Xavier Gendre
email: xavier.gendre@math.univ-toulouse.fr

Model selection and estimation of a component in additive regression

Let Y ∈ R n be a random vector with mean s and covariance matrix σ 2 P n t P n where P n is some known n × n-matrix. We construct a statistical procedure to estimate s as well as under moment condition on Y or Gaussian hypothesis. Both cases are developed for known or unknown σ 2 . Our approach is free from any prior assumption on s and is based on non-asymptotic model selection methods. Given some linear spaces collection {S m , m ∈ M}, we consider, for any m ∈ M, the least-squares estimator ŝm of s in S m . Considering a penalty function that is not linear in the dimensions of the S m 's, we select some m ∈ M in order to get an estimator ŝ m with a quadratic risk as close as possible to the minimal one among the risks of the ŝm 's. Non-asymptotic oracle-type inequalities and minimax convergence rates are proved for ŝ m. A special attention is given to the estimation of a non-parametric component in additive models. Finally, we carry out a simulation study in order to illustrate the performances of our estimators in practice.

Introduction

Additive models

The general form of a regression model can be expressed as

Z = f (X) + σε (1)
where X = (X (1) , . . . , X (k) ) ′ is the k-dimensional vector of explanatory variables that belongs to some product space X = X 1 × • • • × X k ⊂ R k , the unknown function f : X → R is called regression function, the positive real number σ is a standard deviation factor and the real random noise ε is such that E[ε|X] = 0 and E[ε 2 |X] < ∞ almost surely.

In such a model, we are interested in the behavior of Z in accordance with the fluctuations of X. In other words, we want to explain the random variable Z through the function f (x) = E[Z|X = x]. For this purpose, many approaches have been proposed and, among them, a widely used is the linear regression

Z = µ + k i=1 β i X (i) + σε (2)
where µ and the β i 's are unknown constants. This model benefits from easy interpretation in practice and, from a statistical point of view, allows componentwise analysis. However, a drawback of linear regression is its lack of flexibility for modeling more complex dependencies between Z and the X (i) 's. In order to bypass this problem while keeping the advantages of models like [START_REF] Arlot | Choosing a penalty for model selection in heteroscedastic regression[END_REF], we can generalize them by considering additive regression models of the form

Z = µ + k i=1 f i (X (i) ) + σε (3)
where the unknown functions f i : X i → R will be referred to as the components of the regression function f . The object of this paper is to construct a data-driven procedure for estimating one of these components on a fixed design (i.e. conditionally to some realizations of the random variable X). Our approach is based on nonasymptotic model selection and is free from any prior assumption on f and its components. In particular, we do not make any regularity hypothesis on the function to estimate except to deduce uniform convergence rates for our estimators. Models [START_REF] Arlot | Data-driven calibration of penalties for least-squares regression[END_REF] are not new and were first considered in the context of input-output analysis by Leontief [START_REF] Leontief | Introduction to a theory of the internal structure of functional relationships[END_REF] and in analysis of variance by Scheffé [START_REF] Scheffé | The analysis of variance[END_REF]. This kind of model structure is widely used in theoretical economics and in econometric data analysis and leads to many well known economic results. For more details about interpretability of additive models in economics, the interested reader could find many references at the end of Chapter 8 of [START_REF] Härdle | Nonparametric and Semiparametric Models[END_REF].

As we mention above, regression models are useful for interpreting the effects of X on changes of Z. To this end, the statisticians have to estimate the regression function f . Assuming that we observe a sample {(X 1 , Z 1 ), . . . , (X n , Z n )} obtained from model [START_REF] Akaike | Statistical predictor identification[END_REF], it is well known (see [START_REF] Stone | Additive regression and other nonparametric models[END_REF]) that the optimal L 2 convergence rate for estimating f is of order n -α/(2α+k) where α > 0 is an index of smoothness of f . Note that, for large value of k, this rate becomes slow and the performances of any estimation procedure suffer from what is called the curse of the dimension in literature. In this connection, Stone [START_REF] Stone | Additive regression and other nonparametric models[END_REF] has proved the notable fact that, for additive models (3), the optimal L 2 convergence rate for estimating each component f i of f is the one-dimensional rate n -α/(2α+1) . In other terms, estimation of the component f i in (3) can be done with the same optimal rate than the one achievable with the model Z ′ = f i (X (i) ) + σε.

Components estimation in additive models has received a large interest since the eighties and this theory benefited a lot from the the works of Buja et al. [START_REF] Buja | Linear smoothers and additive models (with discussion)[END_REF], Hastie and Tibshirani [START_REF] Hastie | Generalized additive models[END_REF]. Very popular methods for estimating components in [START_REF] Arlot | Data-driven calibration of penalties for least-squares regression[END_REF] are based on backfitting procedures (see [START_REF] Breiman | Estimating optimal transformations for multiple regression and correlations (with discussion)[END_REF] for more details). These techniques are iterative and may depend on the starting values. The performances of these methods deeply depends on the choice of some convergence criterion and the nature of the obtained results is usually asymptotic (see, for example, the works of Opsomer and Ruppert [START_REF] Opsomer | Fitting a bivariate additive model by local polynomial regression[END_REF] and Mammen, Linton and Nielsen [START_REF] Mammen | The existence and asymptotic properties of a backfitting projection algorithm under weak conditions[END_REF]). More recent non-iterative methods have been proposed for estimating marginal effects of the X (i) on the variable Z (i.e. how Z fluctuates on average if one explanatory variable is varying while others stay fixed). These procedures, known as marginal integration estimation, were introduced by Tjøstheim and Auestad [START_REF] Tjøstheim | Nonparametric identification of nonlinear time series: Selecting significant lags[END_REF] and Linton and Nielsen [START_REF] Linton | A kernel method of estimating structured nonparametric regression based on marginal integration[END_REF]. In order to estimate the marginal effect of X (i) , these methods take place in two times. First, they estimate the regression function f by a particular estimator f * , called pre-smoother, and then they average f * according to all the variables except X (i) . The way for constructing f * is fundamental and, in practice, one uses a special kernel estimator (see [START_REF] Ruppert | Multivariate locally weighted least squares regression[END_REF] and [START_REF] Severance-Lossin | Estimation of derivatives for additive separable models[END_REF] for a discussion on this subject). To this end, one needs to estimate two unknown bandwidths that are necessary for getting f * . Dealing with a finite sample, the impact of how we estimate these bandwidths is not clear and, as for backfitting, the theoretical results obtained by these methods are mainly asymptotic.

In contrast with these methods, we are interested here in nonasymptotic procedures to estimate components in additive models. The following subsection is devoted to introduce some notations and the framework that we handle but also a short review of existing results in nonasymptotic estimation in additive models.

Statistical framework

We are interested in estimating one of the components in the model [START_REF] Arlot | Data-driven calibration of penalties for least-squares regression[END_REF] with, for any i, X i = [0, 1]. To focus on it, we denote by s : [0, 1] → R the component that we plan to estimate and by t 1 , . . . , t K : [0, 1] → R the K 1 other ones. Thus, considering the design points (x 1 , y 1 1 , . . . , y K 1 ) ′ , . . . , (x 1 , y 1 1 , . . . , y K 1 ) ′ ∈ [0, 1] K+1 , we observe

Z i = s(x i ) + µ + K j=1 t j (y j i ) + σε i , i = 1, . . . , n , (4)
where the components s, t 1 , . . . , t K are unknown functions, µ in an unknown real number, σ is a positive factor and ε = (ε 1 , . . . , ε n ) ′ is an unobservable centered random vector with i.i.d. components of unit variance.

Let ν be a probability measure on [0, 1], we introduce the space of centered and squareintegrable functions

L 2 0 ([0, 1], ν) = f ∈ L 2 ([0, 1], ν) : 1 0 f (t)ν(dt) = 0 .
Let ν 1 , . . . , ν K be K probability measures on [0, 1], to avoid identification problems in the sequel, we assume s ∈ L 2 0 ([0, 1], ν) and t j ∈ L 2 0 ([0, 1], ν j ) , j = 1, . . . , K .

This hypothesis is not restrictive since we are interested in how Z = (Z 1 , . . . , Z n ) ′ fluctuates with respect to the x i 's. A shift on the components does not affect these fluctuations and the estimation proceeds up to the additive constant µ.

The results described in this paper are obtained under two different assumptions on the noise terms ε i , namely (H Gau ) the random vector ε is a standard Gaussian vector in R n , and (H Mom ) the variables ε i satisfy the moment condition ∃p > 2 such that ∀i,

τ p = E [|ε i | p ] < ∞ . (6)
Obviously, (H Mom ) is weaker than (H Gau ). We consider these two cases in order to illustrate how better are the results in the Gaussian case with regard to the moment condition case.

From the point of view of model selection, we show in the corollaries of Section 2 that we are allowed to work with more general model collections under (H Gau ) than under (H Mom ) in order to get similar results. Thus, the main contribution of the Gaussian assumption is to give more flexibility to the procedure described in the sequel. So, our aim is to estimate the component s on the basis of the observations (4). For the sake of simplicity of this introduction, we assume that the quantity σ 2 > 0 is known (see Section 3 for unknown variance) and we introduce the vectors s = (s 1 , . . . , s n ) ′ and t = (t 1 , . . . , t n ) ′ defined by, for any i ∈ {1, . . . , n},

s i = s(x i ) and t i = µ + K j=1 t j (y j i ) . (7)
Moreover, we assume that we know two linear subspaces

E, F ⊂ R n such that s ∈ E, t ∈ F and E ⊕ F = R n .
Of course, such spaces are not available to the statisticians in practice and, when we handle additive models in Section 4, we will not suppose that they are known. Let P n be the projection onto E along F , we derive from (4) the following regression framework

Y = P n Z = s + σP n ε (8)
where

Y = (Y 1 , . . . , Y n ) ′ belongs to E = Im(P n ) ⊂ R n .
The framework ( 8) is similar to the classical signal-plus-noise regression framework but the data are not independent and their variances are not equal. Because of this uncommonness of the variances of the observations, we qualify [START_REF] Birgé | Minimum contrast estimators on sieves: exponential bounds and rates of convergence[END_REF] as an heteroscedastic framework. The object of this paper is to estimate the component s and we handle [START_REF] Birgé | Minimum contrast estimators on sieves: exponential bounds and rates of convergence[END_REF] to this end. The particular case of P n equal to the unit matrix has been widely treated in the literature (see, for example, [START_REF] Birgé | Gaussian model selection[END_REF] for (H Gau ) and [START_REF] Baraud | Model selection for regression on a fixed design[END_REF] for (H Mom )). The case of an unknown but diagonal matrix P n has been studied in several papers for the Gaussian case (see, for example, [START_REF] Comte | Adaptive estimation of mean and volatility functions in (auto-)regressive models[END_REF] and [START_REF] Gendre | Simultaneous estimation of the mean and the variance in heteroscedastic gaussian regression[END_REF]). By using cross-validation and resampling penalties, Arlot and Massart [START_REF] Arlot | Data-driven calibration of penalties for least-squares regression[END_REF] and Arlot [START_REF] Arlot | Choosing a penalty for model selection in heteroscedastic regression[END_REF] have also considered the framework [START_REF] Birgé | Minimum contrast estimators on sieves: exponential bounds and rates of convergence[END_REF] with unknown diagonal matrix P n . Laurent, Loubes and Marteau [START_REF] Laurent | Testing inverse problems: a direct or an indirect problem[END_REF] deal with a known diagonal matrix P n for studying testing procedure in an inverse problem framework. The general case of a known non-diagonal matrix P n naturally appears in applied fields as, for example, genomic studies (see Chapters 4 and 5 of [START_REF] Robin | DNA, Words and Models[END_REF]).

The results that we introduce in the sequel consider the framework (8) from a general outlook and we do not make any prior hypothesis on P n . In particular, we do not suppose that P n is invertible. We only assume that it is a projector when we handle the problem of component estimation in an additive framework in Section 4. Without loss of generality, we always admit that s ∈ Im(P n ). Indeed, if s does not belong to Im(P n ), it suffices to consider the orthogonal projection π Pn onto Im(P n ) ⊥ and to notice that π Pn Y = π Pn s is not random. Thus, replacing Y by Y -π Pn Y leads to [START_REF] Birgé | Minimum contrast estimators on sieves: exponential bounds and rates of convergence[END_REF] with a mean lying in Im(P n ). For general matrix P n , other approaches could be used. However, for the sake of legibility, we consider s ∈ Im(P n ) because, for the estimation of a component in an additive framework, by construction, we always have Y = P n Z ∈ Im(P n ) as it will be specified in Section 4.

We now describe our estimation procedure in details. For any z ∈ R n , we define the least-squares contrast by

γ n (z) = Y -z 2 n = 1 n n i=0 (Y i -z i ) 2 .
Let us consider a collection of linear subspaces of Im(P n ) denoted by F = {S m , m ∈ M} where M is a finite or countable index set. Hereafter, the S m 's will be called the models.

Denoting by π m the orthogonal projection onto S m , the minimum of γ n over S m is achieved at a single point ŝm = π m Y called the least-squares estimator of s in S m . Note that the expectation of ŝm is equal to the orthogonal projection s m = π m s of s onto S m . We have the following identity for the quadratic risks of the ŝm 's,

Proposition 1.1. Let m ∈ M, the least-squares estimator ŝm = π m Y of s in S m satisfies E s -ŝm 2 n = s -s m 2 n + Tr( t P n π m P n ) n σ 2 (9)
where Tr(•) is the trace operator.

Proof. By orthogonality, we have

s -ŝm 2 n = s -s m 2 n + σ 2 π m P n ε 2 n . ( 10
)
Because the components of ε are independent and centered with unit variance, we easily compute

E π m P n ε 2 n = Tr( t P n π m P n ) n .
We conclude by taking the expectation on both side of [START_REF] Birgé | Gaussian model selection[END_REF].

A "good" estimator is such that its quadratic risk is small. The decomposition given by [START_REF] Birgé | An adaptive compression algorithm in Besov spaces[END_REF] shows that this risk is a sum of two non-negative terms that can be interpreted as follows. The first one, called bias term, corresponds to the capacity of the model S m to approximate the true value of s. The second, called variance term, is proportional to Tr( t P n π m P n ) and measures, in a certain sense, the complexity of S m . If S m = Ru, for some u ∈ R n , then the variance term is small but the bias term is as large as s is far from the too simple model S m . Conversely, if S m is a "huge" model, whole R n for instance, the bias is null but the price is a great variance term. Thus, [START_REF] Birgé | An adaptive compression algorithm in Besov spaces[END_REF] illustrates why choosing a "good" model amounts to finding a trade-off between bias and variance terms.

Clearly, the choice of a model that minimizes the risk (9) depends on the unknown vector s and makes good models unavailable to the statisticians. So, we need a data-driven procedure to select an index m ∈ M such that

E[ s -ŝ m 2 n ] is close to the smaller L 2 -risk among the collection of estimators {ŝ m , m ∈ M}, namely R(s, F) = inf m∈M E s -ŝm 2 n .
To choose such a m, a classical way in model selection consists in minimizing an empirical penalized criterion stochastically close to the risk. Given a penalty function pen : M → R + , we define m as any minimizer over M of the penalized least-squares criterion

m ∈ argmin m∈M {γ n (ŝ m ) + pen(m)} . ( 11
)
This way, we select a model S m and we have at our disposal the penalized least-squares estimator s = ŝ m. Note that, by definition, the estimator s satisfies

∀m ∈ M, γ n (s) + pen( m) γ n (ŝ m ) + pen(m) . (12)
To study the performances of s, we have in mind to upperbound its quadratic risk. To this end, we establish inequalities of the form

E s -s 2 n C inf m∈M s -s m 2 n + pen(m) + R n (13)
where C and R are numerical terms that do not depend on n. Note that if the penalty is proportional to Tr( t P n π m P n )σ 2 /n, then the quantity involved in the infimum is of order of the L 2 -risk of ŝm . Consequently, under suitable assumptions, such inequalities allow us to deduce upperbounds of order of the minimal risk among the collection of estimators {ŝ m , m ∈ M}. This result is known as an oracle inequality

E s -s 2 n CR(s, F) = C inf m∈M E s -ŝm 2 n . ( 14
)
This kind of procedure is not new and the first results in estimation by penalized criterion are due to Akaike [START_REF] Akaike | Statistical predictor identification[END_REF] and Mallows [START_REF] Mallows | Some comments on c p[END_REF] in the early seventies. Since these works, model selection has known an important development and it would be beyond the scope of this paper to make an exhaustive historical review of the domain. We refer to the first chapters of [START_REF] Mcquarrie | Regression and times series model selection[END_REF] for a more general introduction.

Nonasymptotic model selection approach for estimating components in an additive model was studied in few papers only. Considering penalties that are linear in the dimension of the models, Baraud, Comte and Viennet [START_REF] Baraud | Adaptive estimation in autoregression or β-mixing regression via model selection[END_REF] have obtained general results for geometrically βmixing regression models. Applying it to the particular case of additive models, they estimate the whole regression function. They obtain nonasymptotic upperbounds similar to (13) on condition ε admits a moment of order larger than 6. For additive regression on a random design and alike penalties, Baraud [START_REF] Baraud | Model selection for regression on a random design[END_REF] proved oracle inequalities for estimators of the whole regression function constructed with polynomial collections of models and a noise that admits a moment of order 4. Recently, Brunel and Comte [START_REF] Brunel | Adaptive nonparametric regression estimation in presence of right censoring[END_REF] have obtained results with the same flavor for the estimation of the regression function in a censored additive model and a noise admitting a moment of order larger than 8. Pursuant to this work, Brunel and Comte [START_REF] Brunel | Model selection for additive regression models in the presence of censoring, chapter 1 in "Mathematical Methods in Survival Analysis, Reliability and Quality of Life[END_REF] have also proposed a nonasymptotic iterative method to achieve the same goal. Combining ideas from sparse linear modeling and additive regression, Ravikumar et al. [START_REF] Ravikumar | Sparse additive models[END_REF] have recently developed a data-driven procedure, called SpAM, for estimating a sparse high-dimensional regression function. Some of their empirical results have been proved by Meier, van de Geer and Bühlmann [START_REF] Meier | High-dimensional additive modeling[END_REF] in the case of a sub-Gaussian noise and some sparsity-smoothness penalty.

The methods that we use are similar to the ones of Baraud, Comte and Viennet and are inspired from [START_REF] Baraud | Model selection for regression on a fixed design[END_REF]. The main contribution of this paper is the generalization of the results of [START_REF] Baraud | Model selection for regression on a fixed design[END_REF] and [START_REF] Baraud | Adaptive estimation in autoregression or β-mixing regression via model selection[END_REF] to the framework (8) with a known matrix P n under Gaussian hypothesis or only moment condition on the noise terms. Taking into account the correlations between the observations in the procedure leads us to deal with penalties that are not linear in the dimension of the models. Such a consideration naturally arises in heteroscedastic framework. Indeed, as mentioned in [START_REF] Arlot | Choosing a penalty for model selection in heteroscedastic regression[END_REF], at least from an asymptotic point of view, considering penalties linear in the dimension of the models in an heteroscedastic framework does not lead to oracle inequalities for s. For our penalized procedure and under mild assumptions on F, we prove oracle inequalities under Gaussian hypothesis on the noise or only under some moment condition.

Moreover, we introduce a nonasymptotic procedure to estimate one component in an additive framework. Indeed, the works cited above are all connected to the estimation of the whole regression function by estimating simultaneously all of its components. Since these components are each treated in the same way, their procedures can not focus on the properties of one of them. In the procedure that we propose, we can be sharper, from the point of view of the bias term, by using more models to estimate a particular component. This allows us to deduce uniform convergence rates over Hölderian balls and adaptivity of our estimators. Up to the best of our knowledge, our results in nonasymptotic estimation of a nonparametric component in an additive regression model are new.

The paper is organized as follows. In Section 2, we study the properties of the estimation procedure under the hypotheses (H Gau ) and (H Mom ) with a known variance factor σ 2 . As a consequence, we deduce oracle inequalities and we discuss about the size of the collection F. The case of unknown σ 2 is presented in Section 3 and the results of the previous section are extended to this situation. In Section 4, we apply these results to the particular case of the additive models and, in the next section, we give uniform convergence rates for our estimators over Hölderian balls. Finally, in Section 6, we illustrate the performances of our estimators in practice by a simulation study. The last sections are devoted to the proofs and to some technical lemmas. Notations: in the sequel, for any x = (x 1 , . . . , x n ) ′ , y = (y 1 , . . . , y n ) ′ ∈ R n , we define

x 2 n = 1 n n i=1
x 2 i and x, y n = 1 n n i=1

x i y i .

We denote by ρ the spectral norm on the set M n of the n×n real matrices as the norm induced by

• n , ∀A ∈ M n , ρ(A) = sup x∈R n \{0} Ax n x n .
For more details about the properties of ρ, see Chapter 5 of [START_REF] Horn | Matrix analysis[END_REF].

Main results

Throughout this section, we deal with the statistical framework given by (8) with s ∈ Im(P n ) and we assume that the variance factor σ 2 is known. Moreover, in the sequel of this paper, for any d ∈ N, we define N d as the number of models of dimension d in F,

N d = Card {m ∈ M : dim(S m ) = d} .
We

(m) (1 + θ + L m ) Tr( t P n π m P n ) n σ 2 for all m ∈ M , ( 15
)
then the penalized least-squares estimator s given by [START_REF] Birgé | Minimal penalties for gaussian model selection[END_REF] satisfies

E s -s 2 n C inf m∈M s -s m 2 n + pen(m) - Tr( t P n π m P n ) n σ 2 + ρ 2 (P n )σ 2 n R n (θ) (16)
where we have set

R n (θ) = C ′ m∈M exp - C ′′ L m Tr( t P n π m P n ) ρ 2 (P n )
and C > 1 and C ′ , C ′′ > 0 are constants that only depend on θ.

If the errors are not supposed to be Gaussian but only to satisfy the moment condition (H Mom ), the following upperbound on the q-th moment of s -s 2 n holds.

Theorem 2.2. Assume that (H Mom ) holds and take q > 0 such that 2(q + 1) < p. Consider θ > 0 and some collection {L m , m ∈ M} of positive weights. If the penalty function is such that pen(m)

(1 + θ + L m ) Tr( t P n π m P n ) n σ 2 for all m ∈ M , ( 17
)
then the penalized least-squares estimator s given by [START_REF] Birgé | Minimal penalties for gaussian model selection[END_REF] satisfies

E s -s 2q n 1/q C inf m∈M s -s m 2 n + pen(m) + ρ 2 (P n )σ 2 n R n (p, q, θ) 1/q (18)
where we have set R n (p, q, θ) equal to

C ′ τ p   N 0 + m∈M:Sm ={0} 1 + Tr( t P n π m P n ) ρ 2 (π m P n ) L m Tr( t P n π m P n ) ρ 2 (P n ) q-p/2   and C = C(q, θ), C ′ = C ′ (p, q, θ) are positive constants.
The proofs of these theorems give explicit values for the constants C that appear in the upperbounds. In both cases, these constants go to infinity as θ tends to 0 or increases toward infinity. In practice, it does neither seem reasonable to choose θ close to 0 nor very large. Thus this explosive behavior is not restrictive but we still have to choose a "good" θ. The values for θ suggested by the proofs are around the unity but we make no claim of optimality. Indeed, this is a hard problem to determine an optimal choice for θ from theoretical computations since it could depend on all the parameters and on the choice of the collection of models. In order to calibrate it in practice, several solutions are conceivable. We can use a simulation study, deal with cross-validation or try to adapt the slope heuristics described in [START_REF] Birgé | Minimal penalties for gaussian model selection[END_REF] to our procedure.

For penalties of order of Tr( t P n π m P n )σ 2 /n, Inequalities ( 16) and [START_REF] Härdle | Nonparametric and Semiparametric Models[END_REF] are not far from being oracle. Let us denote by R n the remainder term R n (θ) or R n (p, q, θ) according to whether (H Gau ) or (H Mom ) holds. To deduce oracle inequalities from that, we need some additional hypotheses as the following ones:

(A 1 ) there exists some universal constant ζ > 0 such that pen(m) ζ Tr( t P n π m P n ) n σ 2 , for all m ∈ M , (A 2 ) there exists some constant R > 0 such that sup n 1 R n R , (A 3 ) there exists some constant ρ > 1 such that sup n 1 ρ 2 (P n ) ρ 2 .
Thus, under the hypotheses of Theorem 2.1 and these three assumptions, we deduce from ( 16) that

E s -s 2 n C inf m∈M s -s m 2 n + Tr( t P n π m P n ) n σ 2 + Rρ 2 σ 2 n
where C is a constant that does not depend on s, σ 2 and n. By Proposition 1.1, this inequality corresponds to [START_REF] Brunel | Model selection for additive regression models in the presence of censoring, chapter 1 in "Mathematical Methods in Survival Analysis, Reliability and Quality of Life[END_REF] up to some additive term. To derive similar inequality from ( 18), we need on top of that to assume that p > 4 in order to be able to take q = 1. Assumption (A 3 ) is subtle and strongly depends on the nature of P n . The case of oblique projector that we use to estimate a component in an additive framework will be discussed in Section 4. Let us replace it, for the moment, by the following one

(A ′
3 ) there exists c ∈ (0, 1) that does not depend on n such that

cρ 2 (P n ) dim(S m ) Tr( t P n π m P n ) .
By the properties of the norm ρ, note that Tr( t P n π m P n ) always admits an upperbound with the same flavor

Tr( t P n π m P n ) = Tr(π m P n t (π m P n )) ρ(π m P n t (π m P n ))rk(π m P n t (π m P n )) ρ 2 (π m P n )rk(π m ) ρ 2 (P n ) dim(S m ) .
In all our results, the quantity Tr( t P n π m P n ) stands for a dimensional term relative to S m . Hypothesis (A ′

3 ) formalizes that by assuming that its order is the dimension of the model S m up to the norm of the covariance matrix t P n P n .

Let us now discuss about the assumptions (A 1 ) and (A 2 ). They are connected and they raise the impact of the complexity of the collection F on the estimation procedure. Typically, condition (A 2 ) will be fulfilled under (A 1 ) when F is not too "large", that is, when the collection does not contain too many models with the same dimension. We illustrate this phenomenon by the two following corollaries.

Corollary 2.1. Assume that (H Gau ) and (A ′

3 ) hold and consider some finite A 0 such that sup

d∈N:N d >0 log N d d A . ( 19
)
Let L, θ and ω be some positive numbers that satisfy

L 2(1 + θ) 3 cθ 2 (A + ω) .
Then, the estimator s obtained from [START_REF] Birgé | Minimal penalties for gaussian model selection[END_REF] with penalty function given by

pen(m) = (1 + θ + L) Tr( t P n π m P n ) n σ 2 is such that E s -s 2 n C inf m∈M s -s m 2 n + (L ∨ 1) Tr( t P n π m P n ) ∨ (cρ 2 (P n )) n σ 2
where C > 1 only depends on θ, ω and c.

For errors that only satisfy moment condition, we have the following similar result.

Corollary 2.2. Assume that (H Mom ) and (A ′ 3 ) hold with p > 6 and let A > 0 and ω > 0 such that N 0 1 and sup

d>0:N d >0 N d (1 + d) p/2-3-ω A . ( 20
)
Consider some positive numbers L, θ and ω ′ that satisfy

L ω ′ A 2/(p-2) ,
then, the estimator s obtained from [START_REF] Birgé | Minimal penalties for gaussian model selection[END_REF] with penalty function given by

pen(m) = (1 + θ + L) Tr( t P n π m P n ) n σ 2
is such that

E s -s 2 n Cτ p inf m∈M s -s m 2 n + (L ∨ 1) Tr( t P n π m P n ) ∨ (cρ 2 (P n )) n σ 2
where C > 1 only depends on θ, p, ω, ω ′ and c.

Note that the assumption (A ′

3 ) guarantees that Tr( t P n π m P n ) is not smaller than cρ 2 (P n ) dim(S m ) and, at least for the models with positive dimension, this implies Tr( t P n π m P n ) cρ 2 (P n ). Consequently, up to the factor L, the upperbounds of E s -s 2 n given by Corollaries 2.1 and 2.2 are of order of the minimal risk R(s, F). To deduce oracle inequalities for s from that, (A 1 ) needs to be fulfilled. In other terms, we need to be able to consider some L independently from the size n of the data. It will be the case if the same is true for the bounds A.

Let us assume that the collection F is small in the sense that, for any d ∈ N, the number of models N d is bounded by some constant term that neither depends on n nor d. Typically, collections of nested models satisfy that. In this case, we are free to take L equal to some universal constant. So, (A 1 ) is true for ζ = 1 + θ + L and oracle inequalities can be deduced for s. Conversely, a large collection F is such that there are many models with the same dimension. We consider that this situation happens, for example, when the order of A is log n. In such a case, we need to choose L of order log n too and the upperbounds on the risk of s become oracle type inequalities up to some logarithmic factor. However, we know that in some situations, this factor can not be avoided as in the complete variable selection problem with Gaussian errors (see Chapter 4 of [START_REF] Massart | Concentration inequalities and model selection[END_REF]).

As a consequence, the same model selection procedure allows us to deduce oracle type inequalities under (H Gau ) and (H Mom ). Nevertheless, the assumption on N d in Corollary 2.2 is more restrictive than the one in Corollary 2.1. Indeed, to obtain an oracle inequality in the Gaussian case, the quantity N d is limited by e Ad while the bound is only polynomial in d under moment condition. Thus, the Gaussian assumption (H Gau ) allows to obtain oracle inequalities for more general collections of models.

Estimation when variance is unknown

In contrast with Section 2, the variance factor σ 2 is here assumed to be unknown in [START_REF] Birgé | Minimum contrast estimators on sieves: exponential bounds and rates of convergence[END_REF]. Since the penalties given by Theorems 2.1 and 2.2 depend on σ 2 , the procedure introduced in the previous section does not remain available to the statisticians. Thus, we need to estimate σ 2 in order to replace it in the penalty functions. The results of this section give upperbounds for the L 2 -risk of the estimators s constructed in such a way.

To estimate the variance factor, we use a residual least-squares estimator σ2 that we define as follows. Let V be some linear subspace of Im(P n ) such that

Tr( t P n πP n ) Tr( t P n P n )/2 ( 21
)
where π is the orthogonal projection onto V . We define

σ2 = n Y -πY 2 n Tr ( t P n (I n -π)P n ) . (22)
First, we assume that the errors are Gaussian. The following result holds.

Theorem 3.1. Assume that (H Gau ) holds. For any θ > 0, we define the penalty function

∀m ∈ M, pen(m) = (1 + θ) Tr( t P n π m P n ) n σ2 . ( 23
)
Then, for some positive constants C, C ′ and C ′′ that only depend on θ, the penalized leastsquares estimator s satisfies

E s -s 2 n C inf m∈M E s -ŝm 2 n + s -πs 2 n + ρ 2 (P n )σ 2 n Rn (θ) (24)
where we have set

Rn (θ) = C ′ 2 + s 2 n ρ 2 (P n )σ 2 exp - θ 2 Tr( t P n P n ) 32ρ 2 (P n ) + m∈M exp -C ′′ Tr( t P n π m P n ) ρ 2 (P n ) .
If the errors are only assumed to satisfy a moment condition, we have the following theorem.

Theorem 3.2. Assume that (H Mom ) holds. Let θ > 0, we consider the penalty function defined by

∀m ∈ M, pen(m) = (1 + θ) Tr( t P n π m P n ) n σ2 . ( 25
)
For any 0 < q 1 such that 2(q + 1) < p, the penalized least-squares estimator s satisfies

E[ s -s 2q n ] 1/q C inf m∈M E[ s -ŝm 2 n ] + 2 s -πs 2 n + ρ 2 (P n )σ 2 Rn (p, q, θ)
where C = C(q, θ) and

C ′ = C ′ (p, q, θ) are positive constants, Rn (p, q, θ) is equal to R n (p, q, θ) 1/q n + C ′ τ 1/q p κ n s 2 n ρ 2 (P n )σ 2 + τ p ρ 2αp (P n ) Tr( t P n P n ) βp 1/q-2/p
with R n (p, q, θ) defined as in Theorem 2.2, (κ n ) n∈N = (κ n (p, q, θ)) n∈N is a sequence of positive numbers that tends to κ = κ(p, q, θ) > 0 as Tr( t P n P n )/ρ 2 (P n ) increases toward infinity and

α p = (p/2 -1) ∨ 1 and β p = (p/2 -1) ∧ 1 .
Penalties given by ( 23) and ( 25) are random and allow to construct estimators s when σ 2 is unknown. This approach leads to theoretical upperbounds for the risk of s. Note that we use some generic model V to construct σ2 . This space is quite arbitrary and is pretty much limited to be an half-space of Im(P n ). The idea is that taking V as some "large" space can lead to a good approximation of the true s and, thus, Y -πY is not far from being centered and its normalized norm is of order σ 2 . However, in practice, it is known that the estimator σ2 inclined to overestimate the true value of σ 2 as illustrated by Lemmas 8.4 and 8.5. Consequently, the penalty function tends to be larger and the procedure overpenalizes models with high dimension. To offset this phenomenon, a practical solution could be to choose some smaller θ when σ 2 is unknown than when it is known as we discuss in Section 6.

Application to additive models

In this section, we focus on the framework [START_REF] Baraud | Model selection for regression on a fixed design[END_REF] given by an additive model. To describe the procedure to estimate the component s, we assume that the variance factor σ 2 is known but it can be easily generalized to the unknown factor case by considering the results of Section 3. We recall that s ∈ L 2 0 ([0, 1], ν), t j ∈ L 2 0 ([0, 1], ν j ), j = 1, . . . , K, and we observe

Z i = s i + t i + σε i , i = 1, . . . , n , (26)
where the random vector ε = (ε 1 , . . . , ε n ) ′ is such that (H Gau ) or (H Mom ) holds and the vectors s = (s 1 , . . . , s n ) ′ and t = (t 1 , . . . , t n ) ′ are defined in [START_REF] Birgé | From model selection to adaptive estimation[END_REF]. Let S n be a linear subspace of L 2 0 ([0, 1], ν) and, for all j ∈ {1, . . . , K}, S j n be a linear subspace of L 2 0 ([0, 1], ν j ). We assume that these spaces have finite dimensions

D n = dim(S n ) and D (j) n = dim(S j n ) such that D n + D (1) n + • • • + D (K) n < n .
We consider an orthonormal basis {φ 1 , . . . , φ Dn } (resp. {ψ

(j) 1 , . . . , ψ (j)
D (j) n }) of S n (resp. S j n ) equipped with the usual scalar product of L 2 ([0, 1], ν) (resp. of L 2 ([0, 1], ν j )). The linear spans E, F 1 , . . . , F K ⊂ R n are defined by E = Span (φ i (x 1 ), . . . , φ i (x n )) ′ , i = 1, . . . , D n and F j = Span (ψ (j) i (y j 1 ), . . . , ψ (j) i (y j n )) ′ , i = 1, . . . , D (j) n , j = 1, . . . , K .
Let 1 n = (1, . . . , 1) ′ ∈ R n , we also define

F = R1 n + F 1 + • • • + F K
where R1 n is added to the F j 's in order to take into account the constant part µ of (4). Furthermore, note that the sum defining the space F does not need to be direct. We are free to choose the functions φ i 's and ψ j i 's. In the sequel, we assume that these functions are chosen in such a way that the mild assumption E ∩ F = {0} is fulfilled. Note that we do not assume that s belongs to E neither that t belongs to F . Let G be the space (E + F ) ⊥ , we obviously have E ⊕ F ⊕ G = R n and we denote by P n the projection onto E along F + G. Moreover, we define π E and π F +G as the orthogonal projections onto E and F + G respectively. Thus, we derive the following framework from [START_REF] Mammen | The existence and asymptotic properties of a backfitting projection algorithm under weak conditions[END_REF],

Y = P n Z = s + σP n ε (27)
where we have set

s = P n s + P n t = s + (P n -I n )s + P n t = s + (P n -I n )(s -π E s) + P n (t -π F +G t) = s + h .
Let F = {S m , m ∈ M} be a finite collection of linear subspaces of E, we apply the procedure described in Section 2 to Y given by [START_REF] Massart | Concentration inequalities and model selection[END_REF], that is, we choose an index m ∈ M as a minimizer of ( 11) with a penalty function satisfying the hypotheses of Theorems 2.1 or 2.2 according to whether (H Gau ) or (H Mom ) holds. This way, we estimate s by s. From the triangular inequality, we derive that

E[ s -s 2 n ] 2E[ s -s 2 n ] + 2 h 2 n .
As we discussed previously, under suitable assumptions on the complexity of the collection F, we can assume that (A 1 ) and (A 2 ) are fulfilled. Let us suppose for the moment that (A 3 ) is satisfied for some ρ > 1. Note that, for any m ∈ M, π m is an orthogonal projection onto the image set of the oblique projection P n . Consequently, we have Tr

( t P n π m P n ) rk(π m ) = dim(S m ) and Assumption (A 3 ) implies (A ′ 3 ) with c = 1/ρ 2 . Since, for all m ∈ M, s -π m s n s -π m s n + h -π m h n s -π m s n + h n ,
we deduce from Theorems 2.1 or 2.2 that we can find, independently from s and n, two positive numbers C and C ′ such that

E[ s -s 2 n ] C inf m∈M s -π m s 2 n + Tr( t P n π m P n ) n σ 2 + C ′ h 2 n + ρ 2 σ 2 n R . (28)
To derive an interesting upperbound on the L 2 -risk of s, we need to control the remainder term. Because ρ(•) is a norm on M n , we dominate the norm of h by

h n ρ(I n -P n ) s -π E s n + ρ(P n ) t -π F +G t n (1 + ρ(P n ))( s -π E s n + t -π F +G t n ) (1 + ρ)( s -π E s n + t -π F +G t n ) .
Note that, for any m ∈ M, S m ⊂ E and so, s -π E s n s -π m s n . Thus, Inequality (28) leads to

E[ s-s 2 n ] C(1+ρ) 2 inf m∈M s -π m s 2 n + Tr( t P n π m P n ) n σ 2 +C ′ (1+ρ) 2 t -π F +G t 2 n + σ 2 n R .
(29) The space F + G has to be seen as a large approximation space. So, under a reasonable assumption on the regularity of the component t, the quantity t -π F +G t 2 n could be regarded as being neglectable. It mainly remains to understand the order of the multiplicative factor (1 + ρ) 2 . Thus, we now discuss about the norm ρ(P n ) and the assumption (A 3 ). This quantity depends on the design points (x i , y 1 i , . . . , y K i ) ∈ [0, 1] K+1 and on how we construct the spaces E and F , i.e. on the choice of the basis functions φ i and ψ (j) i . Hereafter, the design points (x i , y 1 i , . . . , y K i ) will be assumed to be known independent realizations of a random variable on

[0, 1] K+1 with distribution ν ⊗ ν 1 ⊗ • • •⊗ ν K .
We also assume that these points are independent of the noise ε and we proceed conditionally to them. To discuss about the probability for (A 3 ) to occur, we introduce some notations. We denote by D ′ n the integer

D ′ n = 1 + D (1) n + • • • + D (K)
n and we have dim(F ) D ′ n . Let A be a p × p real matrix, we define

r p (A) = sup    p i=1 p j=1 |a i a j | × |A ij | : p i=1 a 2 i 1    .
Moreover, we define the matrices V (φ) and B(φ) by

V ij (φ) = 1 0 φ i (x) 2 φ j (x) 2 ν(dx) and B ij (φ) = sup x∈[0,1] |φ i (x)φ j (x)| ,
for any 1 i, j D n . Finally, we introduce the quantities

L φ = max r 2 Dn (V (φ)), r Dn (B(φ)) and b φ = max i=1,...,Dn sup x∈[0,1] |φ i (x)| and L n = max L φ , D n D ′ n , b φ nD n D ′ n .
Proposition 4.1. Consider the matrix P n defined in [START_REF] Massart | Concentration inequalities and model selection[END_REF]. We assume that the design points are independent realizations of a random variable on [0, 1] K+1 with distribution ν ⊗ν 1 ⊗•

∀1 i D n , 1 0 φ i (x)ν(dx) = 0 (30)
then, there exists some universal constant C > 0 such that, for any ρ > 1,

P (ρ(P n ) > ρ) 4D n (D n + D ′ n ) exp - Cn L n (1 -ρ -1 ) 2 .
As a consequence of Proposition 4.1, we see that (A 3 ) is fulfilled with a large probability since we choose basis functions φ i in such a way to keep L n small in front of n. It will be so for localized bases (piecewise polynomials, orthonormal wavelets, ...) with L n of order of n 1-ω , for some ω ∈ (0, 1), once we consider D n and D ′ n of order of n

1 3 -3ω
2 (this is a direct consequence of Lemma 1 in [START_REF] Birgé | Minimum contrast estimators on sieves: exponential bounds and rates of convergence[END_REF]). This limitation, mainly due to the generality of the proposition, could seem restrictive from a practical point of view. However the statistician can explicitly compute ρ(P n ) with the data. Thus, it is possible to adjust D n and D ′ n in order to keep ρ(P n ) small in practice. Moreover, we will see in Section 6 that, for our choices of φ i and ψ j i , we can easily consider D n and D ′ n of order of √ n as we keep ρ(P n ) small (concrete values are given in the simulation study).

Convergence rates

The previous sections have introduced various upperbounds on the L 2 -risk of the penalized least-squares estimators s. Each of them is connected to the minimal risk of the estimators among a collection {ŝ m , m ∈ M}. One of the main advantages of such inequalities is that it allows us to derive uniform convergence rates with respect to many well known classes of smoothness (see [START_REF] Birgé | From model selection to adaptive estimation[END_REF]). In this section, we give such results over Hölderian balls for the estimation of a component in an additive framework. To this end, for any α > 0 and R > 0, we introduce the space H α (R) of the α-Hölderian functions with constant R > 0 on [0, 1],

H α (R) = {f : [0, 1] → R : ∀x, y ∈ [0, 1], |f (x) -f (y)| R|x -y| α } .
In order to derive such convergence rates, we need a collection of models F with good approximation properties for the functions of H α (R). We denote by P BM n any oblique projector defined as in the previous section and based on spaces S n and S j n that are constructed as one of the examples given in Section 2 of [START_REF] Birgé | An adaptive compression algorithm in Besov spaces[END_REF]. In particular, such a construction allows us to deal with approximation spaces S n and S j n that can be considered as spaces of piecewise polynomials, spaces of orthogonal wavelet expansions or spaces of dyadic splines on [0, 1]. We consider the dimensions D n = dim(S n ) and, for any j ∈ {1, . . . , K}, D

(j) n = dim(S j n ) = D n /K.
Finally, we take a collection of models F BM that contains subspaces of E = Im(P BM n ) as Baraud did in Section 2.2 of [START_REF] Baraud | Model selection for regression on a random design[END_REF].

Proposition 5.1. Consider the framework ( 4) and assume that (H Gau ) or (H Mom ) holds with p > 6. We define Y in [START_REF] Massart | Concentration inequalities and model selection[END_REF] with P BM n . Let η > 0 and s be the estimator selected by the procedure [START_REF] Birgé | Minimal penalties for gaussian model selection[END_REF] applied to the collection of models F BM with the penalty

pen(m) = (1 + η) Tr( t P BM n π m P BM n ) n σ 2 .
Suppose that (A 3 ) is fulfilled, we define

ζ n = 1 2 log n log D n -1 > 0 .
For any α > ζ n and R > 0, the penalized least-squares estimator s satisfies

sup (s,t 1 ,...,t K )∈Hα(R) K+1 E ε,d s -s 2 n C α n -2α/(2α+1) (31)
where E ε,d is the expectation on ε and on the random design points and C α > 1 only depends on α, ρ, σ 2 , K, L, θ and p (under (H Mom ) only).

Note that the supremum is taken over Hölderian balls for all the components of the regression function, i.e. the regression function is itself supposed to belong to an Hölderian space. As we mention in the introduction, Stone [START_REF] Stone | Additive regression and other nonparametric models[END_REF] has proved that the rate of convergence given by ( 31) is optimal in the minimax sense.

Simulation study

In this section, we study simulations based on the framework given by (4) with K + 1 components s, t 1 , . . . , t K and Gaussian errors. First, we introduce the spaces S n and S j n , j ∈ {1, . . . , K}, and the collections of models that we handle. Next, we illustrate the performances of the estimators in practice by several examples.

Preliminaries

To perform the simulation study, we consider two collections of models. In both cases, we deal with the same spaces S n and S j n defined as follows. Let ϕ be the Haar wavelet's mother function,

∀x ∈ R, ϕ(x) =    1 if 0 x < 1/2 , -1 if 1/2 x < 1 , 0 otherwise.
For any i ∈ N and j ∈ {0, . . . , 2 i -1}, we introduce the functions

ϕ i,j (x) = 2 i/2 ϕ(2 i x -j), x ∈ R .
It is clear that these functions are orthonormal in L 2 0 ([0, 1], dx) for the usual scalar product. Let d n be some positive integer, we consider the space S n ⊂ L 2 0 ([0, 1], dx) generated by the functions ϕ i,j such that 0 i d n and 0 j < 2 i . The dimension of this space is dim(S n ) = D n = 2 dn+1 -1. In the sequel, we denote by Π n the set of all the allowed pairs (i, j),

Π n = (i, j) ∈ N 2 such that 0 i d n , 0 j < 2 i .
Moreover, for any k ∈ {1, . . . , D n } such that k = 2 i + j with (i, j) ∈ Π n , we denote φ k = ϕ i,j .

Let d ′ n be an other positive integer, the spaces S j n ⊂ L 2 0 ([0, 1], dy j ) are all supposed to be generated by the functions defined on [0, 1] by

ψ 2i (y) = ψ
ϕ i,j = (ϕ i,j (x 1 ), . . . , ϕ i,j (x n )) ′ ∈ R n , (i, j) ∈ Π n .
Let m be a subset of Π n , the model S m is defined as the linear subspace of E generated by the vectors ϕ i,j with (i, j) ∈ m.

In the following simulations, we always take D n and D ′ n close to 4 √ n, i.e.

d n = log(2 √ n + 1/2) log(2) and d ′ n = 4 √ n -1 2K
where, for any x ∈ R, ⌊x⌋ denotes the largest integer not greater than x. For such choices, basic computations lead to L n of order of n 5/4 in Proposition 4.1. As a consequence, this proposition does not ensure that (A 3 ) is fulfilled with a large probability. However, ρ(P n ) remains small in practice as we will see and it allows us to deal with larger collections of models.

Collections of models

The first collection of models is the smaller one because the models are nested. Let us introduce the index subsets, for any i ∈ {0, . . . , d n },

m i = (i, j), 0 j < 2 i ⊂ Π n .
Thus, we define F N as

F N = S m such that ∃k ∈ {0, . . . , d n }, m = k i=0 m i .
This collection has a small complexity since, for any d ∈ N, N d 1. According to Corollary 2.1, we can consider the penalty function given by pen

N (m) = (1 + C) Tr( t P n π m P n ) n σ 2 (32)
for some C > 0. In order to compute the selected estimator s, we simply compute ŝm in each model of F N and we take the one that minimizes the penalized least-squares criterion.

The second collection of models is larger than F N . Indeed, we allow m to be any subset of Π n and we introduce

F C = {S m such that m ⊂ Π n } .
The complexity of this collection is large because, for any d ∈ N,

N d = D n d = D n ! d!(D n -d)! eD n d d .
So, we have log N d d(1 + log D n ) and, according to Corollary 2.1, we take a penalty function as

pen C (m) = (1 + C + log D n ) Tr( t P n π m P n ) n σ 2 (33)
for some C > 0. The large number of models in F C leads to difficulties for computing the estimator s. Instead of exploring all the models among F C , we break the penalized criterion down with respect to an orthonormal basis φ 1 , . . . , φ Dn of E and we get

Y - Dn i=1 Y, φ i n φ i 2 n + (1 + C + log D n ) Tr( t P n π E P n ) n σ 2 = Y 2 n - Dn i=1 Y, φ i 2 n -(1 + C + log D n ) t P n φ i 2 n σ 2 .
In order to minimize the penalized least-squares criterion, we only need to keep the coefficients Y, φ i n that are such that

Y, φ i 2 n (1 + C + log D n ) t P n φ i 2 n σ 2 .
This threshold procedure allows us to compute the estimator s in reasonable time.

In accordance with the results of Section 3, in the case of unknown variance, we substitute σ2 for σ 2 in the penalties (32) and (33).

Numerical simulations

We now illustrate our results and the performances of our estimation procedure by applying it to simulated data

Z i = s(x i ) + K j=1 t j (y j i ) + σε i , i = 1, . . . , n ,
where K

1 is an integer that will vary from an experiment to an other, the design points (x i , y 1 i , . . . , y K i ) ′ are known independent realizations of an uniform random variable on [0, 1] K+1 and the errors ε i are i.i.d. standard Gaussian random variables. We handle this framework with known or unknown variance factor σ 2 = 1 according to the cases and we consider a design of size n = 512. The unknown components s, t 1 , . . . , t K are either chosen among the following ones, or set to zero in the last subsection,

f 1 (x) = sin 4π x ∧ 1 2 f 2 (x) = cos 2π x - 1 4 2 -C 2 f 3 (x) = x + 2 exp(-16x 2 ) -C 3 f 4 (x) = sin(2x) + 2 exp(-16x 2 ) -C 4 f 5 (x) = 1 -exp(-10(x -1/2)) 1 + exp(-10(x -1/2)) f 6 (x) = 6x(1 -x) -1
where the constants C 2 , C 3 and C 4 are such that f i ∈ L 2 0 ([0, 1], dx) for any i ∈ {1, . . . , 6}. The first step of the procedure consists in computing the oblique projector P n and taking the data Y = P n Z. Figure 1 gives an example by representing the signal s, the data Z and the projected data Y for K = 6, s = f 1 and t j = f j , j ∈ {1, . . . , 6}. In particular, for this example, we have ρ 2 (P n ) = 1.22. We see that we actually get reasonable value of ρ 2 (P n ) with our particular choices for D n and D ′ n . In order to estimate the component s, we choose m by the procedure [START_REF] Birgé | Minimal penalties for gaussian model selection[END_REF] with penalty function given by [START_REF] Ravikumar | Sparse additive models[END_REF] or [START_REF] Robin | DNA, Words and Models[END_REF] according to the cases. The first simulations deal with the collection F N of nested models. Figure 2 represents the true s and the estimator s for K = 6 parasitic components given by t j = f j , j ∈ {1, . . . , 6} and s = f 1 or s = f 5 . The penalty function [START_REF] Ravikumar | Sparse additive models[END_REF] has been used with a constant C = 1.5. The second set of simulations is related to the large collection F C and to the penalty function [START_REF] Robin | DNA, Words and Models[END_REF] with C = 4.5. Figure 3 illustrates the estimation of s = f 1 and s = f 2 with K = 6 parasitic components t j = f j , j ∈ {1, . . . , 6}.

In both cases, we see that the estimation procedure behaves well and that the norms ρ(P n ) are close to one in spite of the presence of the parasitic components. Moreover, note that the collection F C allows to get estimators that are sharper because they detect constant parts of s. This advantage leads to a better bias term in the quadratic risk decomposition at the price of the logarithmic term in the penalty (33).

Ratio estimation

In Section 4, we discussed about assumptions that ensure a small remainder term in Inequality [START_REF] Meier | High-dimensional additive modeling[END_REF]. This result corresponds to some oracle type inequality for our estimation procedure of a component in an additive framework. We want to evaluate how far E s -s 2 n is from the oracle risk. Thus, we estimate the ratio by repeating 500 times each experiment for various values of K and C. For each set of simulations, the parasitic components are taken such that t j = f j , j ∈ {1, . . . , K}, the values of ρ(P n ) are given and the variance σ 2 is either assumed to be known or not. Table 1 (resp. Table 2) gives the values of r K (s) obtained for s = f 1 (resp. s = f 5 ) with the collection F N and the penalty [START_REF] Ravikumar | Sparse additive models[END_REF]. We clearly see that taking C close to zero or too large is not a good thing for the procedure. In our examples, C = 1.5 give good results and we get reasonable values of r K (s) for other choices of C between 1 and 3 for known or unknown variance. As expected, we also note that the values of ρ(P n ) and r K (s) tend to increase when K goes up but remain acceptable for K ∈ {1, . . . , 6}.

r K (s) = E s -s 2 n inf m∈M s -s m
In the same way, we estimate the ratio r K (s) for s = f 1 and s = f 2 with the collection F C and the penalty [START_REF] Robin | DNA, Words and Models[END_REF]. The results are given in Table 3 and Table 4. We obtain reasonable values of r K (s) for choices of C larger than what we took in the nested case. This phenomenon is related to what we mentioned at the end of Section 2. Indeed, for large collection of models, we need to overpenalize in order to keep the remainder term small enough. Moreover, because σ2 tends to overestimate σ 2 (see Section 3), we see that we can consider smaller values for C when the variance is unknown than when it is known for obtaining equivalent results.

Parasitic components equal to zero

We are now interested in the particular case of parasitic components t j equal to zero in (4), i.e. data are given by 1: Ratio r K (s) for the estimation of s = f 1 with F N . Each pair of lines corresponds to a value of K with the known σ 2 case on the first line and unknown σ 2 case on the second one.

Z i = s(x i ) + σε i , i = 1, . . . ,
If we know that these K components are zero and if we deal with the collection F N and a known variance σ 2 , we can consider the classical model selection procedure given by

m0 ∈ argmin m∈M Z -π m Z 2 n + C dim(S m ) n σ 2 . ( 34
)
Then, we can define the estimator s0 = π m0 Z. This procedure is well known and we refer to [START_REF] Massart | Concentration inequalities and model selection[END_REF] for more details. If we do not know that the K parasitic components are null, we can use our procedure to estimate s by s. In order to compare the performances of s and s0 with respect to the number K of zero parasitic components, we estimate the ratio

r K (s, s0 ) = E[ s -s 2 n ] E[ s -s0 2 n ]
for various values of K and C by repeating 500 times each experiment. The obtained results are given in Tables 5 and6 for s = f 1 and s = f 5 respectively. Obviously, the ratio r K (s, s0 ) is always larger than one because the procedure (34) makes good use of its knowledge about nullity of the t j . Nevertheless, we see that our procedure performs nearly as well as [START_REF] Ruppert | Multivariate locally weighted least squares regression[END_REF] even for a large number of zero components. Indeed, for K ∈ {1, . . . , 9}, do not assuming that we know that the t j are zero only implies a loss between 1% and 10% for the risk. Such a loss remains acceptable in practice and allows us to consider more general framework for estimating s.

Proofs

In the proofs, we repeatedly use the following elementary inequality that holds for any α > 0 and x, y ∈ R, 2: Ratio r K (s) for the estimation of s = f 5 with F N . Each pair of lines corresponds to a value of K with the known σ 2 case on the first line and unknown σ 2 case on the second one. By definition of γ n , for any t ∈ R n , we can write

2|xy| αx 2 + α -1 y 2 . ( 35
s -t 2 n = γ n (t) + 2σ t -Y, P n ε n + σ 2 P n ε 2 n .
Let m ∈ M, since ŝm = s m + σπ m P n ε, this identity and (12) lead to

s -s 2 n = s -s m 2 n + γ n (s) -γ n (s m ) + 2σ s -s m , P n ε n = s -s m 2 n + γ n (s) -γ n (ŝ m ) -σ 2 π m P n ε 2 n -2σ s -s, P n ε n + 2σ s -s m , P n ε n s -s m 2 n + pen(m) -pen( m) + 2σ 2 π mP n ε 2 n (36) -2σ s -s m, P n ε n + 2σ s -s m , P n ε n -σ 2 π m P n ε 2 n .
Consider an arbitrary a m ∈ S ⊥ m such that a m n = 1, we define

u m = (s -s m )/ s -s m n if s = π m s a m otherwise . ( 37
)
Thus, (36) gives

s -s 2 n s -s m 2 n + pen(m) -pen( m) + 2σ 2 π mP n ε 2 n (38) +2σ s -s m n | u m, P n ε n | + 2σ s -s m , P n ε n -σ 2 π m P n ε 2 n .
Take α ∈ (0, 1) that we specify later and we use the inequality [START_REF] Scheffé | The analysis of variance[END_REF], 3: Ratio r K (s) for the estimation of s = f 1 with F C . Each pair of lines corresponds to a value of K with the known σ 2 case on the first line and unknown σ 2 case on the second one.

(1 -α) s -s 2 n s -s m 2 n + pen(m) -pen( m) (39)
+(2 -α)σ 2 π mP n ε 2 n + α -1 σ 2 u m, P n ε 2 n +2σ s -s m , P n ε n -σ 2 π m P n ε 2 n . C 0.0 0.
We choose α = 1/(1 + θ) ∈ (0, 1) but for legibility we keep using the notation α. Let us now introduce two functions p 1 , p 2 : M → R + that will be specified later to satisfy, for all m ∈ M,

pen(m) (2 -α)p 1 (m) + α -1 p 2 (m) . ( 40
)
We use this bound in [START_REF] Bahr | Inequalities for the rth absolute moment of a sum of random variables 1 r 2[END_REF] to obtain Taking the expectation on both sides, it leads to

(1 -α) s -s 2 n s -s m 2 n + pen(m) + (2 -α) σ 2 π mP n ε 2 n -p 1 ( m) +α -1 σ 2 u m, P n ε 2 n -p 2 ( m) + 2σ s -s m , P n ε n -σ 2 π m P n ε 2 n s -s m 2 n + pen(m) + 2σ s -s m , P n ε n -σ 2 π m P n ε 2 n +(2 -α) sup m ′ ∈M σ 2 π m ′ P n ε 2 n -p 1 (m ′ ) + +α -1 sup m ′ ∈M σ 2 u m ′ , P n ε 2 n -p 2 (m ′ ) + . C 0.0 0.
(1 -α)E s -s 2 n s -s m 2 n + pen(m) -Tr( t P n π m P n )σ 2 /n +(2 -α)E sup m ′ ∈M σ 2 π m ′ P n ε 2 n -p 1 (m ′ ) + +α -1 E sup m ′ ∈M σ 2 u m ′ , P n ε 2 n -p 2 (m ′ ) + s -s m 2 n + pen(m) -Tr( t P n π m P n )σ 2 /n +(2 -α) m ′ ∈M E σ 2 π m ′ P n ε 2 n -p 1 (m ′ ) + +α -1 m ′ ∈M E σ 2 u m ′ , P n ε 2 n -p 2 (m ′ ) + s -s m 2 n + pen(m) -Tr( t P n π m P n )σ 2 /n +(2 -α) m ′ ∈M E 1,m ′ + α -1 m ′ ∈M E 2,m ′ . C 0.0 0.5 1.0 1.
= f 5 with F N .
Because the choice of m is arbitrary among M, we can infer that

(1 -α)E s -s 2 n inf m∈M s -s m 2 n + pen(m) -Tr( t P n π m P n )σ 2 /n +(2 -α) m∈M E 1,m + α -1 m∈M E 2,m . (41)
We now have to upperbound E 1,m and E 2,m in (41). Let start by the first one. If S m = {0}, then π m P n = 0 and p 1 (m) 0 suffices to ensure that E 1,m = 0. So, we can consider that the dimension of S m is positive and π m P n = 0. The Lemma 8.2 applied with A = π m P n gives, for any x > 0,

P n π m P n ε 2 n Tr( t P n π m P n ) + 2 ρ 2 (P n )Tr( t P n π m P n )x + 2ρ 2 (P n )x e -x (42)
because ρ(π m P n ) ρ(π m )ρ(P n ) ρ(P n ). Let β = θ2 /(1 + 2θ) > 0, [START_REF] Scheffé | The analysis of variance[END_REF] and (42) lead to

P n π m P n ε 2 n (1 + β)Tr( t P n π m P n ) + (2 + β -1 )ρ 2 (P n )x e -x . (43)
Let δ = θ 2 /((1 + θ)(1 + 2θ + 2θ 2 )) > 0, we set np 1 (m) = ((1 + β) + (2 + β -1 )δL m )Tr( t P n π m P n )σ 2
and (43) implies

E m,1 = ∞ 0 P σ 2 π m P n ε 2 n -p 1 (m) + ξ dξ = ∞ 0 P n π m P n ε 2 n -np 1 (m)/σ 2 nξ/σ 2 dξ ∞ 0 exp - δL m Tr( t P n π m P n ) ρ 2 (P n ) - nξ (2 + β -1 )ρ 2 (P n )σ 2 dξ (2 + β -1 )ρ 2 (P n )σ 2 n exp - δL m Tr( t P n π m P n ) ρ 2 (P n ) . (44)
We now focus on E m,2 . The random variable u m , P n ε n = t P n u m , ε n is a centered Gaussian variable with variance t P n u m inequality gives

P (| u m , P n ε n | x) exp - nx 2 2 t P n u m 2 n exp - nx 2 2ρ 2 (P n ) that is equivalent to P n u m , P n ε 2 n 2ρ 2 (P n )x e -x . (45)
We set

np 2 (m) = 2δL m Tr( t P n π m P n )σ 2
and (45) leads to

E m,2 = ∞ 0 P σ 2 u m , P n ε 2 n -p 2 (m) + ξ dξ = ∞ 0 P u m , P n ε 2 n -np 2 (m)/σ 2 nξ/σ 2 dξ ∞ 0 exp - δL m Tr( t P n π m P n ) ρ 2 (P n ) - nξ 2ρ 2 (P n )σ 2 dξ 2ρ 2 (P n )σ 2 n exp - δL m Tr( t P n π m P n ) ρ 2 (P n ) . (46)
We inject (44) and ( 46) in (41) and we replace α, β and δ to obtain

θ θ + 1 E s -s 2 n inf m∈M s -s m 2 n + pen(m) -Tr( t P n π m P n )σ 2 /n + ρ 2 (P n )σ 2 n R θ
where we have set

R θ = c θ m∈M exp - L m Tr( t P n π m P n ) c θ ρ 2 (P n )
and

c θ = 2θ 4 + 8θ 3 + 8θ 2 + 4θ + 1 θ 2 (1 + θ) .
Finally, (40) gives a penalty as [START_REF] Buja | Linear smoothers and additive models (with discussion)[END_REF] and the announced result follows.

Proof of Theorem 2.2

In order to prove Theorem 2.2, we show the following stronger result. Under the assumptions of the theorem, there exists a positive constant C that only depends on p and θ, such that, for any z > 0,

P θ θ + 2 H + ρ 2 (P n )σ 2 n z Cτ p N 0 1 ∧ z -p/2 + R Pn,p (F, z) (47)
where the quantity H is defined by

H = s -s 2 n - θ + 4 θ inf m∈M s -s m 2 n + 2(θ + 2) θ + 4 pen(m)
and we have set R Pn,p (F, z) equal to

m∈M:Sm ={0} 1 + Tr( t P n π m P n ) ρ( t P n π m P n ) L m Tr( t P n π m P n ) ρ 2 (P n ) + z -p/2
.

Thus, for any q > 0 such that 2(q + 1) < p, we integrate (47) via Lemma 8.1 to get

E H q + = ∞ 0 qt q-1 P (H + t) dt = (θ + 2)ρ 2 (P n )σ 2 θn q ∞ 0 qz q-1 P θ θ + 2 H + ρ 2 (P n )σ 2 n z dz C ′ (p, q, θ)τ p ρ 2 (P n )σ 2 n q R p,q Pn,θ (F) (48)
where we have set

R p,q Pn,θ (F) = N 0 + m∈M:Sm ={0} 1 + Tr( t P n π m P n ) ρ( t P n π m P n ) L m Tr( t P n π m P n ) ρ 2 (P n ) q-p/2 . Since E s -s 2q n 1/q E θ + 8 θ inf m∈M s -s m 2 n + 2(θ + 4) θ + 8 pen(m) + H + q 1/q
, it follows from Minkowski's Inequality when q 1 or convexity arguments when 0 < q < 1 that

E s -s 2q n 1/q 2 (q -1 -1) + C ′′ (θ) inf m∈M s -s m 2 n + pen(m) + E H q + 1/q . ( 49
)
Inequality [START_REF] Härdle | Nonparametric and Semiparametric Models[END_REF] directly follows from (48) and (49). We now turn to the proof of (47). Inequality [START_REF] Bahr | Inequalities for the rth absolute moment of a sum of random variables 1 r 2[END_REF] does not depend on the distribution of ε and we start from here. Let α = α(θ) ∈ (0, 1), for any m ∈ M we have

(1 -α) s -s 2 n s -s m 2 n + pen(m) -pen( m) + (2 -α)σ 2 π mP n ε 2 n +α -1 σ 2 u m, P n ε 2 n + 2σ s -s m , P n ε n
where u m is defined by [START_REF] Stone | Additive regression and other nonparametric models[END_REF]. Use again [START_REF] Scheffé | The analysis of variance[END_REF] with α to obtain

(1 -α) s -s 2 n s -s m 2 n + pen(m) -pen( m) + (2 -α)σ 2 π mP n ε 2 n +α -1 σ 2 u m, P n ε 2 n + 2σ s -s m n | u m , P n ε n | (1 + α) s -s m 2 n + pen(m) -pen( m) (50)
+(2 -α)σ 2 π mP n ε 2 n +α -1 σ 2 u m, P n ε 2 n + α -1 σ 2 u m , P n ε 2 n .
Let us now introduce two functions p1 , p2 : M → R + that will be specified later and that satisfy,

∀m ∈ M, pen(m) (2 -α)p 1 (m) + α -1 p2 (m) . (51)
Thus, Inequality (50) implies

(1 -α) s -s 2 n (1 + α) s -s m 2 n + pen(m) + α -1 p2 (m) +(2 -α) σ 2 π mP n ε 2 n -p1 ( m) +α -1 σ 2 u m, P n ε 2 n -p2 ( m) +α -1 σ 2 u m , P n ε 2 n -p2 (m) (1 + α) s -s m 2 n + 2pen(m)/(1 + α) +(2 -α) sup m ′ ∈M σ 2 π m ′ P n ε 2 n -p1 (m ′ ) + +2α -1 sup m ′ ∈M σ 2 u m ′ , P n ε 2 n -p2 (m ′ ) + .
Because the choice of m is arbitrary among M, we can infer that, for any ξ > 0,

P ((1 -α)H + ξ) P (2 -α) sup m∈M σ 2 π m P n ε 2 n -p1 (m) + ξ 2 +P 2α -1 sup m∈M σ 2 u m , P n ε 2 n -p2 (m) + ξ 2 m∈M P σ 2 π m P n ε 2 n p1 (m) + ξ 2(2 -α) + m∈M P σ 2 u m , P n ε 2 n p2 (m) + αξ 4 m∈M P 1,m (ξ) + m∈M P 2,m (ξ) . (52)
We first bound P 1,m (ξ). For m ∈ M such that S m = {0} (i.e. π m = 0), p1 (m) 0 leads obviously to P 1,m (ξ) = 0. Thus, it is sufficient to bound P 1,m (ξ) for m such that π m is not null. This ensures that the symmetric nonnegative matrix à = t P n π m P n lies in M n \ {0}. Thus, under hypothesis (6), Corollary 5.1 of [START_REF] Baraud | Model selection for regression on a fixed design[END_REF] gives us, for any x m > 0,

P n π m P n ε 2 n Tr( Ã) + 2 ρ( Ã)Tr( Ã)x m + ρ( Ã)x m C 1 (p)τ p Tr( Ã) ρ( Ã)x p/2 m
where C 1 (p) is a constant that only depends on p. The properties of the norm ρ imply

ρ( Ã) = ρ( t (π m P n )(π m P n )) = ρ(π m P n ) 2 ρ 2 (P n ) . (53)
By the inequalities (53) and ( 35) with θ/2 > 0, we obtain

P n π m P n ε 2 n 1 + θ 2 Tr( Ã) + 1 + 2 θ ρ 2 (P n )x m C 1 (p)τ p Tr( Ã) ρ( Ã)x p/2 m . ( 54
)
We take α = 2/(θ + 2) ∈ (0, 1) but for legibility we keep using the notation α. Moreover, we choose

np 1 (m) = 1 + θ 2 + L m 2(θ + 1)
Tr( t P n π m P n )σ 2 and

x m = θ 2(θ + 1)(θ + 2) × L m Tr( t P n π m P n ) + nξ/σ 2 ρ 2 (P n ) .
Thus, Inequality (54) leads to

P 1,m (ξ) = P σ 2 π m P n ε 2 n p1 (m) + ξ 2(2 -α) = P σ 2 π m P n ε 2 n p1 (m) + (θ + 2)ξ 4(θ + 1) P n π m P n ε 2 n 1 + θ 2 Tr( t P n π m P n ) + 1 + 2 θ ρ 2 (P n )x m C 2 (p, θ) Tr( t P n π m P n )τ p ρ( t P n π m P n ) L m Tr( t P n π m P n ) + nξ/σ 2 ρ 2 (P n ) -p/2 . ( 55
)
We now focus on P 2,m (ξ). Let y m be some positive real number, the Markov Inequality leads to

P (| u m , P n ε n | y m ) y -p m E [| u m , P n ε n | p ] = y -p m E t P n u m , ε n p . ( 56
)
Since p > 2, the quantity τ p is lower bounded by 1,

τ p = E [|ε 1 | p ] E ε 2 1 p/2 = 1 . (57)
Moreover, we can apply the Rosenthal inequality (see Chapter 2 of [START_REF] Petrov | Limit theorems of probability theory: sequences of independent random variables[END_REF]) to obtain Thus, the Inequality (58) becomes

E t P n u m ,
E t P n u m , ε n p 2C 3 (p)ρ p (P n )τ p n -p/2
and, putting this inequality in (56), we obtain

P (| u m , P n ε n | y m ) 2C 3 (p)ρ p (P n )τ p n -p/2 y -p m . ( 59
)
We take np 2 (m) = 1 2(θ + 1)

σ 2 L m Tr( t P n π m P n )
and

y 2 m = 1 2(θ + 2)n L m Tr( t P n π m P n ) + nξ σ 2 .
Finally, (59) gives

P 2,m (ξ) = P σ 2 u m , P n ε 2 n p2 (m) + αξ 4 = P σ 2 u m , P n ε 2 n p2 (m) + ξ 2(θ + 2) P u m , P n ε 2 n y 2 m C 4 (p, θ)τ p L m Tr( t P n π m P n ) + nξ/σ 2 ρ 2 (P n ) -p/2 . ( 60
)
Taking

R ′ (ξ) = m∈M:Sm ={0} 1 + Tr( t P n π m P n ) ρ( t P n π m P n ) L m Tr( t P n π m P n ) + nξ/σ 2 ρ 2 (P n ) -p/2
and putting together Inequalities (52), ( 55) and (60) lead us to

P ((1 -α)H + ξ) m∈M P 1,m (ξ) + m∈M P 2,m (ξ) m∈M:Sm={0} P 2,m (ξ) + m∈M:Sm ={0} P 1,m (ξ) + m∈M:Sm ={0} P 2,m (ξ) m∈M:Sm={0} 1 ∧ C 4 (p, θ)τ p nξ σ 2 ρ 2 (P n ) -p/2 + C 5 (p, θ)τ p R ′ (ξ) N 0 (1 ∨ C 4 (pθ))τ p 1 ∧ nξ ρ 2 (P n )σ 2 -p/2 + C 5 (p, θ)τ p R ′ (ξ) .
For z > 0, take ξ = ρ 2 (P n )σ 2 z/n to obtain (47). We conclude the proof by computing the lowerbound (51) on the penalty function,

(2 -α)p 1 (m) + α -1 p2 (m) = 2(θ + 1) θ + 2 p1 (m) + θ + 2 2 p2 (m) = 1 + θ + θ 2 + 8θ + 8 4(θ + 1)(θ + 2) L m Tr( t P n π m P n ) n σ 2 .
Since (θ 2 + 8θ + 8)/(4(θ + 1)(θ + 2)) 1, the penalty given by ( 17) satisfies the condition (51).

7.2 Proofs of Theorems 3.1 and 3. Given θ > 0, we can find two positive numbers δ = δ(θ) < 1/2 and η = η(θ) such that (1 + θ)(1 -2δ) (1 + 2η). Thus we define

Ω n = σ2 > (1 -2δ)σ 2 .
On Ω n , we know that

∀m ∈ M, pen(m) (1 + 2η) T r( t P n π m P n ) n σ 2 .
Taking care of the random nature of the penalty, we argue as in the proof of Theorem 2.1 with L m = η to get

E s -s 2 n 1l Ωn η + 1 η inf m∈M s -s m 2 n + E[pen(m)] - Tr( t P n π m P n ) n σ 2 + ρ 2 (P n )σ 2 n R ′′ Pn,η (F) (61) where R ′′ Pn,η (F) is defined by R ′′ Pn,η (F) = C η m∈M exp - C ′ η Tr( t P n π m P n ) ρ 2 (P n ) .
We use Lemma 8.3 and (21) to get an upperbound for E[pen(m)],

E[pen(m)]

(1 + θ)

Tr( t P n π m P n ) n σ 2 + (1 + θ) Tr( t P n π m P n ) s -πs 2 n Tr ( t P n (I n -π)P n ) (1 + θ) Tr( t P n π m P n ) n σ 2 + (1 + θ) Tr( t P n P n ) s -πs 2 n Tr ( t P n (I n -π)P n ) (1 + θ) Tr( t P n π m P n ) n σ 2 + 2(1 + θ) s -πs 2 n .
The Proposition 1.1 and (61) give

E s -s 2 n 1l Ωn C(θ) inf m∈M E s -ŝm 2 n + 2(θ + 1) s -πs 2 n + ρ 2 (P n )σ 2 n R ′′ Pn,η (F) (62)
where

C(θ) > 1. We now bound E[ s -s 2 n 1l Ω c n ]. Note that s -s 2 n = s -s m 2 n + σ 2 π mP n ε 2 n s 2 n + σ 2 P n ε 2 n
and thus, by the Cauchy-Schwarz Inequality,

E[ s -s 2 n 1l Ω c n ] s 2 n P (Ω c n ) + σ 2 E[ P n ε 2 n 1l Ω c n ] s 2 n + σ 2 E[ P n ε 4 n ] 1/2 P (Ω c n ) 1/2 .
Moreover, the eigenvalues of the matrix P n t P n are nonnegative and so

E[ P n ε 4 n ] 1/2 = Var( P n ε 2 n ) + E[ P n ε 2 n ] 2 1/2
P n ) + ρ 2 (P n ) n .
Finally, the Lemma 8.4 gives

E[ s -s 2 n 1l Ω c n ] C ′ (θ) s 2 n + Tr( t P n P n ) + ρ 2 (P n ) n σ 2 exp - θ 2 Tr( t P n P n ) 32ρ 2 (P n ) C ′ (θ) s 2 n + ρ 2 (P n )(n + 1) n σ 2 exp - θ 2 Tr( t P n P n ) 32ρ 2 (P n ) C ′ (θ) s 2 n + 2ρ 2 (P n )σ 2 exp - θ 2 Tr( t P n P n ) 32ρ 2 (P n ) (63)
where C ′ (θ) > 1. The inequality [START_REF] Linton | A kernel method of estimating structured nonparametric regression based on marginal integration[END_REF] follows by collecting (62) and (63).

Proof of Theorem 3.2

Given θ > 0, we can find two positive numbers δ = δ(θ) < 1/3 and η = η(θ) such that

(1 + θ)(1 -3δ) (1 + 2η
). Thus we define

Ω ′ n = σ2 > (1 -3δ)σ 2 .
On Ω ′ n , we know that

∀m ∈ M, pen(m) (1 + 2η) T r( t P n π m P n ) n σ 2 .
Let m be any element of M that minimize s -s m ′ 2 n + σ 2 Tr( t P n π m ′ P n )/n among m ′ ∈ M. Taking care of the random nature of the penalty, we argue as in the proof of Theorem 2.2 with L m = η to get

E s -s 2q n 1l Ω ′ n 1/q C(q, θ)E s -s m 2 n + Tr( t P n π mP n ) n σ2 q 1/q + ρ 2 (P n )σ 2 n R n (p, q, θ) 1/q
where R n (p, q, θ) is equal to

C ′ (p, q, θ)τ p   N 0 + m∈M:S ={0} 1 + Tr( t P n π m P n ) ρ( t P n π m P n ) Tr( t P n π m P n ) ρ 2 (P n ) q-p/2   .
Since q 1, by a convexity argument and Jensen's inequality we deduce

E s -s 2q n 1l Ω ′ n 1/q C(q, θ) s -s m 2 n + Tr( t P n π mP n ) n E[σ 2 ] + ρ 2 (P n )σ 2 n R n (p, q, θ) 1/q .
(64) Lemma 8.3 and (21) give

Tr( t P n π mP n ) n E[σ 2 ] = Tr( t P n π mP n ) n σ 2 + nTr( t P n π mP n ) s -πs 2 n nTr( t P n (I n -π)P n ) Tr( t P n π mP n ) n σ 2 + 2 s -πs 2 n .
Thus, by the definition of m and Proposition 1.1, (64) becomes

E s -s 2q n 1l Ω ′ n 1/q C(q, θ) inf m∈M E[ s -ŝm 2 n ] + 2 s -πs 2 n + ρ 2 (P n )σ 2 n R n (p, q, θ) 1/q . ( 65
) We now bound E[ s -s 2q n 1l Ω ′c n ]. Note that s -s 2 n = s -s m 2 n + σ 2 π mP n ε 2 n s 2 n + σ 2 P n ε 2 n .
Since q 1, we have

E[ s -s 2q n 1l Ω ′c n ] s 2q n P(Ω ′ c n ) + σ 2q E[ P n ε 2q n 1l Ω ′c n ] .
Hölder's Inequality with exponent p/2q > 1 gives

E[ P n ε 2q n 1l Ω ′c n ] E[ P n ε p n ] 2q/p P(Ω ′ c n ) 1-2q/p
and, since

E[ P n ε p n ] 2q/p ρ 2q (P n )E[ ε p n ] 2q/p ρ 2q (P n )τ 2q/p p ,
we obtain by using Lemma 8.5 that

E[ s -s 2q n 1l Ω ′c n ] ( s 2q n + σ 2q ρ 2q (P n )τ 2q/p p )P(Ω ′ c n ) 1-2q/p C(p, q, θ)κ ′ n (p, q, θ)( s 2q n + σ 2q ρ 2q (P n )τ 2q/p p ) τ p ρ αp (P n )Tr( t P n P n ) -βp 1-2q/p
where α p = (p/2 -1) ∨ 1 and β p = (p/2 -1) ∧ 1 .

Thus, we get

E[ s -s 2q n 1l Ω ′c n ] 1/q C ′ (p, q, θ)κ n (p, q, θ)τ 1/q p ( s 2 n + τ p ρ 2 (P n )σ 2 ) ρ 2αp (P n ) Tr( t P n P n ) βp 1/q-2/p . (66)
The announced result follows from (65) and (66).

Proofs of Corollaries and Propositions

m = L, E s -s 2 n (1 + θ -1 ) inf m∈M s -s m 2 n + (θ + L) Tr( t P n π m P n ) n σ 2 + ρ 2 (P n )σ 2 n R n (θ) .
(67) We now upperbound the remainder term. Assumption (A ′

3 ) and bounds on N d and L lead to

R n (θ) 2(1 + θ) 4 θ 3 m∈M exp - θ 2 L 2(1 + θ) 3 × Tr( t P n π m P n ) ρ 2 (P n ) 2(1 + θ) 4 θ 3 m∈M exp - cθ 2 L 2(1 + θ) 3 dim(S m ) 2(1 + θ) 4 θ 3 d∈N N d e -(A+ω)d 2(1 + θ) 4 θ 3 d∈N e -ωd .
The last bound is clearly finite and we denote it by R = R(θ, ω). Thus, we derive from (67)

θ θ + 1 E s -s 2 n inf m∈M s -s m 2 n + (θ + L)Tr( t P n π m P n ) + Rρ 2 (P n )(dim(S m ) ∨ 1) σ 2 n and hypothesis (A ′ 3 ) gives θ θ + 1 E s -s 2 n inf m∈M s -s m 2 n + (θ + L + R/c) Tr( t P n π m P n ) ∨ cρ 2 (P n ) σ 2 n
that concludes the proof.

Proof of Corollary 2.2

Since p > 6, we can take q = 1 and apply Theorem 2.2 with constant weights L m = L to get

E s -s 2 n C inf m∈M s -s m 2 n + (1 + θ + L) Tr( t P n π m P n ) n σ 2 + ρ 2 (P n )σ 2 n R n (p, 1, θ) .
(68) To upperbound the remainder term, we use Assumption (A ′

3 ) and bounds on N d and L to get

R n (p, 1, θ) C ′ τ p   1 + m∈M:Sm ={0} 1 + Tr( t P n π m P n ) ρ( t P n π m P n ) LTr( t P n π m P n ) ρ 2 (P n ) 1-p/2   C ′ τ p   1 + m∈M:Sm ={0} (1 + dim(S m ))(Lc dim(S m )) 1-p/2   C ′ τ p 1 + (cω) ′1-p/2 A d>0 N d (1 + d)d 1-p/2 C ′ τ p 1 + (cω) ′1-p/2 d>0 (1 + d) p/2-2-ω d 1-p/2 .
The last bound is clearly finite and we denote it by Rτ p = R(θ, p, ω, ω ′ , c)τ p . Thus, as we did in the previous proof, we derive from (68) and

(A ′ 3 ) 1 C ′′ E s -s 2 n inf m∈M s -s m 2 n + (1 + θ + L + Rτ p /c) Tr( t P n π m P n ) ∨ cρ 2 (P n ) σ 2 n .
Since τ p 1, the announced result follows.

Proof of Proposition 4.1

The design points (x i , y 1 i , . . . , y K i ) are all assumed to be independent realizations of a random variable in [0, 1] K+1 with distribution ν ⊗ ν 1 ⊗ • • • ⊗ ν K . We denote by I k the unit k × k matrix and, for any a = (a 1 , . . . , a k ) ′ ∈ R k , we define the usual norm

|a| 2 = k i=1 a 2 i 1/2 . We also consider δ n = dim(F ) D (1) n + • • • + D (K) n + 1 and N n = n -D n -δ n .
The quantities δ n and N n are random and only depend on the y j i 's and not on the x i 's. The space E is generated by the vectors e (i) = (φ i (x 1 ), . . . , φ i (x n )) ′ , for i = 1, . . . , D n . Let {f (1) , . . . , f (δn) } be an orthonormal basis of F and {g (1) , . . . , g (Nn) } be an orthonormal basis of G = (E + F ) ⊥ . In the basis b of R n given by the e (i) 's, the f (i) 's and the g (i) 's, the projection P n onto E along F + G can be expressed as

M = I Dn 0 0 0 ∈ M n (R) .
Considering the matrix C that transforms b into the canonical basis, we can decompose

P n = CM C -1
. By the properties of the norm ρ, we get

ρ 2 (P n ) ρ 2 (C)ρ 2 (M )ρ 2 (C -1 ) = 1 n ρ( t CC) nρ( t C -1 C -1 ) .
For any ρ > 1, we deduce from the previous inequality that

P(ρ(P n ) > ρ) P ρ t CC n > ρ + P ρ(n t C -1 C -1 ) > ρ . (69)
Note that for any invertible matrix

A ∈ M n (R) and λ > 1, if ρ(A -I n ) < 1 -λ -1
, then ρ(A -1 ) < λ. Thus, Inequality (69) leads to

P(ρ(P n ) > ρ) P ρ t CC n > ρ + P ρ t CC n -I n > 1 -ρ -1 2P ρ t CC n -I n > 1 -ρ -1 . (70)
Let us denote by Φ the D n × D n Gram matrix associated to the vectors e (1) , . . . , e (Dn) . If we define the D n × δ n matrix Ω by

∀1 i D n , ∀1 j δ n , Ω ij = e (i) , f (j) n ,
then we can write the following decomposition by blocks,

1 n t CC =   Φ Ω 0 t Ω I δn 0 0 0 I Nn   ∈ M n (R) .
Consequently, by the definition of ρ(•), we obtain

ρ t CC n -I n ρ(Φ -I Dn ) + ρ(Ω ′ ) (71)
where we have set

Ω ′ = 0 Ω t Ω 0 .
Using (71) in (70) leads to

P(ρ(P n ) > ρ) 2P ρ(Φ -I Dn ) > 1 -ρ -1 2 + 2P ρ(Ω ′ ) > 1 -ρ -1 2 = 2P 1 + 2P 2 . (72)
First, we upperbound P 1 . Let x > 0, we consider the event

E x = ∀1 i, j D n , e (i) , e (j) n - 1 0 φ i (u)φ j (u)ν(du) V ij (φ) √ 2x + B ij (φ)x .
Because Φ -I Dn is symmetric, we know that, on the event a i a j e (i) , e (j) n -

1 0 φ i (u)φ j (u)ν(du) 1 sup a∈R Dn , |a| 2 1 Dn i=1 Dn j=1 |a i a j | |V ij (φ)| √ 2x + |B ij (φ)|x 2xL φ + xL φ .
Thus, for any x > 0 such that

2xL φ + xL φ 1 -ρ -1 2 (73)
we deduce

P 1 P ∃(i, j) : e (i) , e (j) n - 1 0 φ i (u)φ j (u)ν(du) > V ij (φ) √ 2x + B ij (φ)x Dn i=1 Dn j=1 P e (i) , e (j) n - 1 0 φ i (u)φ j (u)ν(du) > V ij (φ) √ 2x + B ij (φ)x . ( 74
)
The choice x = (1 -ρ -1 ) 2 /(12L(φ)) satisfies (73) and we apply Bernstein Inequality (see Lemma 8 of [START_REF] Birgé | Minimum contrast estimators on sieves: exponential bounds and rates of convergence[END_REF]) to the terms of the sum in (74) to obtain

P 1 2D 2 n exp - n(1 -ρ -1 ) 2 12L φ . (75)
It remains to upperbound the probability P 2 . Let x > 0, we consider the event

E ′ x = ∀1 i D n , ∀1 j δ n , e (i) , f (j) n √ 2x + b φ √ nx .
By definition of the norm ρ(•), we know that, on the event

E ′ x , ρ(Ω ′ ) =
|a i b j | e (i) , f (j) n 2 D n δ n √ 2x + b φ √ nx .
Thus, for any x > 0 such that

2 D n δ n √ 2x + b φ √ nx 1 -ρ -1 2 , (76)
we apply Bernstein Inequality conditionally to the y j i 's to deduce

P y ρ(Ω ′ ) > 1 -ρ -1 2 P y ∃(i, j) : e (i) , f (j) n > √ 2x + b φ √ nx Dn i=1 δn j=1 P y e (i) , f (j) n > √ 2x + b φ √ nx 2D n δ n e -nx 2D n D ′ n e -nx (77)
where P y is the conditional probability given the y j i 's. Indeed, under P y and (30), the variables e (i) , f (j) n are centered with unit variance. The choice 76) and (77) leads to

x = (1 -ρ -1 ) 2 16 max 4D n δ n , b φ √ nD n δ n satisfies (
P 2 = E P y ρ(Ω ′ ) > 1 -ρ -1 2 2D n D ′ n E exp - n(1 -ρ -1 ) 2 16 max 4D n δ n , b φ √ nD n δ n 2D n D ′ n exp - n(1 -ρ -1 ) 2 16 max 4D n D ′ n , b φ nD n D ′ n . (78)
The announced result follows from (72), ( 75) and (78).

Proof of Proposition 5.1

The collection F BM is nested and, for any d ∈ N, the quantity N d is bounded independently from d. Consequently, Condition [START_REF] Hastie | Generalized additive models[END_REF] is satisfied in the Gaussian case and ( 20) is fulfilled under moment condition. In both cases, we are free to take L = θ = η/2 and (A 1 ) is true for K = η. Assumption (A ′ 3 ) is fulfilled with c = 1/ρ 2 and, since dim(S m ) > 0 for any m ∈ M, we can apply Corollary 2.1 or 2.2 according to whether (H Gau ) or (H Mom ) holds. Moreover, we denote by E ε (resp. E d ) the expectation on ε (resp. the design points). So

E ε,d [•] = E ε [E d [•]].
We argue in the same way than in Section 4 and we use (A 3 ) to get

E ε,d s -s 2 n C inf m∈M E d s -s m 2 n + Tr( t P n π m P n ) n σ 2 + C ′ (1 + ρ) 2 E d [ t -π F +G t 2 n ] + R n σ 2 C inf m∈M E d [ s -s m 2 n ] + dim(S m ) n ρ 2 σ 2 + C ′ (1 + ρ) 2 E d [ t -π F +G t 2 n ] + R n σ 2 .
The definition of the norm • n implies that, for any f ∈ L 2 ([0, 1], ν),

E d 1 n n i=1 f (x i ) 2 = 1 0 f (x) 2 ν(dx) .
Since s ∈ H α (R), it is easy to see that this function lies in a Besov ball. Thus, we can apply Theorem 1 of [START_REF] Birgé | An adaptive compression algorithm in Besov spaces[END_REF] and we get, for any m ∈ M,

E d [ s -s m 2 n ] C(α, R) dim(S m ) -2α .
Arguing in the same way for the t j ∈ L 2 ([0, 1], ν j ) and, since F ⊥ G, we obtain

E d [ t -π F +G t 2 n ] C(K) K j=1 E d [ t j -π F +G t j 2 n ] C(K) K j=1 E d [ t j 2 n -π F t j 2 n -π G t j 2 n ] C(K) K j=1 E d [ t j -π F t j 2 n -t j -π E+F t j 2 n ] C(K) K j=1 E d [ t j -π F t j

Lemmas

This section is devoted to some technical results and their proofs.

Lemma 8.1. Let p, q > 0 be two real numbers such that 2q < p. For any θ > 0, the following inequality holds ∞ 0 qz q-1 (θ + z) p/2 dz C(p, q)θ q-p/2 where C(p, q) = p/(p -2q).

Proof. By splitting the integral around θ, we get ∞ 0 qz q-1 (θ + z) p/2 dz = θ 0 qz q-1 (θ + z) p/2 dz + ∞ θ qz q-1 (θ + z) p/2 dz θ -p/2 θ 0 qz q-1 dz + ∞ θ qz q-1-p/2 dz 1 + 2q p -2q θ q-p/2 .

The Proof. It is known that Aε is a centered Gaussian vector of R n of covariance matrix given by the positive symmetric matrix A t A. Let us denote by a 1 , . . . , a n 0 the eigenvalues of the A t A. Thus, the distribution of n Aε 2 n is the same as the one of n i=1 a i ε 2 i . We have and we can apply the Lemma 1 of [START_REF] Laurent | Adaptive estimation of a quadratic functional by model selection[END_REF] to obtain the announced inequalities.

We now introduce some properties that are satisfied by the estimator σ2 defined in [START_REF] Laurent | Adaptive estimation of a quadratic functional by model selection[END_REF].

Lemma 8.3. In the Gaussian case or under moment condition, the estimator σ2 satisfies

E σ2 = σ 2 + n s -πs 2 n
Tr ( t P n (I n -π)P n ) .

Proof. We have the following decomposition Y -πY 2 n = s -πs 2 n + σ 2 (I n -π)P n ε 2 n + 2σ s -πs, P n ε n .

The components of ε are independent and centered with unit variance. Thus, taking the expectation on both side, we obtain E Y -πY 2 n = s -πs 2 n + σ 2 Tr( t P n (I n -π)P n ) n .

Obviously, this inequality can be extended to x ∈ R, P n π ′ P n ε

To conclude, we use ( 84) and ( 87) in (83).

Lemma 8.5. Consider the estimator σ2 defined under moment condition. For any 0 < δ < 1/3, there exists a sequence (κ δ,n ) n∈N of positive numbers that tends to a positive constant κ δ as Tr( t P n P n )/ρ 2 (P n ) tends to infinity, such that P σ2 (1 -3δ)σ 2 C(p, δ)κ δ,n τ p ρ (p-2)∨2 (P n )Tr( t P n P n ) -((p/2-1)∧1) .

Proof. We define the vector u ∈ V ⊥ and the projection matrix π ′ as we did in the proof of Lemma 8.4. The lowerbound (82) does not depend on the distribution of ε and gives P σ2 (1 -3δ)σ To conclude, we use (91), ( 92) and ( 93) in (89).

  ) = sin(iπy) and ψ 2i-1 (y) = ψ (j) 2i-1 (y) = cos(iπy) for any i ∈ {1, . . . , d ′ n } and j ∈ {1, . . . , K}. Thus, we have dim(S j n ) = D (j) n = 2d ′ n and D ′ n = 2Kd ′ n + 1. As previously, we define P n as the oblique projector onto E along F + (E + F ) ⊥ . The image set E = Im(P n ) is generated by the vectors

Figure 1 :

 1 Figure 1: Plot in (x, z) of the signal s (dashed line), the data Z (dots) and the projected data Y (plain line).

Figure 2 :

 2 Figure 2: Estimation of s (dashed) by s (plain) with F N , K = 6 and t j = f j , j ∈ {1, . . . , 6}, for s = f 1 (left, ρ(P n ) = 1.24) and for s = f 5 (right, ρ(P n ) = 1.25).

2 n+Figure 3 :

 23 Figure 3: Estimation of s (dashed) by s (plain) with F C , K = 6 and t j = f j , j ∈ {1, . . . , 6}, for s = f 1 (left, ρ(P n ) = 1.23) and for s = f 2 (right, ρ(P n ) = 1.27).

7. 1

 1 Proofs of Theorems 2.1 and 2.

2

 2

7. 1 . 1

 11 Proof of Theorem 2.1

  (s) for the estimation of s = f 2 with F C . Each pair of lines corresponds to a value of K with the known σ 2 case on the first line and unknown σ 2 case on the second one. 1.11 1.11 1.09 1.06 1.04 1.03 1.03 1.02 1.01 1.02 1.02 K = 2 1.12 1.08 1.08 1.06 1.04 1.03 1.02 1.01 1.01 1.01 1.01 K = 3 1.13 1.09 1.07 1.07 1.05 1.03 1.01 1.01 1.02 1.02 1.02 K = 4 1.08 1.08 1.06 1.05 1.04 1.02 1.02 1.01 1.01 1.01 1.01 K = 5 1.10 1.05 1.06 1.06 1.03 1.02 1.02 1.01 1.01 1.01 1.01 K = 6 1.08 1.07 1.06 1.05 1.03 1.02 1.01 1.01 1.01 1.01 1.01 K = 7 1.11 1.09 1.08 1.05 1.03 1.02 1.01 1.01 1.01 1.01 1.01 K = 8 1.09 1.06 1.08 1.05 1.04 1.02 1.01 1.01 1.01 1.01 1.01 K = 9 1.10 1.08 1.07 1.05 1.03 1.02 1.01 1.01 1.01 1.01 1.01 Table 5: Ratio r K (s, s0 ) for the estimation of s = f 1 with F N .

2

 2

7. 2 . 1

 21 Proof of Theorem 3.1

7. 3 . 1

 31 Proof of Corollary 2.1 Let us begin by applying Theorem 2.1 with constant weights L

1 | 1

 11 E x , ρ(Φ -I Dn ) = sup a∈R Dn , |a| 2 t a(Φ -I Dn )a| = sup a∈R Dn , |a| 2 Dn

2 nC 2 nC

 22 , R, K) dim(S m ) -2α .Consequently, for any m ∈ M, we obtainE ε,d s -s ′′ dim(S m ) -2α + dim(S m )Since α > ζ n , we can consider some model S m in F BM with dimension of order n 1/(2α+1) and derive thatE ε,d s -s ′′ 2n -2α/(2α+1) + 1 n C α n -2α/(2α+1) .

ρ(A) 2 2

 22 = max i=1,...,n |a i | and Tr(A t A) = n i=1 a i .Because the a i 's are nonnegative, Tr(A t A)

  first introduce general model selection theorems under hypotheses (H Gau ) and (H Mom ).

	Theorem 2.1. Assume that (H Gau ) holds and consider a collection of nonnegative numbers
	{L m , m ∈ M}. Let θ > 0, if the penalty function is such that
	pen

  4.08 1.52 1.22 1.20 1.27 1.35 1.45 1.56 1.64 1.70 1.79 3.44 1.58 1.36 1.26 1.30 1.37 1.45 1.55 1.64 1.72 1.81

	K = 2,	4.07 1.66 1.28 1.26 1.32 1.40 1.49 1.57 1.66 1.74 1.82
	ρ(P n ) = 1.23	2.29 1.69 1.36 1.32 1.36 1.44 1.53 1.60 1.65 1.73 1.82
	K = 3,	4.17 1.65 1.36 1.34 1.42 1.50 1.60 1.67 1.77 1.89 2.01
	ρ(P n ) = 1.25	2.24 1.70 1.41 1.41 1.48 1.55 1.61 1.71 1.80 1.92 2.01
	K = 4,	4.42 1.88 1.43 1.34 1.36 1.45 1.53 1.61 1.69 1.77 1.86
	ρ(P n ) = 1.26	3.80 1.75 1.51 1.42 1.44 1.50 1.56 1.66 1.75 1.84 1.93
	K = 5,	4.57 1.82 1.43 1.37 1.39 1.46 1.53 1.60 1.67 1.76 1.83
	ρ(P n ) = 1.26	2.33 1.77 1.51 1.43 1.44 1.50 1.54 1.64 1.74 1.82 1.89
	K = 6,	4.98 2.08 1.59 1.47 1.45 1.49 1.57 1.66 1.77 1.86 1.96
	ρ(P n ) = 1.27	2.57 1.91 1.62 1.52 1.54 1.57 1.65 1.73 1.84 1.93 2.02
	Table

Table 4 :

 4 Ratio r K

		5	1.0	1.5	2.0	2.5	3.0	3.5	4.0	4.5	5.0
	K = 1,	2.01 1.92 1.86 1.80 1.76 1.74 1.70 1.70 1.68 1.67 1.68
	ρ(P n ) = 1.28	2.03 1.93 1.87 1.81 1.77 1.72 1.68 1.65 1.65 1.66 1.67
	K = 2,	2.02 1.93 1.85 1.79 1.75 1.71 1.68 1.66 1.66 1.66 1.66
	ρ(P n ) = 1.22	1.95 1.88 1.82 1.78 1.75 1.71 1.68 1.67 1.65 1.64 1.64
	K = 3,	2.04 1.93 1.86 1.81 1.76 1.71 1.68 1.64 1.62 1.62 1.62
	ρ(P n ) = 1.26	1.96 1.87 1.80 1.74 1.68 1.66 1.63 1.63 1.61 1.62 1.62
	K = 4,	2.12 2.00 1.90 1.81 1.73 1.67 1.64 1.62 1.60 1.61 1.60
	ρ(P n ) = 1.25	1.99 1.90 1.80 1.73 1.68 1.65 1.62 1.60 1.60 1.60 1.60
	K = 5,	2.47 2.34 2.23 2.17 2.10 2.05 1.99 1.95 1.91 1.88 1.86
	ρ(P n ) = 1.24	2.30 2.20 2.11 2.03 1.97 1.92 1.88 1.83 1.82 1.80 1.80
	K = 6,	2.45 2.32 2.21 2.11 2.03 1.99 1.95 1.91 1.89 1.86 1.84
	ρ(P n ) = 1.26	2.17 2.06 1.99 1.94 1.89 1.85 1.84 1.80 1.79 1.79 1.75

Table 6 :

 6 K = 1 1.08 1.09 1.07 1.07 1.09 1.09 1.08 1.07 1.06 1.09 1.07 K = 2 1.09 1.05 1.08 1.09 1.09 1.08 1.08 1.08 1.06 1.06 1.05 K = 3 1.12 1.12 1.11 1.07 1.09 1.10 1.09 1.08 1.07 1.06 1.07 K = 4 1.09 1.11 1.08 1.10 1.10 1.09 1.07 1.06 1.07 1.06 1.07 K = 5 1.10 1.08 1.09 1.09 1.09 1.06 1.06 1.06 1.07 1.07 1.05 K = 6 1.08 1.04 1.06 1.07 1.08 1.07 1.07 1.09 1.06 1.06 1.06 K = 7 1.06 1.05 1.07 1.08 1.10 1.09 1.07 1.09 1.08 1.07 1.06 K = 8 1.08 1.13 1.08 1.09 1.09 1.08 1.06 1.07 1.07 1.06 1.06 K = 9 1.13 1.05 1.09 1.09 1.07 1.07 1.07 1.06 1.07 1.07 1.06 Ratio r K (s, s0 ) for the estimation of s

	5	2.0	2.5	3.0	3.5	4.0	4.5	5.0

  1 n Tr( t P n P n ) (Tr( t P n P n ) + 2ρ 2 (P n )) Tr( t P n P n ) + (Tr( t P n P n ) + 2ρ 2 (P n )) 2n Tr( t P n

  2 sup a∈R Dn , b∈R δn |a| 2 +|b| 2 1

			t aΩb
			Dn		δn
	2	sup			a i b j e (i) , f (j)	n
		a∈R Dn , b∈R δn |a| 2 1, |b| 2 1	i=1	j=1
			Dn		δn
	2	sup
		a∈R Dn , b∈R δn |a| 2 1, |b| 2 1	i=1	j=1

  next lemma is a variant of a lemma due to Laurent and Massart.Lemma 8.2. Let A ∈ M n \ {0} and ε = (ε 1 , . . . , ε n ) ′ be a standard Gaussian vector of R n .For any x > 0, we haveP n Aε 2 n Tr(A t A) + 2 ρ(A) 2 Tr(A t A)x + 2ρ(A) 2 x

	e -x	(79)
	and
	P n Aε 2

n Tr(A t A) -2 ρ(A) 2 Tr(A t A)x e -x .

(80)

  Tr( t P n P n ) 8(δ + 1)ρ 2 (P n ) ∧ 1 .

	2 n	(1 + δ/2)Tr( t P n πP n ) + (1 + δ/2)ρ 2 (P n ) + (2 + 2/δ)x	exp -	x ∨ 0 ρ 2 (P n )
									(86)
	and we take
	x =	δ 2(δ + 1)	δTr( t P n (I n -π)P n ) -	δ 2	Tr( t P n πP n ) -1 +	δ 2	ρ 2 (P n )
	=	δ 2(δ + 1)	δTr( t P n P n ) -	3δ 2	Tr( t P n πP n ) -1 +	δ 2	ρ 2 (P n )
		δ 2(δ + 1)	δTr( t P n P n ) 4	-1 +	δ 2	ρ 2 (P n ) .
	Finally, we get
	P 2		exp -	δ 2(δ + 1)ρ 2 (P n )	δTr( t P n P n ) 4	-1 +	δ 2	ρ 2 (P n )	+
			exp -	δ(δ + 2) 4(δ + 1)	δTr( t P n P n ) 2(δ + 2)ρ 2 (P n )	-1	+
		=	exp		δ(δ + 2) 4(δ + 1)	× exp -	δ 2

C

  Since the matrix t P n P n is symmetric, we have the following decompositionn P n ε 2 n -Tr( t P n P n ) = n t P n P n ε, ε n -Tr( t P n P n ) P n P n ) ij ε i ε j -Tr( t P n P n ) P n P n ) ij ε i ε j . P n P n ) ii (ε 2 i -1) -δTr( t P n (I n -π)P n ) , P n P n ) ij ε i ε j -δTr( t P n (I n -π)P n ) P n π ′ P n ε 2 n -Tr( t P n πP n ) δTr( t P n (I n -π)P n )) .Note that P1 concerns a sum of independent centered random variables. By Markov's inequality and (21), we getP n P n ) ii (ε 2 i -1) δTr( t P n (I n -π)P n ) δ -p/2Tr( t P n (I n -π)P n ) -p/2 E Since, for any i ∈ {1, . . . , n}, ( t P n P n ) ii ρ 2 (P n ), by a convexity argument, we get 2C ′ (p)τ p ρ p/2 (P n )Tr( t P n P n ) p/4 . If 2 < p < 4, we refer to [39] for the following inequality C ′′ (p)τ p ρ p-2 (P n )Tr( t P n P n ) . -p/2 τ p ρ p/2 (P n )Tr( t P n P n ) -β Tr( t P n P n )ρ p-2 (P n ) (δTr( t P n P n )/4 -(1 + δ/2) ρ 2 (P n )) ′′ (p, δ)τ p Tr( t P n P n ) 1-p/2 ρ p-2 (P n ) (1 -2 (1 + 2/δ) ρ 2 (P n )/Tr( t P n P n ))

	where we have set then we obtain
				P1 = P P3 ( P2 = P n i=1 C ′ (p, δ)τ p   2 n	p/2 + (  ∧ 1  p/2 + ∧ 1	(93)
							i=1 j>i
	and			P3 = P1	P	n i=1	( 	n
														i=1	( p/2	
													 .	(90)
	If p 4 then we use the Rosenthal Inequality (see Chapter 2 of [31]) and (57) to obtain
	E	 	n	( p/2	  C ′ (p)τ p	 	n	( t P n P n ) p/2 ii +	n	( t P n P n ) 2 ii	p/4	  .
			i=1									i=1	i=1
					n
	(88) P1 C(p)δ (91) = n i=1 n j=1 n n p/2   2 n i=1 ( t P n P n ) ii (ε 2 p/2 i -1) In both cases, (90) becomes E  i=1 ( E   n i=1 ( t P n P n ) ii (ε 2 i -1) ( = i=1 ( t P n P n ) ii (ε 2 i -1) + 2 i=1 j>i ( Thus, (88) leads to P σ2 (1 -3δ)σ 2 (89) P1 + P2 + P3 with β = (p/2 -1) ∧ p/4.

2 P n P n ε 2 n -n π ′ P n ε 2 n (1 -3δ)Tr( t P n (I n -π)P n ) . t t t t t t P n P n ) ii (ε 2 i -1) p/2   2 p/2 δ -p/2 Tr( t P n P n ) -p/2 E   n i=1 ( t P n P n ) ii (ε 2 i -1) t P n P n ) ii (ε 2 i -1) t P n P n ) ii (ε 2 i -1) p/2  

n /n. For any x > 0, the standard Gaussian deviation

 Lemma 8.4. Consider the estimator σ2 defined in the Gaussian case. For any 0 < δ < 1/2, P σ2 (1 -2δ)σ 2 C δ exp -δ 2 Tr( t P n P n ) 16ρ 2 (P n )

where C δ > 1 only depends on δ.

Proof. Let a ∈ V ⊥ such that a 2 n = 1, we set u = (s -πs)/ s -πs n if s = πs , a otherwise .

We have

and we deduce from (81)

where π ′ is the orthogonal projection onto V ⊕ Ru. Consequently,

The Inequality (80) and [START_REF] Laurent | Testing inverse problems: a direct or an indirect problem[END_REF] give us the following upperbound for P 1 ,

By the properties of the norm ρ, we deduce that

where we have defined π u as the orthogonal projection onto Ru. We now apply (79) with A = π ′ P n to obtain, for any x > 0,

Let us now bound P2 . By Chebyshev's inequality, we get

Note that, by independence between the components of ε, the expectation in the last sum is not null if and only if i = p and j = q (in this case, its value is 1). Thus, we have

We finally focus on P3 . Recalling (85), we apply Corollary 5.1 of [START_REF] Baraud | Model selection for regression on a fixed design[END_REF] with à = t P n π ′ P n to obtain, for any x > 0,

Thus, for any x ∈ R, we define

and ψ(x) is an upperbound for