to estimate. We replace (g mg) 2 (x 0 ) by sup j∈Jn,D j ≥Dm (g jg m ) 2 (x 0 ). Indeed, those two terms have the same order according to (11) in H dens (β):

sup j∈Jn,Dm≤D j (g j -g m ) 2 (x 0 ) ≤ sup j∈Jn,Dm≤D j ≤Mn (g j -g) 2 (x 0 ) + (g m -g) 2 (x 0 ) ≤ sup j∈Jn,Dm≤D j C 0 D -2β j + C 0 D -2β m ≤ 2C 0 D -2β m (17)
Then we define the best theoretical model as:

(18) m opt := arg min m∈Jn [ sup j∈Jn,D j ≥Dm (g j (x 0 ) -g m (x 0 )) 2 + pen(m)] := arg min m∈Jn [Crit(m)]
where pen(m) := AK 2 x m ν n Dm n , A is a positive constant and x m a weight of order ln(D m ). More precisely (19) x

m := max{B 1 ln(1 + D m ); B 2 ν n ln 2 (1 + D m ) D m n } where (B 1 , B 2 ) are constants with B 1 > 16/A and B 2 > 128K 2 /A. Asymptotically x m = B 1 ln(1 + D m ).
Then, the natural idea would be to replace (g jg m ) 2 (x 0 ) by ( g jg m ) 2 (x 0 ), one can notice that this estimator is biased. In fact:

E[( g m -g j ) 2 (x 0 )] = (g j -g m ) 2 (x 0 ) + E[(( g j -g m )(x 0 ) -(g j -g m )(x 0 )) 2 ]
and the last term is a variance-type term. Therefore we use the following bound:

E[(( g j -g m )(x 0 ) -(g j -g m )(x 0 )) 2 ] ≤ 2(E[( g j -g j ) 2 (x 0 )] + E[( g m -g m ) 2 (x 0 )]) ≤ 2K 2 g ∞ D j + D m n
The last inequality is established by the same upper bounds as ( 16). So (g jg m ) 2 (x 0 ) is replaced by the positive part of ( g mg j ) 2 (x 0 ) -AK 2 ν n x j,m D j +Dm n where x j,m is a weight of order ln(1 + D m + D j ):

(20)

x j,m := max{2B 1 ln(1

+ D j + D m ); B 2 ν n ln 2 (1 + D j + D m ) D j + D m n }.
Finally the selected model m is m = arg min m∈Mn Crit(m) where:

(21) Crit(m) = sup j∈Jn,D j ≥Dm [( g j -g m ) 2 (x 0 ) -AK 2 ν n x j,m D j + D m n ] + + pen(m).
Our estimator of g is g b m .

3.3. Estimation of ν. In this section, we propose an estimator ν n of ν = g ∞ constructed from the sample Z 1 . Let m 0 be a medium-size model. More precisely, let γ ∈]1/3, 1/2[ and m 0 = min{m ∈ J n : D m 0 ≥ n γ } and p 0 = D m 0 . We define:

ν n := g m 0 ∞
The following results hold:

Proposition 3.1. Suppose that Assumption H dens (β) holds, and that for every model m ∈ M n the functions {χ λ ) λ∈Im are continuous. Then for every n such that:

(A 1 ) C 0 p -β 0 < ν/6 Then there exists a numerical constant C 1 such that:

(22) P [ ν n ≤ ν/2] ≤ 2 exp(- C 1 K 2 ν n p 0 )
If in addition:

(A 2 ) p 0 √ n ≤ ν 3K 2 then there exist numerical constants C 2 , C 3 and C 4 such that: (23) P [ ν n ≥ 2ν] ≤ exp(- C 2 K 2 ν n p 0 ) + exp(- C 3 K 4 ν 2 n 3/2 p 2 0 ) + exp(- C 4 K 2 n p 2 0 )
Comments:

1) There exists an integer N which depends on (K, β, C 0 ) such that for every n ≥ N , (A 1 ) and (A 2 ) hold.

2) The condition of continuity of the (χ λ ) prohibits the piecewise continuous bases, like for example the histograms. Nevertheless, similar upper bounds can be obtained with localised bases, included the histograms. Besides, the collections of model in which ν n and g b m are computed can be different. Theorem 3.1. Assume that H dens (β), (A 1 ) and (A 2 ) hold, there exists a constant κ which depends on (α, B 1 , B 2 ) such that the following inequality holds:

E[( g b m -g) 2 (x 0 )] ≤ κ(1 + ν)( n ln n ) -2β/(2β+1) + R n with: R n = (ν+K 2 M n ) 2 exp(-C 1 ν K 2 n p 0 )+( n ln n ) 2β/(2β+1) K 2 p 0 [exp(-C 2 K 2 ν n p 0 )+exp(-C 3 K 4 ν 2 n 3/2 p 2 0 )+ exp(-C 4 K 2 n p 2 0 )]
Comments: 1) According to (13), p 0 is of order O(n γ ) with γ ∈]1/3, 1/2[. Thus we get immediately that R n ≤ C/n for some constant C depending on (ν, K, p 0 ).

2) If g ∈ H(β, L) then g satisfies H dens (β). Besides, [START_REF] Stone | Optimal rates of convergence for nonparametric estimators[END_REF] proves that the minimax rate of convergence over the set of k times continuously differentiable functions in density estimation is n -2k/(2k+1) . Tsybakov generalized this result to Hölder classes of functions for every β > 0 (see [START_REF] Tsybakov | Introduction l'estimation non paramtrique[END_REF]). Moreover [START_REF] Lepski | Optimal pointwise adaptive methods in nonparametric estimation[END_REF] show that the adaptive minimax rate of convergence over Hölder classes for the white noise model is (n/ ln n) -2β/(2β+1) . This allows to believe that the adaptive rate of convergence over Hölder classes for density is also (n/ ln n) -2β/(2β+1) . So our estimator seems to be adaptive over Hölder classes.

According to the comments about Theorem 2.1, our estimator is also adaptive over Sobolev classes.

3.5. Application to the estimation of the error density. Now we go back to the initial issue, the estimation of the error density, and clarify the estimator defined in Section 2. Let us recall that our goal is to build an estimator of the error density f out of a sample (X i , Y i ) {i=-n,...,-1}∪{1,...2n} from regression framework (1). The sample is split into three independent samples Z -, Z + 0 and Z + 1 defined in (5). Let b be an estimator of b computed from the sample Z -and ǫ i = Y ib(X i ) for i ∈ {1, . . . , 2n} the residuals from the two other samples. Given Z -, b is fixed and the ( ǫ i ) have a density f -. Let us give f -explicitly. Let F be any function, then:

E[F ( ǫ 1 )|Z -] = E[F (ǫ 1 + (b -b)(X 1 ))|Z -] = t∈R 1 x=0 F (t + (b -b)(x)µ(x)f (t)dxdt = u∈R F (u) 1 x=0 f (u -(b -b)(x))µ(x)dxdu.
Hence:

f -(t) = 1 0 f (t -(b -b)(x)µ(x)dx.
We can easily deduce from this expression that f -is upper bounded by ν -:= f -∞ ≤ ν for every Z -. Now, we apply the density estimation procedure presented in Section 3 to the sample ( ǫ i ). For every model m, let f - m be the projection estimator defined in (6). Let m 0 = min{m : D m ≥ n γ }, and:

ν - n := f - m 0 ∞
. Then the density estimation procedure is applied to the residuals and provides the estimator f - b m , where m is the selected model ( 7).

Let us explain the basic guidelines of this result. The risk of the estimator f - b m comes from two consecutive approximations of different nature: the first one consists in replacing the errors (ǫ i ) by the residuals, and the second one is a density estimation error. These two approximations appear in the following inequality:

(24) E[( f - b m -f ) 2 (x 0 )] ≤ 2{E[( f - b m -f -) 2 (x 0 )] + E[(f --f ) 2 (x 0 )]}.
On the one hand, for a fixed sample Z -, we prove (see Lemma 6.3) that f -satisfies the Assumption H dens (β) so

E[( f - b m -f -) 2 (x 0 )|Z -] ≤ κ( n ln n
)n -2β/(2β+1) + C n By taking the expectation over Z -, we get the first term in Theorem 2.1. Actually, the constant C depends on f -and so on Z -and we need to study it more carefully to obtain this result (see Section 5).

On the other hand f -is the density of ǫ i = ǫ i + (bb)(X i ), so the difference between f and f -can be expressed in function of (bb). More precisely, we will prove that:

E[(f -f -) 2 (x 0 )] ≤ CE[ b -b 2 ].

An adaptive estimator of the regression function

In this section, we briefly exhibit an estimator b of b which suits to our setting. This is the estimator which is implemented in the simulations. The regression function estimator presented here results from Baraud's works (see [START_REF] Baraud | Model selection for regression on a random design[END_REF] and [START_REF] Baraud | Model selection for regression on a fixed design[END_REF]), gathered in [START_REF] Plancade | Non parametric estimation of the density of the regression noise[END_REF]. Consider the following assumption:

H 3 : The density µ of X 1 is supported on a compact J, and is lower bounded by a m 0 > 0 and upper bounded by m 1 < +∞. Let us consider a collection of finite dimensional models Σ n which satisfies the following assumptions:

H b : Σ n is included in a global model S n with dimension smaller than n 1/2-d for some d > 0. Furthermore, there exists some nonnegative constants Γ and R such that

|{m ∈ M n (resp.Σ n ) : D m = n} ≤ ΓD R
for every D ∈ N * . Finally, there exists a constant K such that:

t ∞ ≤ K N n t , ∀t ∈ S n .
For every model m ∈ Σ n , let b m be the least squares estimator of b:

b m := arg min t∈Sm γ n (t) where γ n (t) := 1 n -1 i=-n (Y i -t(X i )) 2 ,
and the selected model is

m = arg min m∈Σn [γ n ( b m ) + σ 2 n Dm n ]
where σ 2 n is an estimator of the variance of ǫ 1 : let V n be a space of dimension E(n/2) which includes the global model S n , then:

σ 2 n = 1 n -E(n/2) inf t∈Sn (Y i -t(X i )) 2 Let us define b = b b m if b b m ≤ n and b = 0 otherwise then: E[ b -b 2 µ ] ≤ C inf m∈Σn [ b -b m 2 + σ 2 D m n ].
Finally, classical results about approximation theory in Besov spaces lead to the following statement: if b belong to the Besov space 2α+1) . This entails the following Corollary:

B α,∞ 2 , then E[ b -b 2 µ ] ≤ Cn -2α/(
Corollary 4.1. Suppose that Assumptions H 0 (f ), H b and H 1 (β) or H 2 (β) hold, and suppose that b ∈ B α,∞ 2 with α ≥ β -1/2 then: E[( f - b m -f ) 2 (x 0 )] ≤ κ( n ln n ) -2β 2β+1
for some constant κ independent from n.

In other words, if b is smoother than f , the rate of convergence of f - b m is the optimal rate we would get if the (ǫ i ) were directly observed.

Simulations

5.1. Density estimation. This section illustrates the density estimation procedure presented in Section 3, with the sine-cardinal collection of models A n described in (2). We choose B = 10 and M n = √ n. We draw 50 samples (V 1 , . . . , V n ) of size n = 200, 500, 2000 of i.i.d. variables with gaussian distribution (denoted by N (0, 1)) and with Laplace density g(x) = 1 2 exp(-|x|) (denoted by L(1)). Let J be the set of 150 regularly spaced points on [-5, 5]. For each sample and for every point x ∈ J we compute an estimator g b m (x) as follows, assuming that the maximum of the density ν is known:

• First we compute the projection density estimators ( g m (x)) for every m ∈ 1 10 N, m ≤ M n and every x ∈ J (cf (15)).

• Then for every x ∈ J, we select the best model as:

m = arg min{sup j≥m [( g j -g m ) 2 (x) -αν ln(1 + j + m) j + m n ] + + βν ln(1 + m) m n
with α = 5 and β = 15. • We plot the set of points {(x, g b m (x)), x ∈ J} In Figure 1, each graph presents the 50 curves of g b m for a given density g i and a given n. 5.2. Error density estimation. This section proposes illustrations of the error density estimator described in Section 2, with the following procedure:

V i ∼ N (0, 1) n=200 n=500 n=2000 V i ∼ L(1) n=200
• We draw a sample (X 1 , . . . , X 2n ) with common density f X uniform on [0, 1] and χ 2 (3). We draw also a sample (ǫ 1 , . . . , ǫ 2n ) with common density f from a distribution N (0, 1) and L(1) . We 2008)). • We compute the residuals from the second sample ( ǫ i ) i=n+1,...,2n , where ǫ i = Y ib(X i ). • Let J be a set of 150 regularly spaced points on [-5, 5] and apply the density estimation procedure described in Section 5.1 to the residuals ( ǫ i ) i=n+1,...,2n . Figure 3 presents the error density estimator (dotted blue line) and the theoretical estimator we get by applying the density estimation procedure of Section 5.1 directly to the sample (ǫ i ) i=n+1,...,2n . The thick line is the true density of ǫ 1 . We have also checked that the error density estimator hardly depends on the designs' distribution. Claim 1. If Assumption H dens (β) holds, there exist constants κ and κ ′ which depend on (C 0 , B 1 , B 2 , K) such that the following inequality holds:

X i ∼ U[0, 1], ǫ i ∼ N (0, 1), b(x) = x 3 + 5x n=200 n=500 n=2000 X i ∼ χ 2 (3), ǫ i ∼ L(
(25) E 1 [( g b m -g) 2 (x 0 )] × 1I {b νn≥ν/2} ≤ κ[Crit(m opt ) + (g mopt (x 0 ) -g(x 0 )) 2 ] + κ ′ (1 + ν) n Proof of Claim 1.
For every j, m ∈ J n , we denote by:

H(j, m) := AK 2 x j,m ν n D m + D j n .
The basic idea of the proof is to upper bound E 1 [( g b mg) 2 (x 0 )]1I {b νn≥ν/2} by the sum of two terms:

(26) E 1 [( g b m -g) 2 (x 0 )1I {b νn≥ν/2} ] ≤ 2(E 1 [(( g b m -g) 2 (x 0 ) -U opt ) + ]1I {b νn≥ν/2} + E 1 [U opt ]
) where E[U opt ] is a quantity with same order as Crit(m opt ). Besides:

(27) E 1 [(( g b m -g) 2 (x 0 ) -U opt ) + ] ≤ +∞ 0 P 1 [( g b m -g) 2 (x 0 ) -U opt ≥ x]dx
and the quantity U opt will be chosen such that the probability under the integral decreases exponentially in n. Let us consider a first result: Lemma 6.1. For every δ > 0, x > 0 and for every model m:

P 1 [ Crit(m) ≥ (1 + δ)Crit(m) + x] ≤ j∈Jn,D j ≥Dm exp[-C(x, j, m)] where C(x, j, m) = min{ u νK 2 nx D j + D m +Aux j,m ν n ν ; u ′ n √ x K 2 (D j + D m ) + u ′ √ A K x j,m ν n n D j + D m } and u = 1/(8(1 + 1 δ )) and u ′ = 1/(4 √ 2 1 + 1 δ ).
Proof of Lemma 6.1: The empirical criterion Crit(m) (defined in ( 21)) is built from Crit(m) (defined in (18)) by replacing the unknown (g j ) by their empirical means ( g j ), so the deviation between Crit(m) and Crit(m) is upper bounded with Bernstein Inequality (see Appendix). More precisely:

P 1 [ Crit(m) ≥ (1 + δ)Crit(m) + x] ≤ P 1 [ sup j∈Jn,D j ≥Dm (( g j -g m ) 2 (x 0 ) -H(j, m)) + ≥ (1 + δ) sup j∈Jn,D j ≥Dm (g j -g m ) 2 (x 0 ) + x]
As sup j∈Jn,D j ≥Dm (g jg m ) 2 (x 0 ) + x is positive, we omit the positive part (.) + :

P 1 [ Crit(m) ≥ (1 + δ)Crit(m) + x] ≤ P 1 [ sup j∈Jn,D j ≥Dm (( g j -g m ) 2 (x 0 ) -H(j, m)) ≥ (1 + δ) sup j∈Jn,D j ≥Dm (g j -g m ) 2 (x 0 ) + x] ≤ j∈Jn,D j ≥Dm P 1 [( g j -g m ) 2 (x 0 ) ≥ (1 + δ)(g j -g m ) 2 (x 0 ) + x + H(j, m)] := j∈Jm,D j ≥Dm P j,m
and for every (j, m):

P j,m = P 1 [( g j -g m ) 2 (x 0 ) ≥ (1 + δ)(g j -g m ) 2 (x 0 ) + (1 + 1 δ )( x + H(j, m) (1 + 1 δ ) ) 2 ]
It follows from the inequality (x + y) 2 ≤ x 2 (1 + 1/a) + y 2 (1 + a), ∀ x, y ∈ R, a > 0 that:

P j,m ≤ P 1 [( g j -g m ) 2 (x 0 ) ≥ (|(g j -g m )(x 0 )| + x + H(j, m) 1 + 1 δ ) 2 ] = P 1 [|( g j -g m )(x 0 )| ≥ |(g j -g m )(x 0 )| + x + H(j, m) 1 + 1 δ ] ≤ P 1 [|( g j -g m )(x 0 ) -(g j -g m )(x 0 )| + |(g j -g m )(x 0 )| ≥ |(g j -g m )(x 0 )| + x + H(j, m) 1 + 1 δ ] = P 1 [| 1 n n i=1 (U i -E(U i ))| ≥ x + H(j, m) 1 + 1 δ ]
where

U i = λ∈I j χ λ (V i )χ λ (x 0 ) -λ∈Im χ λ (V i )χ λ (x 0 ) and E(U i ) = (g j -g m )(x 0
). Let us compute the terms v and c involved in Bernstein Inequality (Theorem 7.1). By the same methods as in ( 16) we get:

E 1 (U 2 1 ) ≤ 2E 1 [ g 2 j (x 0 ) + g 2 m (x 0 )] ≤ 2νK 2 (D j + D m ) := v Let ℓ be
an integer greater than 2, then:

E 1 [(U 1 ) ℓ + ] ≤ E 1 [U 2 1 ] × U 1 ℓ-2 ∞ ≤ v 2 [ λ∈Im χ λ (V 1 )χ λ (x 0 ) ∞ + λ∈I j χ λ (V 1 )χ λ (x 0 ) ∞ ] ℓ-2 ≤ v 2 [ λ∈Im χ 2 λ (V 1 ) λ∈Im χ 2 λ (x 0 ) ∞ + λ∈I j χ 2 λ (V 1 ) λ∈I j χ 2 λ (x 0 ) ∞ ] ℓ-2
Finally, Assumption H con leads to:

(28) E 1 [(U 1 ) l + ] ≤ v 2 [K 2 (D j + D m )] l-2
and c = K 2 (D j + D m ). Let us denote:

ǫ = x + H(j, m) 1 + 1 δ ≥ 1 2(1 + 1 δ ) ( √ x + H(j, m))
Then Bernstein Inequality provides the following upper bound for P j,m

P j,m ≤ 2 exp[-min( nǫ 2 4v ; nǫ 4c )]
Moreover:

nǫ 2 4v = nx 8(1 + 1 δ )νK 2 (D j + D m ) + AK 2 x j,m ν n (D j + D m ) 8(1 + 1 δ )νK 2 (D j + D m ) = u νK 2 nx D j + D m + Aux j,m ν n ν nǫ 4c ≥ n √ x 4 √ 2 1 + 1 δ K 2 (D j + D m ) + n Ax j,m ν n D j + D m n 1 4 √ 2 1 + 1 δ K 2 (D j + D m ) = u ′ n √ x K 2 (D j + D m ) + u ′ √ A K x j,m ν n n D j + D m
which provides the upper bound of Lemma 6.1. ✷

We deduce from Lemma 6.1 an upper bound for a quantity of the kind P 1 [( g b m -g) 2 (x 0 )-U opt ≥ x]: Lemma 6.2. Let δ and x some positive numbers, then for every Z 1 :

1) P 1 [{( g b m -g) 2 (x 0 ) ≥ (1 + δ)( sup j∈Jn,D j ≥Dm opt (g j -g) 2 (x 0 ) + Crit(m opt )) + x} ∩ { m > m opt }] ≤ j∈Jn,D j ≥Dm opt exp[-C(x, j, m opt )] + 2 m∈Jn exp[-C(x, m)] where C(x, m) = min{ 2u K 2 nx νD m + 2Au x m ν n ν ; u ′ K n √ x D m + u ′ √ A K nx m ν n D m }. 2) P 1 [{( g b m -g) 2 (x 0 ) ≥ 2(1 + δ)Crit(m opt ) + 2 sup j∈Jn,D j ≤Dm opt H(m opt , j) + 2( g mopt -g) 2 (x 0 ) +2x} ∩ { m ≤ m opt }] ≤ j∈Jn,D j ≥Dm opt exp[-C(x, j, m)]
Proof of Lemma 6.2:

• Let us prove inequality 1).

P 1 [{( g b m -g) 2 (x 0 ) ≥ (1 + δ)( sup j∈Jn,D j ≥Dm opt (g j -g) 2 (x 0 ) + Crit(m opt )) + x} ∩ { m > m opt }] ≤ P 1 [{( g b m -g) 2 (x 0 ) ≥ (1 + δ) sup j∈Jn,D j ≥Dm opt (g j -g) 2 (x 0 ) + Crit( m) + x} ∩ { m > m opt }] +P 1 [ Crit( m) ≥ (1 + δ)Crit(m opt )]
By definition of m, Crit( m) = inf m∈Jn Crit(m) ≤ Crit(m opt ) thus we get from Lemma 6.1:

P 1 [ Crit( m) ≥ (1 + δ)Crit(m opt )] ≤ P [ Crit(m opt ) ≥ (1 + δ)Crit(m opt ) ≤ j∈Jn,D j ≥Dm opt exp[-C(x, j, m opt )]
Besides for every model m, Crit(m) ≥ pen(m) according to the definition of Crit(m), and if m ≤ m opt , sup j∈Jn,D j ≥Dm opt (g jg) 2 (x 0 ) ≥ (g b mg) 2 (x 0 ), thus:

P 1 [{( g b m -g) 2 (x 0 ) ≥ (1 + δ) sup j∈Jn,D j ≥Dm opt (g j -g) 2 (x 0 ) + Crit( m) + x} ∩ { m > m opt }] ≤ P 1 [( g b m -g) 2 (x 0 ) ≥ (1 + δ)(g b m -g) 2 (x 0 ) + pen( m) + x] ≤ m∈Jn P 1 [( g m -g) 2 (x 0 ) ≥ (1 + δ)(g m -g) 2 (x 0 ) + pen(m) + x] := m∈Jn P m
The quantities P m are upper bounded in the same way as P j,m in the proof of Lemma 6.1, so we only give the outline of the proof. First of all we have for every model m:

P m ≤ P 1 [|( g m -g m )(x 0 )| ≥ 1 1 + 1 δ pen(m) + x] = P 1 [| 1 n n i=1 U i -E(U i )| ≥ 1 1 + 1 δ pen(m) + x]
where U i = λ∈Im χ λ (V i )χ λ (x 0 ). We apply Bernstein Inequality with the following quantities v and c:

E[U 2 1 ] ≤ νK 2 D m := v 2
For every integer l ≥ 2, similarly to inequality (28) we have:

E[(U 1 ) l + ] ≤ v 2 (K 2 D m ) l-2 thus c = K 2 D m .
Then Bernstein Inequality provides inequality 1), exactly like in the proof of Lemma 6.1.

• Let us prove now inequality 2) in Lemma 6.2. If m ≤ m opt given that pen(m) is always positive:

Crit( m) ≥ sup j∈Jn,D j ≥D b m [( g j -g b m ) 2 (x 0 ) -H(j, m)] ≥ ( g mopt -g b m ) 2 (x 0 ) -H(m opt , m) Moreover ( g b m -g) 2 (x 0 ) ≤ 2[( g b m -g mopt ) 2 (x 0 ) + ( g mopt -g) 2 (x 0 )], thus: Crit( m) ≥ 1 2 ( g b m -g) 2 (x 0 ) -( g mopt -g) 2 (x 0 ) -H(m opt , m) ≥ 1 2 ( g b m -g) 2 (x 0 ) -( g mopt -g) 2 (x 0 ) - sup j∈Jn,D j ≤Dm opt H(m opt , j)
Hence:

P 1 [{ 1 2 ( g b m -g) 2 (x 0 ) ≥ (1 + δ)Crit(m opt ) + sup j∈Jn,D j ≤Dm opt H(m opt , j) +( g mopt -g) 2 (x 0 ) + x} ∩ { m ≤ m opt }] ≤ P [ Crit( m) ≥ (1 + δ)Crit(m opt ) + x] ≤ j∈Jn,D j ≥Dm opt exp[-C(x, j, m opt )]
which concludes the proof of Lemma 6.2. ✷ Let us define:

U opt := 2( g b m -g) 2 (x 0 )+2(1+δ)Crit(m opt )+2 sup j∈Jn,D j ≤Dm opt H(m opt , j)+ sup j∈Jn,D j ≥Dm opt (g j -g) 2 (x 0 )
for some constant δ > 0 defined later. According to inequalities 1) and 2) in Lemma 6.2, we have:

P 1 [( g b m -g) 2 (x 0 ) ≥ U opt + x] ≤ 2 m∈Jn exp(-C(x, m)) + 2
j∈Jn,D j ≥Dm opt exp(-C(x, j, m))

Hence:

E 1 [(( g b m -g) 2 (x 0 ) -U opt ) + ] ≤ +∞ 0 P 1 [( g b m -g) 2 (x 0 ) ≥ U opt + x]dx ≤ 2 +∞ 0 [ m∈Jn exp(-C(x, m)) + j∈Jn,D j ≥Dm opt exp(-C(x, j, m))]dx (29)
Besides, for every constant C:

+∞ 0 exp(-Cx)dx = 1 C , +∞ 0 exp(-C √ x)dx = 2 C 2 . Thus +∞ 0 j∈Jn,D j ≥Dm opt exp(-C(x, j, m opt ))dx ≤ j∈Jn,D j ≥Dm opt exp(-Aux j,mopt ν n ν ) K 2 ν u D j + D mopt n + exp(- u ′ √ A K x j,mopt ν n n D j + D mopt ) K 4 u ′ 2 (D j + D mopt ) 2 n 2
Moreover, for every j ∈ J n , (D j + D mopt )/n ≤ 2, and if ν n /ν ≥ 1/2 then:

+∞ 0 j∈Jn,D j ≥Dm opt exp(-C(x, j, m opt ))dx ≤ 2 n   K 2 ν u j∈Jn,D j ≥Dm opt exp(- 1 2 Aux j,mopt )(D j + D mopt ) + 2K 4 u ′ 2 j∈Jn,D j ≥Dm opt exp(- u ′ √ A K x j,mopt n ν n D j + D mopt )(D j + D mopt )  
The term +∞ 0 j∈Jn,j≥mopt exp(-C(x, j, m opt ))dx has order 1/n as soon as:

   exp(-1 2 Aux j,mopt ) ≤ (1 + D j + D mopt ) -(2+a) exp(-u ′ √ A K x j,m opt nb νn D j +Dm opt ) ≤ (1 + D j + D mopt ) -(2+a)
for some a > 0 which is equivalent to:

x j,mopt ≥ 2(2+a) Au ln(1 + D j + D m ) x j,mopt ≥ (2+a) 2 K 2 u ′ 2 Ab νn × D j +Dm opt n ln 2 (1 + D j + D mopt )
This is guaranteed if:

x j,m ≥ max{ 16 A (2 + a)(1 + 1 δ ) 2 ln(1 + D j + D m ); 32K 2 A ν n (1 + 1 δ ) 2 (2 + a) 2 ln 2 (1 + D j + D m ) D j + D m n } (30)
Let B 1 > 32/A and B 2 > 128K 2 /A be the constants involved in the definition (20) of the (x j,m ) and let consider δ > 0 and a > 0 such that 2B 1 ≥ 32

A (2 + a)(1 + 1 δ ) 2 and B 2 ≥ 32K 2 A(1 + 1 δ ) 2
(2 + a) 2 . Then x j,mopt satisfies inequality (30), and there exists a constant C which depends on (A, B 1 , B 2 , K) such that:

+∞ 0   j∈Jn,D j ≥Dm opt exp(-C(x, j, m opt ))   dx ≤ (1 + ν) C n
The same type of computation yields:

+∞ 0 m∈Jn exp(-C(x, m)) dx ≤ (1 + ν) C n
Then inequality (29) leads to:

(31) E 1 [(( g b m -g) 2 (x 0 ) -U opt ) + ] ≤ (1 + ν) C n
Besides, for every D j ≤ D mopt , H(m opt , j) ≤ 2pen(m opt ). Moreover:

sup j∈Jn,D j ≥Dm opt (g j -g) 2 (x 0 ) ≤ 2[ sup j∈Jn,D j ≥Dm opt (g j -g mopt ) 2 (x 0 ) + (g mopt -g) 2 (x 0 )]
Hence:

E 1 [U opt ] ≤ 3(g mopt -g) 2 (x 0 ) + 2(1 + δ)Crit(m opt ) + 4pen(m opt ) + sup j∈Jn,D j ≥Dm opt (g j -g mopt ) 2 (x 0 ) ≤ C ′ Crit(m opt ) + 3(g mopt -g) 2 (x 0 ) (32)
By gathering inequalities ( 26), ( 31) and (32), we get inequality (25).

Proof of Theorem 3.1. We prove the following claim: Claim 2. If Assumption H dens (β) holds, there exist constants κ and κ ′ which depend on (β, B 1 , B 2 ), and a universal constant

C 1 such that (33) E[( g b m -g) 2 (x 0 )1I {b νn≥ν/2} ] ≤ κ(1+2ν)( n ln n ) -2β/(2β+1) +κ ′ K 2 p 0 ( n ln n ) -2β/(2β+1) P [ ν n > 2ν] (34) E[( g b m -g) 2 (x 0 )1I {b νn<ν/2} ] ≤ (ν + K 2 M n ) 2 P [ ν n < 1 2 ν].
Proof of Claim 2 • Let us prove inequality (33). First of all, we notice that if ν n ≥ ν/2 then for every model m:

x m ≤ max{B 1 ln(1 + D m ), 2B 2 ln(1 + D m ) D m n ≤ B 3 ln(1 + D m )} with B 3 = max(B 1 , 2B 2 ). Thus: Crit(m)1I {b νn≥ν/2} ≤ 2[ sup D j ≥Dm (g j -g) 2 (x 0 ) + (g m -g) 2 (x 0 )] + x m ν n ln(1 + D m ) D m n ≤ 2C 0 D -2β m + B 3 ln(1 + D m ) ν n D m n ≤ C(1 + ν n )[D -2β m + ln(1 + D m ) D m n ]
for some constant C depending on (A, K, β, C 0 ). Let us denote

F (m) = D -2β m + ln(1 + D m ) Dm
n and m 1 = arg min F (m). Then:

F (m 1 ) ≤ F (( n ln(1 + n) ) 1/(2β+1) ) ≤ 2( n ln n ) -2β)/(2β+1)
Remark 1. We give here an upper bound for D m 1 , which will be useful in the proof of Theorem 2.1. The model m 1 satisfies:

2β D 2β+1 m 1 = 1 n [( D m 1 1 + D m 1 + ln(1 + D m 1 )]
Besides, the function m → ( Dm 1+Dm + ln(1 + D m ) is increasing so: 2β

D 2β+1 m 1 ≥ α 1 n ⇒ D m 1 ≤ α 1/(2β+1) 1 α 2 n -1/(2β+1)
where α 1 and α 2 are defined in (8) and (9).

Besides:

-If D mopt ≤ D m 1 , then:

( g mopt -g) 2 (x 0 ) ≤ C 0 D -2β mopt ≤ C 0 D -2β m 1 ≤ F (m 1 ) -If D mopt > D m 1 : (g mopt -g) 2 (x 0 ) ≤ 2[(g mopt -g m 1 ) 2 (x 0 ) + (g m 1 -g) 2 (x 0 )] ≤ 2 sup j∈Jn,j≥mopt (g mopt -g j ) 2 (x 0 ) + C(β, L)D -(2β-1) m 1 ≤ Crit(m opt ) + F (m 1 ) ≤ Crit(m 1 ) + F (m 1 ) ≤ C(1 + ν n )F (m 1 )
Hence in these two cases:

Crit(m opt ) + (g -g mopt ) 2 (x 0 ) ≤ C(1 + ν n )F (m 1 ).
Thus according to inequality (25) in Claim 1 we have:

E 1 [( g b m -g) 2 (x 0 )]1I {b νn≥ν/2} ≤ max(κ, κ ′ )(1 + ν n )( n ln n ) -2β/(2β+1)
And by integrating this result over the sample Z 1 we get:

(35) E[( g b m -g) 2 (x 0 )1I {b νn≥ν/2} ] ≤ C(1 + E[ ν n ])( n ln n ) -2β/(2β+1) .
Moreover we have proved in (36) that ν n = g m 0 ∞ ≤ K 2 p 0 . Thus

E[ ν n ] = E[ ν n 1 {b νn≤2ν} ] + E[ ν n 1 {b νn>2ν} ] ≤ 2ν + K 2 p 0 P [ ν n > 2ν]
By reporting this result in (35), we get inequality (33).

• Let us prove inequality (34). For every model m ∈ J n , ( g m -g) 2 (x 0 ) ≤ (| g m (x 0 )|+ν) 2 . Besides:

( g m ) 2 (x 0 ) = λ∈Im ( 1 n n i=1 χ λ (V i ))χ λ (x 0 )) 2 ≤ 1 n n i=1 ( λ∈Im χ λ (V i )χ λ (x 0 )) 2 ≤ λ∈Im χ 2 λ 2 ∞ ≤ K 4 D 2 m ( 36
)
Hence:

P [( g b m -g) 2 (x 0 )1I b νn<ν/2 ] ≤ (KM n + ν) 2 P [ ν n < 1 2 ν]
and inequality (22) in Proposition 3.1 ends the proof of (34). ✷ Theorem 3.1 results directly from Claim 2: P [ ν n > 2ν] and P [ ν n < ν/2] are upper bounded by Proposition (3.1):

E[( g b m -g) 2 (x 0 )1I {b νn≥ν/2} ] ≤ κ(1 + 2ν) n ln n ) -2β/(2β+1) +κ ′ K 2 p 0 ( n ln n ) -2β/(2β+1) )[exp(- C 2 K 2 ν n p 0 ) + exp(- C 3 K 4 ν 2 n 3/2 p 2 0 ) + exp(- C 4 K 2 n p 2 0 )]
the combination of inequalities ( 37) and ( 34) ends the proof of Theorem 3.1 ✷ 6.2. Proof of Proposition 3.1. • We prove inequality (22). Let x 1 ∈ I be such that g(x 1 ) ≥ 5ν/6, then by definition of ν n :

P [ ν n ≤ ν/2] ≤ P [ g m 0 (x 1 ) ≤ ν/2] = P [( g m 0 -g m 0 )(x 1 ) ≤ 5ν/6 -g m 0 (x 1 ) -ν/3] ≤ P [( g m 0 -g m 0 )(x 1 ) ≤ (g -g m 0 )(x 1 ) -ν/3]
According to (11), we get:

P [ ν n ≤ ν/2] ≤ P [( g m 0 -g m 0 )(x 1 ) ≤ C 0 p -β 0 -ν/3
] and with condition (A 1 ):

P [ ν n ≤ ν/2] ≤ P [( g m 0 -g m 0 )(x 1 ) ≤ -ν/6] ≤ P [|( g m 0 -g m 0 )(x 1 )| ≥ ν/6] = P [| 1 n n i=1 U i -E(U i )| ≥ ν/6] (37)
where

U i = λ∈Im 0 χ λ (V i )χ λ (x 1
). This term is upper bounded with Bernstein Inequality, with the following parameters:

E[U 2 1 ] = E[( λ∈Im 0 χ λ (V 1 )χ λ (x 1 )) 2 ] = I ( λ∈Im 0 χ λ (x)χ λ (x 1 )) 2 g(x)dx ≤ ν λ,λ ′ ∈Im 0 [ I (χ λ (x)χ λ ′ (x)dx]χ λ (x 1 )χ λ ′ (x 1 )) = ν λ∈Im 0 χ 2 λ (x 1 )
as the {χ λ } are orthonormal. Finally, Assumption (13) in H dens (β) leads to:

E[U 2 1 ] ≤ νK 2 p 0 := v 2
Let l be an integer greater than 2, then:

E[(X 1 ) l + ] ≤ E[U 2 1 ] × U 1 l-2 ∞ ≤ v 2 λ∈Im 0 χ λ (V 1 )χ λ (x 0 ) l-2 ∞ ≤ v 2 [ λ∈Im 0 χ 2 λ (V 1 ) ∞ λ∈Ip 0 χ 2 λ (x 0 )] l-2 ≤ v 2 (K 2 p 0 ) l-2
thus c = K 2 p 0 . Bernstein Inequality applied to (37) provides ( 22).

• We prove inequality (23). Let x 1 be such that g m 0 ( x 1 ) ≥ 5 ν n /6, then:

P [ ν n ≥ 2ν] ≤ P [ 6 5 g m 0 ( x 1 ) ≥ 2ν] = P [ 6 5 ( g m 0 -g m 0 )( x 1 ) ≥ 4 5 ν + 6 5 (ν -g m 0 ( x 1 ))].
By definition of ν,

P [ ν n ≥ 2ν] ≤ P [ 6 5 ( g m 0 -g m 0 )( x 1 ) ≥ 4 5 ν + 6 5 (g -g m 0 )( x 1 ))].
According to (11) of H dens (β):

P [ ν n ≥ 2ν] ≤ P [( g m 0 -g m 0 )( x 1 ) ≥ 2 3 ν -C 0 p -β 0 ]
and Assumption (A 1 ) leads to:

P [ ν n ≥ 2ν] ≤ P [( g m 0 -g m 0 )( x 1 ) ≥ 1 2 ν] ≤ P [sup x∈I ( g m 0 -g m 0 )(x) ≥ 1 2 ν].
Let:

Z = sup x∈I ( g m 0 -g m 0 )(x) = sup x∈I 1 n n i=1 { λ∈Im 0 (χ λ (V i )χ λ (x) -E[χ λ (V i )χ λ (x)])}
We upper bound P [Z ≥ ν/2] with Talagrand Inequality (see Theorem 7.2 in Section 7), but the set of functions:

F = {ϕ x : u → λ∈Im 0 χ λ (x)χ λ (u) -E[χ λ (x)χ λ (V 1 )], x ∈ I}
is not countable. Nevertheless, for every u the application x → ϕ x (u) is continuous (as the (χ λ ) are continuous), so :

Z = sup x∈I 1 n n i=1 ϕ x (V i ) = sup x∈I∩Q 1 n n i=1 ϕ x (V i )
by density of Q ∩ I in I, and Q ∩ I is countable, which allows us to apply Talagrand Inequality. For every x ∈ I:

λ∈Im 0 [ 1 n n i=1 (χ λ (V i )χ λ (x) -E[χ λ (V i )χ λ (x)])] 2 ≤ { λ∈Im 0 χ 2 λ (x)} × { λ∈Im 0 [ 1 n n i=1 (χ λ (V i ) -E[χ λ (V i )])] 2 } ≤ K 2 p 0 λ∈Im 0 [ 1 n n i=1 (χ λ (V i ) -E[χ λ (V i )])] 2
Thus:

E[Z] ≤ K 2 p 0 λ∈Im 0 E[( 1 n n i=1 χ λ (V i ) -E[χ λ (V i )]) 2 ] = K 2 p 0 λ∈Im 0 V ar( 1 n n i=1 χ λ (V i )) = K 2 p 0 n λ∈Im 0 V ar(χ λ (V 1 )) ≤ K 2 p 0 n λ∈Im 0 E[χ 2 λ (V 1 )] Hence E[Z] ≤ K 2 p 0 n
Let us compute the variance term v. For every x ∈ I:

V ar(

λ∈Im 0 χ λ (V 1 )χ λ (x)) ≤ E[( λ∈Im 0 χ λ (V 1 )χ λ (x)) 2 ] = I ( λ∈Im 0 χ λ (u)χ λ (x)) 2 g(u)du ≤ ν I ( λ∈Im 0 χ λ (u)χ λ (x)) 2 du = ν λ,λ ′ ∈Im 0 [ I χ λ (u)χ λ ′ (u)du]χ λ (x)χ λ ′ (x)
As the family {χ λ , λ ∈ I m 0 } is orthonormal:

V ar(

λ∈Im 0 χ λ (V 1 )χ λ (x)) ≤ ν λ∈Im 0 χ 2 λ (x) ≤ νK 2 p 0 := v
Finally, for every x ∈ I:

λ∈Im 0 χ λ (x)χ λ ∞ ≤ λ∈Im 0 χ 2 λ (x) × λ∈Im 0 χ 2 λ ∞ ≤ K 2 p 0 := b
Besides, by Assumption (A 2 ) we have:

P [Z ≥ ν 2 ] = P [Z ≥ H + ( ν 2 - K 2 p 0 √ n )] ≤ P [Z ≥ H + ν 6 ],
and Talagrand Inequality provides the following upper bound:

P [Z ≥ ν 2 ] ≤ exp[-C nν 2 νK 2 p 0 + K 4 p 2 0 / √ n + K 2 p 2 0 ν 2 ] ≤ exp(- C 2 K 2 ν n p 0 ) + exp(- C 3 K 4 ν 2 n 3/2 p 2 0 ) + exp(- C 4 K 2 n p 2
(f -f -) 2 (x 0 ) = ( 1 0 [f (x 0 ) -f (x 0 -(b -b)(x))]µ(x)dx) 2 ≤ 1 0 [f (x 0 ) -f (x 0 -(b -b)(x))] 2 µ(x)dx ≤ L 1 0 [(b -b)(x)] 2β-1 µ(x)dx ≤ L 1 0 [(b -b) 2 (x)µ(x)dx ≤ L b -b 2 µ So (38) E[(f --f ) 2 (x 0 )] ≤ LE[ b -b 2 µ ]
To study the second term we need the following preliminary lemma.

Lemma 6.3. 1) Let Z -be fixed. If f satisfies Assumption H 1 (β) (resp. H 2 (β)), so does f -.
2) ν -≤ ν almost everywhere (a.e.).

3) For every m ∈ {1, . . . , M n }:

( f - m (x 0 ) -f -(x 0 )) 2 ≤ (m + ν) 2 ≤ (M n + ν) 2 4) Let us consider a sequence (α n ) of positive number such that α n = o(1/ √ ln n). Then for every n ∈ N such that: 2 √ ln n + σ 2 α 2 n ln n ≤ 1 2
where σ 2 = E[ǫ 2 1 ], we have:

P [ν -≤ α n ] ≤ 2 ln nα 2 n E[ b -b 2 µ ]
The proof of Lemma 6.3 is given at the end of Section 6.3.

According to 1) in Lemma 6.3, for every Z -, f -satisfies Assumption H dens (β) for some β ≥ 1. Thus, according to Remark 1 in the proof of Claim 2, the result of Claim 2 remains true if we restrict ourselves to a maximum size of model M n = E(α 3 n 1/3 ). Indeed, let's go back to the proof of Claim 2. The maximum size M n is involved when we state that:

Crit(m opt ) ≤ Crit(m 1 ) where m opt = arg min m∈Jn Crit(m)
And this holds as soon as m 1 ∈ J n . Besides, we have proved that

D m 1 ≤ α 1/(1β+1) 1 α 2 n 1/(2β+1) ≤ α 1/3 1 α 2 n 1/2 ≤ M n
So Claim 2 provides the following upper bound:

E 1 [( f --f -) 2 (x 0 )] ≤ κ(1 + ν)( n ln n ) -2β/(2β+1) +κ ′ p 0 ( n ln n ) -2β/(2β+1) P 1 [ ν - n > 2ν -] + (ν -+ M n ) 2 P 1 [ ν - n < 1 2 ν -] (39)
Let us define the sets:

A - 1 := {C 0 p -β 0 < 1 6 ν -} A - 2 := { p 0 √ n ≤ ν - 3K 2 }
Then:

E[P 1 ( ν - n > 2ν -)] ≤ E[P 1 ( ν - n > 2ν -)1 A - 1 ∩A - 2 ] + P [(A - 1 ) c ] + P [(A - 2 ) c ] ≤ E[exp(- C 2 K 2 ν -n p 0 + exp(- C 3 K 2 (ν -) 2 n 3/2 p 2 0 )] + exp(- C 4 K 2 n p 2 0 ) +P [(A - 1 ) c ] + P [(A - 2 ) c ] And: E[P 1 ( ν - n < ν -/2)] ≤ E[P 1 ( ν - n < nu -/2)1 A - 1 ] + P [(A - 1 ) c ] ≤ 2E[exp(- C 1 K 2 ν -n p 0 )] + P [(A - 1 ) c ]
Thus inequality (39) leads to:

E 1 [( f --f -) 2 (x 0 )] ≤ C(1 + ν)[( n ln n ) -2β/(2β+1) +p 0 ( n ln n ) -2β/(2β+1) (E[exp(- C 2 K 2 ν -n p 0 ) + exp(- C 3 K 2 (ν -) 2 n 3/2 p 2 0 )] + (40) exp(- C 4 K 2 n p 2 0 ) + P [(A - 1 ) c ] + P [(A - 2 ) c ]) + M 2 n (P [(A - 1 ) c ] + E[exp(- C 1 K 2 ν -n p 0 )])]
Now we upper bound each term in the right side of the above expression.

E[exp(- C 1 K 2 ν -n p 0 )] = E[exp(- C 1 K 2 ν -n p 0 )1 {ν -≥2 ln n K 2 p 0 C 2 n } ] + E[exp(- C 1 K 2 ν -n p 0 )1 {ν -<2 ln n K 2 p 0 C 2 n } ] ≤ E[ 1 n 2 1 {ν -≥2 ln n K 2 p 0 C 2 n } ] + E[1 {ν -<2 ln n K 2 p 0 C 2 n } ] ≤ 1 n 2 + P [ν -< 2 ln n K 2 p 0 C 2 n ] ln 4 n{ p 3 0 n 2+2/3 + p 3 0 n 3/2+2/3 + 1 p 0 n 2/3 + p 3 0 n 1+2/3 + p 2 0 n 2-2/3 + n 2/3 p 2 0 }
is upper bounded by a constant. So:

(44) E[( f --f -) 2 (x 0 )] ≤ C(1 + ν)( n ln n ) -2β/(2β+1) + C ′ (1 + ν)E[ b -b 2 µ ]
Inequalities ( 38) and ( 44

|(f --(f -) m )(u)| ≤ 1 x=0 f -f m ∞ µ(x)dx ≤ f -f m ∞ ≤ C 0 D -β m
• Let f ∈ H(β, L) and r the greater integer smaller than β. f is r times differentiable and its r-th derivative is upper bounded, then with classical analysis results we get: 3) A calculus similar to (36) implies that |f - m (x 0 )| ≤ m, which proves the third point.

4) Given Z -, ǫ 1 and (bb)(X 1 ) are independent, which entails:

E[ ǫ 2 1 |Z -] = E[ǫ 2 1 |Z -] + E[(b -b) 2 (X 1 )|Z -] + 2E[ǫ 1 (b -b)(X 1 )|Z -] Moreover, E[ǫ 1 |Z -] = 0 hence: E[ ǫ 2 1 |Z -] = σ 2 + b -b 2 µ
Thus for every A n > 0: 2) The family {φ m,k , k ∈ Z} is orthonormal for the L 2 -norm.

3) For every m > 0, k∈Z φ 2 m,k ∞ ≤ m 4) A m = {t ∈ L 2 (R), Supp(t * ) ⊂ [-πm, πm]} = span(φ m,k , k ∈ Z). 5) For every h ∈ L 2 (R), the Fourier transform of the projection h m of h on A m is h * m (t) = h * (t)1 [-πm,πm] (t). A simple calculus proves that the Fourier transform of 1 [-π,π] is 2πφ, then φ * = 1 [-π,π] and 1) follows by a change of variable. Next, for every k, l ∈ Z, according to the Parseval formula, we have: ). Assertion 4) follows from Meyer (1990, p.22), and 5) is an immediate consequence of 4). Indeed, for every h ∈ L 2 (R): Proof of Proposition 2.2 • Let f ∈ W (α, L), and x ∈ R:

(f -f m ) 2 (x) = [ R (f * -f * m )(t)e itx dt] 2 = [ |t|>πm f * (t)e itx dt] 2 ≤ |t|>πm |f * (t)| 2 t 2α dt × |t|>πm 1 t 2α dt ≤ L 2 × 1 (2α -1)(πm) 2α-1 = C 0 (L, α) m 2α-1 ✷
• Suppose that α > 3/2, then for every x, y ∈ R:

|f (x) -f (y)| = | 1 2π R f * (

3. 4 .

 4 Upper bound for the pointwise risk of g b m .

n=500 n=2000 Figure 1 .

 n=20001 Figure 1. Beam of 50 density estimators curves (blue dotted lines) built from i.i.d. samples of size n=200, 500 and 2000 of density N (0, 1) and L(1) (red thick line), in sine-cardinal bases.

Figure 2 Figure 2 .

 22 Figure 2 presents a comparison between our pointwise model selection estimator, and a global model selection estimator, computed following the procedure developped by Massart (2007), Section 7, for sample of size n = 500, 2000 with common density χ 2 (3). The global model selection estimator (dotted blue line) is computed in a mixed piecewise polynomial an trionometric polynomial basis using matlab programs available on Yves Rozenholc's web page (http://www.math-info.univ-paris5.fr/ rozen/). The pointwise model selection estimator (solid blue line) is built following the procedure described above, on the set J of 150 regularly spaced points on [-1, 15]. We observe that the pointwise model selection estimator (in solid blue line) fits the true density (in red thick line) for a smaller sample size than the global model selection estimator.

  choose a regression function b(x) = x 3 + 5x and b(x) = exp(-|x|) and compute the sample (Y 1 , . . . , Y 2n ) where Y i = b(X i ) + ǫ i . • From the sample {(X i , Y i )} i=1...n , we compute an estimator b of b following the procedure described in Section 4, using mixed piecewise polynomial and trigonometric polynomial basis (see Comte et al. (

Figure 3 .

 3 Figure 3. Error density estimator (solid blue line), theoretical estimator we would get if the errors were observed (dotted blue line) and true density (thick red line).

  Proof of Theorem 3.1. Let Z 1 be fixed. Let us denote by E 1 [.] the conditional expectation E[.|Z 1 ] and P 1 [.] the conditional probability P [.|Z 1 ]. We first prove the following Claim:

  ) conclude the proof of Theorem 2.1 ✷ Proof of Lemma 6.3:1) • Suppose that H 1 (β) holds. Let u ∈ R and m ∈ N * . (f --(f -) m )(u) = 1 x=0 f (u -(bb)(x)µ(x)dx -k∈Z f -, φ m,k φ m,k (u) -(bb)(x)µ(x)dx)φ m,k (t)dt]φ m,k (u) -(bb)(x))φ m,k (t)dtφ m,k (u)]µ(x)dx Let x be fixed in [0, 1] and f x (u) := f (u -(bb)(x)), then: f (u -(bb)(x)) -k∈Z t∈R f (t -(bb)(x))φ m,k (t)dtφ m,k (u) = f x (u) -(f x ) m (u)Besides, according to 5) in Proposition 7.1:(f x -(f x ) m )(u) = 1 2π |θ|>πm (f x ) * (θ)e iθu dθ = 1 2π |θ|>πm f * (θ)e iθ(b-b b(x)) e iθu dθ = (ff m )(u + (bb)(x)) So for every u ∈ R |(f x -(f x ) m )(u)| ≤ ff m ∞ , hence:

1 0

 1 ) (t -(bb)(x))µ(x)dx For every (t, u) ∈ [-1, 1] 2 : |(f -) (r) (t) -(f -) (r) (u)| = | 1 0 [f (r) (t -(bb)(x))f (r) (u -(bb)(x))]µ(x)dx ≤ 1 0 |f (r) (t -(bb)(x))f (r) (u -(bb)(x))|µ(x)dx ≤ L|t -u| β-r µ(x)dx = L|t -u| β-r so f -∈ H(β, L).2) Let t ∈ R, according to the expression of f -:|f -(t)| ≤ 1 0 |f (t -(bb)(x))|µ(x)dx ≤ ν thus ν -:= f -∞ ≤ ν.

2 n

 2 y)dy ≥ 1 -σ 2 + bb 2 µ A On the other hand, |y|≤An f -(y)dy ≤ 2ν -A n , by definition of ν -. Hence: Some results about projection on sine-cardinal bases. Let m be a positive number, A m the set of functions generated by {φ m,k , k ∈ Z} described in Section 2.2. The following Proposition holds: Proposition 7.1. 1) For every m > 0, k ∈ Z, the Fourier transform of φ m,k is φ * m,k (t) = (1/ √ m)e -ikt/m 1 [-πm,πm] (t)

  e iuxm du, so that k∈Z φ 2 m,k (x) = (m/2π) π -π |e iuxm | 2 du = m. This gives 3

  h m = arg min t∈Am ht 2 = arg min Supp(t * )⊂[-πm,πm] h *t * = h * 1 [-πm,πm]

  Proof of Theorem 2.1. Let consider the decomposition (24). First of all, we study the first term E[(ff -) 2 (x 0 in the right hand side of (24). If Assumption 1 (β) holds, f is Lipschitz. It is easy to check that f is Lipschitz as well if Assumption H 2 (β) holds for some β ≥ 1. Let us denote L the Lipschitz constant of f .

	) ✷
	0
	6.3.

  t)(e itxe ity )dt| (t)||t| 2α dt ×where C is a constant depending on α. Thus:|f (x)f (y)| ≤ |x -y| 2 + L 2 C 2π which proves that f is Lipschitz. ✷

	= | ≤ 2π R 1 2π R 1 |f * (t)|2| sin( f * (t)e it(x+y)/2 2i sin( t(x -y) 2 ≤ 1 2π R -1 |f * (t)||t|dt ≤ 2 Besides, with Schwarz Inequality, we have: |f 1 |t|>1 t(x -y) 2 )|dt |t| 2(1-α) dt )dt| ≤ L 2 C

* (t)||t(xy)|dt f is a density so for every t ∈ R, |f * (t)| ≤ 1, thus: |t|>1 |f * (t)||t|dt ≤ |t|>1 |f *

Acknowledgments. The author is grateful to Fabienne Comte for her helpful and constructive advices, and to Yves Rozenholc for his help with matlab programming.

Besides, as p 0 = n γ with γ ∈]1/3, 1/2[, there exists N 1 which depends on γ, K 2 and σ 2 such that for every n ≥ N 1 , the following inequality holds:

Then, according to 4) from Lemma 6.3:

Similarly, there exists an integer N 3 which depends on γ, K 2 and σ 2 such that, for every n greater than N 3 :

And:

The combination of inequalities ( 40), ( 41), ( 42) and ( 43) leads to:

Besides, we suppose that M n ≤ α 3 n 1/3 and β ≥ 1 which entails that

Moreover, n/p 2 0 = n 1-2γ and 1 -2γ > 0 then:

We have chosen p 0 such that n γ ≤ p 0 ≤ 2n γ with γ in ]1/3, 1/2[, which entails that the quantity: for every A n > 0. Thus:

Let us consider A n = 1/(α n √ ln n), then condition (C) gives: Theorem 7.1. Let (X 1 , . . . , X n ) be independent random variables. Let us suppose that:

. 1) For every ǫ > 0:

2) Similarly, for every ǫ > 0:

).

Theorem 7.2. Let (X 1 , . . . , X n ) be i.i.d., F a class of function and:

Let us consider H, v and b such that:

Then for every λ > 0:

This Theorem results directly from Theorem 1.1 in T. [START_REF] Rio | Concentration around the mean of empirical processes[END_REF].


Scoping Review Protocol ADMINISTRATIVE INFORMATION:

Title:

Impacts of emotional regulation and inhibition on Emotional Eating (EE) and Binge Eating Disorder (BED): Protocol for a scoping review.

Authors

Mahé Arexis 1,2 (mahe.arexis@univ-fcomte.fr) ; Gilles Feron 1 (gilles.feron@inrae.fr) ; Marie-Claude Brindisi 3 (marie-claude.brindisi@chu-dijon.fr) ; Pierre-Édouard Billot 2 (pierreedouard.billot@univ-fcomte.fr) ; Stéphanie Chambaron 1 (stephanie.chambaronginhac@inrae.fr)

INTRODUCTION:

Background / Rationale

Our scoping review mainly focuses on Emotional Eating (EE) and Binge Eating Disorder (BED). EE is an eating behavior, it might be defined as "the tendency to overeat in response to negative emotions such as anxiety or irritability" (van Strien et al., 2007, p. 106) but in a nonpathological way. It differs from BED, which has been formally indexed in 2013 in the DSM-5 as a discrete eating disorder. BED symptoms include recurrent binge-eating episodes (i.e. eating a larger amount of food than most people do during a discrete period of time, with at least one episode per week for three months), "a sense of lack of control over eating during the episode" and "marked distress regarding binge eating", but without compensatory behaviors as in anorexia nervosa or bulimia nervosa (American Psychiatric Association, 2013).

Although different, both EE and BED appear to be affected by deficits in emotional regulation and inhibition [START_REF] Crockett | Boredom proneness and emotion regulation predict emotional eating[END_REF][START_REF] Dingemans | Emotion Regulation in Binge Eating Disorder: A Review[END_REF][START_REF] Ferrell | Emotion regulation difficulties and impaired working memory interact to predict boredom emotional eating[END_REF][START_REF] Giel | Food-related impulsivity in obesity and binge eating disorder-a systematic update of the evidence[END_REF][START_REF] Leehr | Emotion regulation model in binge eating disorder and obesity -a systematic review[END_REF][START_REF] Waltmann | Loss of control over eating: A systematic review of task based research into impulsive and compulsive processes in binge eating[END_REF][START_REF] Zhang | Wanting to eat matters: Negative affect and emotional eating were associated with impaired memory suppression of food cues[END_REF]. Indeed, both individuals with EE and BED present with overeating behaviors caused by emotion regulation difficulties and a lack of inhibition/greater impulsivity. For example, it has been shown that negative emotions act as a trigger for binge-eating episodes in BED [START_REF] Greeno | Binge antecedents in obese women with and without binge eating disorder[END_REF]Leerh et al., 2015;[START_REF] Nicholls | The association between emotions and eating behaviour in an obese population with binge eating disorder[END_REF]. Binge eating is then used as a way to regulate negative emotions (but it is yet uncertain if this strategy successfully improves mood, see [START_REF] Leehr | Emotion regulation model in binge eating disorder and obesity -a systematic review[END_REF][START_REF] Stein | What's driving the binge in binge eating disorder?: A prospective examination of precursors and consequences[END_REF].

Objectives

The objectives of the present scoping review are multiple. The first question we would like to address is the possible existence of a continuum between EE (nonpathological eating behavior) and BED (pathological eating behavior). Moreover, as EE and BED both show deficits in emotion regulation and inhibition, our second question relates to the possibility of a gradation of the deficits in emotion regulation and inhibition. The third question we would like to address in this scoping review is about the ill-defined concept of overeating. Indeed, is overeating a symptom, an eating behavior, a synonym concept of EE or a synonym for binge eating? Finally, this scoping review will hopefully enable to identify gaps in knowledge.

METHOD:

This scoping review will be conducted following the PRISMA-ScR statements which are described in the Preferred Reporting Items for Systematic reviews and Meta-Analyses extension for Scoping Reviews (PRISMA-ScR, [START_REF] Tricco | PRISMA Extension for Scoping Reviews (PRISMA-ScR): Checklist and Explanation[END_REF].

Eligibility criteria (inclusion and exclusion criteria)

Studies will be selected based on the following criteria: PICO criteria / keywords: Participants: People (adult human subjects, 18 years and +) with Binge Eating Disorder (BED) (and meeting full DSM-IV-TR or DSM-5 criteria for BED) or subthreshold BED, and people (adults human subjects, 18 years and +) presenting with Emotional Eating (EE) or Emotional Overeating (EO).

Our review will focus on the impact of "emotional regulation" and "inhibitory control" on BED and EE.

Comparisons: Our review will not focus on studies with specific comparisons.

Outcomes: Our review will consider all types of outcomes related to emotion regulation and inhibitory control in BED, EE and EO (e.g. deficit, level of attention, response impairment, degree of compulsivity…).

Studies: All types of journal articles, published in peer-reviewed scientific journals, except all types of reviews, book chapters, preprints, and thesis.

Languages: English and French

Period / context: Only papers published between January 2009 and January 2022 were eligible for inclusion. The Binge Eating Disorder's MeSH (Medical Subject Headings) was introduced in 2010. Nevertheless, 2009 was chosen because it was a "year of transition" between the previous indexing as Bulimia Nervosa and the BED MeSH introduction in 2010.

Exclusion criteria: No reviews, book chapters, preprints, and thesis.

Information sources

Bibliographic databases: PubMed and PsycINFO.

The literature search will be limited to the eligibility criteria outlined above.

Date of search: January 2022.

Search strategy

The search strategy focused on articles published between January 2009 and January 2022. The following search equation was used in both databases: ("Binge-Eating Disorder"[Mesh] OR BED OR Binge eater OR Emotional Eating OR Emotional Overeating OR Overeater OR Emotional eater OR Overeating) AND ("Emotional Regulation"[Mesh] OR Emotion regulation OR Reappraisal OR Rumination OR Attentional deployment OR Mood regulation OR "Inhibition, Psychological"[Mesh] OR Inhibitory control).

Duplicates will be removed, and all references will be imported on Rayyan, an online application for systematic reviews.

Study selection process

1 st step (screening): Using Rayyan, an online IA software (https://www.rayyan.ai/) [START_REF] Ouzzani | Rayyan-a web and mobile app for systematic reviews[END_REF], the screening will be conducted by at least two reviewers (MA, and SC or PEB) up to four reviewers (i.e. MA, SC, PEB, MCB). Discrepancies will be resolved by a third, or even fourth investigator(s). On this screening step, for each paper, inclusion criteria described in the Eligibility criteria (inclusion and exclusion criteria) section will be applied on both title and abstract.

2 nd step (eligibility): Eligibility will be conducted by at least two reviewers (MA, and SC or PEB) (up to four reviewers, i.e. MA, SC, PEB, MCB). Discrepancies will be resolved by a third or even fourth investigator(s). On this screening for eligibility step, for each paper, inclusion criteria described in the Eligibility criteria (inclusion and exclusion criteria) section will be applied on the complete reading of the article.

At least two reviewers (up to four reviewers) will work on steps 2 and 3. In case of disagreement regarding the selection of an article for the next step, one or two more reviewers will be called on and discussions will take place between the reviewers.

In order to identify articles missed from the selection process, a final search will be conducted on the references lists / bibliographies of the included studies.

Screening and eligibility steps will be performed using Rayyan web app for systematic reviews [START_REF] Ouzzani | Rayyan-a web and mobile app for systematic reviews[END_REF].

Data extraction

The investigators (MA, and SC or PEB) will extract the following data for each study: authors, title, year, country of the study, journal, type of study, study design/method, populations and samples sizes, type of interventions, possible comparisons, outcomes/results. The data will be presented in a form of a table, presenting the main summarized characteristics, PICO criteria and summary of the findings, for each selected study in step 2. This table will be included in the final manuscript of the Scoping Review.

Data analysis

Data will be analyzed qualitatively. In particular, a distinction between the type of population (BED, EE or EO) will be done first in order to evaluate the degree of emotion regulation and of inhibition depending on the type of population. Then, a distinction between interventions (inhibitory control or emotion regulation) will be performed. If relevant, a description of the methodology (questionnaires, EEG…) will also be presented and discussed.

Critical appraisal

Critical appraisal is not planned at the moment. However, an evaluation of the methodologies will be conducted in case of significant discrepancies regarding the conclusions of the different papers.

meetings of this project. Stéphanie Chambaron (PhD in cognitive psychology, Research Director) works on cognition and nutrition. She participated in the meetings, read a part of the articles, managed and funded this project with the ANR JCJC (ANR-17-CE21-0001).

Appendices

None

  1. Centre des Sciences du Goût et de l'Alimentation, CNRS, INRAE, Institut Agro, Université de Bourgogne Franche-Comté, F-21000 Dijon, France. 2. Laboratoire de recherches Intégratives en Neurosciences et psychologie Cognitive (LINC), Université Bourgogne Franche-Comté, Besançon, France 3. Centre Hospitalier Universitaire Dijon Bourgogne, France Corresponding author: Stéphanie Chambaron, Centre des Sciences du Goût et de l'Alimentation, CNRS, INRAE, Institut Agro, Université de Bourgogne Franche-Comté, 17 rue Sully, F-21000 Dijon, France.

Competing of interest

None

Authors' contributions

Mahé Arexis (PhD in cognitive psychology) is the Research Associate on this project, she reads all the articles and writes the article/Scoping review. Pierre-Édouard Billot (PhD in neurosciences / Research Associate) reads all the articles and contributes to the writing of the article/scoping review. Gilles Feron (Research Director) was trained with SLR and Scoping reviews. He participated in the meetings of this project. Marie-Claude Brindisi (Professor in Nutrition and Hospital Practitioner) works on nutrition and obesity and participated in the