F Comte
email: comte@ccr.jussieu.fr

Y Rozenholc

Adaptive estimation of mean and volatility functions in (auto-)regressive models

Keywords: Nonparametric regression, Least-squares estimator, Adaptive estimation, Autoregression, Variance estimation, Mixing processes. 1991 Mathematical Subject Classification. Primary 62G07 Secondary 62J02

In this paper, we study the problem of non parametric estimation of the mean and variance functions b and σ 2 in a model: X i+1 = b(X i ) + σ(X i )ε i+1 . For this purpose, we consider a collection of finite dimensional linear spaces . We estimate b using a mean squares estimator built on a data driven selected linear space among the collection. Then an analogous procedure estimates σ 2 , using a possibly different collection of models. Both data driven choices are performed via the minimization of penalized mean squares contrasts. The penalty functions are random in order not to depend on unknown variancetype quantities. In all cases, we state non asymptotic risk bounds in IL 2 empirical norm for our estimators and we show that they are both adaptive in the minimax sense over a large class of Besov balls. Lastly, we give the results of intensive simulation experiments which show the good performances of our estimator.

Introduction 1.Presentation of the problem

In this paper, we study the following model:

X i+1 = b(X i ) + σ(X i )ε i+1 ,
(1.1) with ε i i.i.d. centered random variables with unit variance. It can be considered as a particular case of the standard regression model:

Y i = b(X i ) + σ(X i )u i , (1.2)
with i.i.d. centered u i 's, Var(u 1 ) = 1, where the (X i , Y i )'s are not assumed to be independent but can be β-mixing. Our results hold for this model.

If f is an estimator of f , where f is b or σ 2 , then we measure the risk of f via the IL 2 -empirical norm:

IE[ f -f 2 n ] = IE 1 n n i=1 ( f (X i ) -f (X i )) 2 .
For a discussion about the choice of this measure of risk, see [START_REF] Baraud | Adaptive estimation in an autoregression or a β-mixing regression[END_REF]. Roughly speaking, the reason for this choice is as follows: let fS be a minimizer of

γ n (t) = 1 n n i=1 [X i+1 -t(X i )] 2 ,
for t in a linear space S ⊂ IL 2 (IR, dx); then the vector ( fS (X 1 ), . . . , fS (X n )) is uniquely defined, but of course not the global function fS at any point. We shall nevertheless talk about "the" mean squares estimator since only the associated vector ( fS (X 1 ), . . . , fS (X n )) of IR n is involved in the computations.

In addition, under suitable assumptions, this risk can be decomposed into bias + variance terms via:

IE[ f -fS 2 n ] ≤ κ f -f S 2 µ + dim(S) n , (1.3)
where f S is the IL 2 (dx)-orthogonal projection of f on S, t 2 µ = IE(t 2 (X 1 )) and κ depends on constants of the problem. To see how (1.3) is obtained, consider a strictly stationary sequence (X i ) drawn from model (1.1) with σ ≡ 1 and stationary [0, 1]-supported density, and let S be generated by ϕ 1 , . . . , ϕ D , the histogram orthonormal basis of IL 2 ([0, 1]): ϕ j (x) = √ D1 I [(j-1)/D,j/D[ (x). Simple algebra leads to

γ n (t) -γ n (s) = b -t 2 n -b -s 2 n + 2 s -t, ε n (1.4)
where t, ε n = (1/n) n i=1 t(X i )ε i+1 . Then we find from (1.4

) bS -b 2 n ≤ b S -b 2 n + 2 b S -bS , ε n ≤ b S -b 2 n + 2 b S -bS sup t∈S, t =1 | t, ε n | ≤ b S -b 2 n + 1 4a b S -bS 2 + 4a sup t∈S, t =1 t, ε 2 n .
Assume that, for some a > 1,

∀t ∈ S, t 2 ≤ a t 2 n , (1.5) then IE( bS -b 2 n ) ≤ 3 b S -b 2 µ + 8aIE sup t∈S, t =1
t, ε 2 n .

(1.6)

Besides, using Cauchy Schwarz inequality yields

IE sup t∈S, t =1 t, ε 2 n = IE   sup P j a 2 j ≤1   D j=1 a j ϕ j , ε n   2   ≤ D j=1 IE ϕ j , ε 2 n 2 = 1 n 2 D j=1 IE n i=1 ϕ j (X i )ε i+1 2 = 1 n 2 D j=1 n i=1 IE(ϕ 2 j (X i ))IE(ε 2 i+1 ) = 1 n IE   D j=1 ϕ 2 j (X 1 )   = D n .
(1.7)

Therefore (1.6) and (1.7) lead to (1.3), provided that (1.5) holds, which is generally true with large probability.

In view of these considerations, here is now our estimation procedure. We start with two finite collections of models denoted by {S

(i) m , m ∈ M (i) n } for b if i = 1 and σ 2 if i = 2; each S (i)
m is a finite dimensional subspace of IL 2 (IR, dx). The functions b and σ 2 are not required to belong to any of the models. Let bm denote the least squares estimator of b on S

(1) m associated to

γ (1) n (t) = 1 n n i=1 (X i+1 -t(X i )) 2
based on the observations X 1 , . . . , X n+1 arising from model (1.1). We use a procedure that chooses m1 in M

(1)

n as the minimizer γ

(1) n ( bm ) + pen (1) (m) among all m in M

n , where pen (1) is a known penalty function specified later. The key point is that this procedure is entirely based on the data and not on any prior information on b, and that it realizes a good trade-off between the bias and variance terms, namely: m and C is a multiplicative constant depending on some quantities of the problem. This means that, up to the constant C, the estimator chooses an optimal model among the collection. In the second step, σ2 is based in an analogous way on the contrast:

γ (2) n (t) = 1 n n i=1 X 2 i+1 -b2 m1 (X i ) -t(X i ) 2 , for t ∈ S (2)
m , with an aim similar to (1.8) and b replaced by σ 2 , using a penalty function pen (2) (m). Both penalty functions pen (i) (m), i = 1, 2 are found of order dim(S (i) m )/n. This model selection criterion is closely related to the classical C p criterion of [START_REF] Mallows | Some comments on C p[END_REF]. It is important to notice that estimators satisfying inequalities as (1.8) have interesting properties on the collections of models that we have in mind (piecewise polynomials, wavelets, trigonometric polynomials). In particular, such estimators are adaptive in the minimax sense with respect to many well known classes of smoothness (see [START_REF] Barron | Risks bounds for model selection via penalization[END_REF] and [START_REF] Birgé | From model selection to adaptive estimation[END_REF]).

Some bibliographic remarks

The autoregressive model has been extensively studied in the literature in view of applications to Finance and Econometrics in particular. People first modeled the conditional mean of the variable of interest X t given its past as a linear function of past X t 's, the conditional variance being constant, see [START_REF] Lütkepohl | Introduction to Multiple Time Series Analysis[END_REF] and the ARMA (Autoregressive Moving Average) models of the time series literature. Then many financial variables were experimented to have non constant conditional variance, and specifications of it as a linear function of the squared values of the past innovations were developed with ARCH (Autoregressive Conditionally Heteroskedastic) models introduced by [START_REF] Engle | Autoregressive conditional heteroskedasticity with estimates of the variance of United Kingdom inflation[END_REF] and generalized by [START_REF] Bollerslev | Generalized autoregressive conditional heteroskedasticity[END_REF]. Lastly, nonlinear extensions of both types of functions (conditional mean and conditional variance) were studied: step functions in [START_REF] Gouriéroux | Qualitative threshold ARCH models[END_REF], general non linear functions in Mc [START_REF] Mc Keague | Identification of nonlinear time series from first order cumulative characteristics[END_REF] or [START_REF] Härdle | Local polynomial estimators of the volatility function in nonparametric regression[END_REF]. This is the reason why statistical methods for nonparametric estimation of variance functions were recently developed. On the other hand, adaptive estimation methods have been studied in some frameworks that can be related to the present one. In particular, several studies related to penalization criteria as Akaike's or BIC criterion for regressive models, by [START_REF] Akaike | Information theory and an extension of the maximum likelihood principle[END_REF], [START_REF] Shibata | Selection of the order of an autoregressive model by Akaike's information criterion[END_REF], [START_REF] Li | Asymptotic optimality for C p , C l cross-validation and generalized crossvalidation: discrete index set[END_REF], [START_REF] Polyak | A family of asymptotically optimal methods for choosing the order of a projective regression estimate[END_REF], have lead to asymptotic results. More recently, a general approach to model selection has been developed by [START_REF] Birgé | From model selection to adaptive estimation[END_REF] and [START_REF] Barron | Risks bounds for model selection via penalization[END_REF] with many applications to adaptive estimation. Their viewpoint is non asymptotic, and so is ours. The procedure we use has been studied for fixed design regressive models by [START_REF] Baraud | Model selection for regression on a fixed design[END_REF] and for β-mixing random design and autoregressive models by [START_REF] Baraud | Adaptive estimation in an autoregression or a β-mixing regression[END_REF]; the variance function is constant in all of these works and thus only the mean function is estimated. Our results here are an extension of the latter to the estimation of the mean when the variance function is not constant, and to the estimation of the variance function as well. Variance estimation has been first studied in fixed design regression models, see for instance [START_REF] Müller | Estimation of heteroscedasticity in regression analysis[END_REF] who apply to this problem a difference-based estimator. [START_REF] Hall | Variance function estimation in regression: the effect estimation of the mean[END_REF] build a residual-based estimator and show that they pointwise reach the optimal rate of convergence even with an unknown mean function b, provided that b has a smoothness order larger than 1/2. Dependent models (autoregressive models or regressive models with mixing random design) have been handled by [START_REF] Härdle | Local polynomial estimators of the volatility function in nonparametric regression[END_REF], [START_REF] Härdle | Nonparametric vector autoregression[END_REF] and [START_REF] Fan | Efficient estimation of conditional variance functions in stochastic regression[END_REF]. [START_REF] Härdle | Local polynomial estimators of the volatility function in nonparametric regression[END_REF] study the estimation of b and σ 2 using local polynomial estimators; they prove pointwise asymptotic normality with standard rates but their procedure is not adaptive. [START_REF] Fan | Efficient estimation of conditional variance functions in stochastic regression[END_REF] describe a data driven procedure with automatic bandwidth selection but their theoretical results provide only a pointwise Central Limit Theorem for a non adaptive estimator. Lastly, adaptive procedures for variance estimation have been studied by [START_REF] Neumann | Fully data-driven nonparametric variance estimators[END_REF] and [START_REF] Hoffmann | On nonparametric estimation in nonlinear AR(1)-models[END_REF]. [START_REF] Neumann | Fully data-driven nonparametric variance estimators[END_REF] builds an adaptive kernel (with random bandwidth) residual-based estimator, but in a fixed design model with a noise admitting moments of any order. He proves optimal rates for the mean integrated squared error of his estimator, provided that the mean function has a smoothness order α1. The framework the most related to the present work is Hoffmann (1999)'s who proposes an adaptive wavelet thresholding procedure in an autoregressive framework. He requires that the noise admits moments of any order and obtains for the general IL p -integrated risk the optimal rates up to some logarithmic factors. The rates for b and σ 2 do not depend on each other, but he assumes that both orders of smoothness are larger than 3/2. To enhance the comparison, let us say that our procedure is adaptive, deals with random and dependent regression variables including the autoregressive framework, requires for the noise the finiteness of moments of a given order, 16 in many cases (and not any order p) and reaches the optimal rate (without any loss) provided that the mean function is smoother than the variance function (namely, α ≥ 2β + 1/2 if α and β are the smoothness orders of b and σ 2 respectively). On the one hand, this condition is less attractive than Neumann's (α > 1) in his independent framework or Hoffmann's and is only a technical loss with no other structural reason than the use of a unique first step estimator of b to estimate σ 2 . Note that, contrary to Hoffmann's result, it allows to reach low orders of smoothness for for b (α > 1/2) and for σ 2 (namely 1/2 < β < 3/2). On the other hand, to separate the variance of the noise from the mean function, it is empirically natural to ask that the latter is much smoother than the former, otherwise it is hard to distinguish between them.

The plan of the paper is as follows. Section 2 presents the whole estimation procedure, namely the building of both estimators of b and σ 2 and the assumptions on the functions, the variables and the collections of models. The results in terms of inequality of type (1.8) and of minimax rates on Besov balls are given in Section 3. Section 4 explains our simulation methods and describes the results of intensive simulation experiments. We used in particular models recently studied by [START_REF] Härdle | Local polynomial estimators of the volatility function in nonparametric regression[END_REF] and [START_REF] Fan | Efficient estimation of conditional variance functions in stochastic regression[END_REF] but also many others. Lastly, almost all proofs are gathered in Section 5 while section 6 contains some complementary informations about the simulations.

2 The estimation procedure

Assumptions on the linear spaces of estimation

We assume that we aim to estimate the functions on a given compact set A. We consider families of linear subspaces S m of IL 2 (A, dx) and we call those families collections of models. It is standard to set the following assumptions on the collections (S

(i) m ) m∈M (i) n , i = 1, 2: (H Φ i ) 1. Each S (i) m is a finite dimensional linear subspace of IL 2 (A, dx) with dimension dim(S (i) m ) = D (i)
m and maximal dimension denoted by

D (i) n .
2. There exists a constant Φ i such that for any pair (m, m ′ ) ∈ (M

(i) n ) 2 , and any t ∈ S (i) m + S (i) m ′ t ∞ ≤ Φ i dim(S (i) m + S (i) m ′ ) t (2.9)
where t = t 2 (x)dx = A t 2 (x)dx.

3. There exists a constant K such that

D (i) n ≤ K √ n/ ln(n) in the general case, D (i)
n ≤ Kn/ ln 2 (n) for wavelets (family (W) below) and for piecewise polynomials (families (DP) and (RP) below).

(H (a i ,b i ) ) There exist some nonnegative constants a i , b i , Σ i , T i such that

m∈M (i) n (D (i) m ) -a i ≤ Σ i < ∞ and |M (i) n | ≤ T i n b i . Comments. 1. Assumption (H Φ i ) 2
. is an assumption of connection between the two norms . ∞ and . . It implies in particular that for all t ∈ S

(i) m , t ∞ ≤ Φ i D (i) m t .
It follows from [START_REF] Barron | Risks bounds for model selection via penalization[END_REF], equations (3.2) and (3.3), that, for any orthonormal basis (ϕ λ ) λ∈Λ of S

(i) m + S (i) m ′ : λ∈Λ ϕ 2 λ ∞ = sup t∈S (i) m +S (i) m ′ ,t =0 t ∞ t .
(2.10)

2. Assumption (H (a i ,b i )
) is a limitation on the number of models which have the same dimension and consequently on the global number of models. It garantees in particular that we do not consider too many models. Note also that the choice b i = a i , T i = Σ i suits. Indeed, since

D (i) m ≤ n, for any m ∈ M (i) n , Σ i ≥ m∈M (i) n (D (i) m ) -a i ≥ m∈M (i) n n -a i = |M (i) n |n -a i
which implies that:

|M (i) n | ≤ Σ i n a i .
In other words the number of models is at most polynomial with respect to n.

We shall essentially consider in the sequel three kinds of specific families of models (S

(i) m ) m∈M (i) n satisfying (H Φ i ) and (H (a i ,b i ) )
: trigonometric polynomials, wavelets and piecewise polynomials that can be described as follows. n is in that case the set of all possible dimensions such that (H Φ i ) 2. holds:

M (i) n = {1, . . . , √ n}. Here Φ i = √ 2 in (H Φ i ) 2., a i = 1 + ǫ, ∀ǫ > 0, and b i = 1/2 in (H a i ,b i ).
(RP) Regular piecewise polynomials: we consider the regular partitions I m defined by I m = {[j/m, (j + 1)/m), j = 0, 1, . . . , m -1}. Given some positive integer r, we define

S (i)
m to be the space of piecewise polynomials with degree bounded by r -1 on the partition I m . Then D

(i) m = rm. The maximal value of m, m(n) is the greatest integer such that rm ≤ n/ ln 2 (n), i.e. m(n) = [n/(r ln 2 (n))] = |M (i) n | where [z]
denotes the integer part of z. Here Φ i = (r + 2)(2r + 1) (see [START_REF] Barron | Risks bounds for model selection via penalization[END_REF] p. 323), a i = 1 + ǫ, ∀ǫ > 0 and b i = 1 suit.

(DP) Dyadic piecewise polynomials: we consider now dyadic partitions I m = {[j/2 m , (j + 1)/2 m ), j = 0, . . . 2 m -1}. Given some positive integer r, we define S (i)

m to be the space of piecewise polynomials with degree bounded by r -1 on the partition Barron et al. (1999) p. 323), but now any positive a i , b i suit.

I m . Then D (i) m = r2 m . The maximal value of m, m(n) is the greatest integer such that r2 m ≤ n/ ln 2 (n), i.e. m(n) = [ln(n/(r ln 2 (n)))/ ln(2)] = |M (i) n |. Again Φ i = (r + 2)(2r + 1) (see
(W) Compactly supported wavelets: Let Λ(j) = {(j, k), k = 1, . . . , 2 j } and let

{φ J 0 ,k , (J 0 , k) ∈ Λ(J 0 )} ∪ {ϕ j,k , (j, k) ∈ +∞ J=J 0 Λ(J)}
be an IL 2 ([0, 1], dx)-orthonormal system of compactly supported wavelets of regularity r built by [START_REF] Cohen | Wavelet and fast wavelet transform on an interval[END_REF]; for a precise description, see Donoho and Jonstone (1998). These new functions derive from [START_REF] Daubechies | Ten lectures on wavelets[END_REF]'s wavelets at the interior of [0, 1] and are boundary corrected at the "edges". For any J n > J 0 , let S n be the space spanned by the φ J 0 ,k 's for (J 0 , k) ∈ Λ(J 0 ) and by the ϕ j,k 's for

(j, k) ∈ ∪ Jn J=J 0 Λ(J). It follows that dim(S n ) = 2 Jn ≤ n if J n ≤ ln 2 (n). For any m ∈ M n = {J 0 , . . . , J n -1}, we take for S (i) m the linear span of the φ J 0 ,k 's for (J 0 , k) ∈ Λ(J 0 ) and of the ϕ j,k 's for (j, k) ∈ ∪ m J=J 0 Λ(J). This implies that D (i)
m ≤ 2 m . We know from [START_REF] Barron | Risks bounds for model selection via penalization[END_REF], p.322, that Φ i = 2 + √ 2 suits and any positive a i , b i suit.

The assumptions on the model

All along the paper, we consider model (1.1) with ε i i.i.d., IE(ε 1 ) = IE(ε 3 1 ) = 0 and Var(ε 1 ) = 1. We assume that the process (X i ) is strictly stationary. Let us recall that a stationary process (X i ) is said to be absolutely regular or β-mixing [START_REF] Kolmogorov | On the strong mixing conditions for stationary gaussian sequences[END_REF] if

1 2 sup    I i=1 J j=1 |IP(A i ∩ B j ) -IP(A i )IP(B j )|    = β k → 0 when k → +∞.
where the supremum is taken over all finite partitions (A i ) 1≤i≤I and (B j ) 1≤j≤J of the probability space Ω respectively F 0 -∞ and F ∞ k measurable where F k i is the σ-algebra generated by {X j , i ≤ j ≤ k}. The mixing is said to be geometrical if there exist positive M and θ such that β k ≤ M e -θk . The mixing is said to be arithmetical if there exist positive M and θ such that β k ≤ M k -θ . We work under the following assumptions: A1 (X t ) t∈Z is geometrically β-mixing.

A2(p) X, b(X), σ(X) and ε admit moments until order p, p ≥ 4. A3 b and σ are bounded on compact sets. A4 X admits a density h X such that for any compact set A in the support of h X , there exist h 0 , h 1 (depending on A) such that

∀x ∈ A, 0 < h 0 ≤ h X (x) ≤ h 1 . (2.11) Under A2(p), we denote by m 4 = IE[(ε 2 -1) 2 ](< ∞)
and by σ q q = IE(|ε 1 | q ) for q ∈ (0, p]. Note that assumptions A1, A2(p), A3 are fulfilled under standard assumptions given by Ango Nze (1992), Proposition 3, (see also Doukhan (1994) p. 107). More precisely, here is a set of assumptions implying A1-A4: B1 There exists constants C 1 > 0 and C 2 > 0 such that, for all y ∈ IR,

|b(y)| ≤ C 1 (1 + |y|), |σ(y)| ≤ C 2 (1 + |y|). B2 The function σ satisfies inf y∈IR σ(y) > C 3 for a C 3 > 0. B3(p) IE(|ε 1 | p ) < +∞ for some p ≥ 4 and IE[C 1 + C 2 |ε 1 |] p < 1.
B4 The density h ε of ε exists and h ε is continuous on its support.

Those assumptions are quite near of those required by [START_REF] Härdle | Local polynomial estimators of the volatility function in nonparametric regression[END_REF]. Under B1-B4, the Markov chain (X i ) given by (1.1) is geometrically ergodic and the stationary law is geometrically absolutely regular; this ensures A1. Under B3(p), we know (see [START_REF] Duflo | Méthodes récursives aléatoires[END_REF], p.178) that for any initial condition X 0 in IL p independent of ε, the X i 's admit moments of the same order as the ε i 's (and thus, so do σ(X i ) and b(X i ) with B1). Thus B3(p) ensures A2(p). As a consequence of B1, it is clear that b and σ are bounded on compact sets, which gives A3. Note that we estimate b and σ on the compact set A only, the same for both functions. Moreover, if µ denotes the stationary law of X 1 (which exists under B1-B4), we know with B2 and B4 that dµ(x) = h X (x)dx with:

h X (x) = h ε x -b(u) σ(u) 1 σ(u) dµ(u).
Indeed the positivity of σ ensures that the change of variable can be done and the continuity of h ε implies the continuity of h X . Thus h X is positive on its support and continuous which ensures A4 for any compact set A in the support of h X .

In other words B1-B2-B3(p)-B4 imply A1-A2(p)-A3-A4.

Since the random variables X i are geometrically β-mixing, this will allow to apply some results established in [START_REF] Baraud | Adaptive estimation in an autoregression or a β-mixing regression[END_REF].

Comments: 1. Ango Nze (1998) gives also conditions on autoregressive models to generate arithmetically mixing variables still admitting a stationary ergodic law. Moreover the results of [START_REF] Baraud | Adaptive estimation in an autoregression or a β-mixing regression[END_REF] also allow to consider arithmetically mixing variables. This implies some robustness of the results with respect to stronger types of dependence. But such results lead to much stronger conditions on the errors and on the size of the collections of models. 2. All the given results would hold for model (1.2) with u i i.i.d., IE(u 1 ) = IE(u 3 1 ) = 0 and var(u 1 ) = 1, (see for such extensions [START_REF] Baraud | Adaptive estimation in an autoregression or a β-mixing regression[END_REF]) under the assumptions A3, A4 and C1 (Y t , X t ) t∈Z is geometrically β-mixing.

C2(p) Y , b(X), σ(X) and ε admit moments until order p, p ≥ 4.

3.

Lastly the real valued random variables X i could be replaced by a r-dimensional random vector

X i = (X (1) i , . . . , X (r)
i ) under the same kind of assumptions and the autoregression of order one can in the same way be generalized into an autoregression of order r. For the extension of assumptions B1-B4 ensuring A1-A4, see Ango [START_REF] Ango Nze | Critères d'ergodicité de quelques modèles à représentation markovienne[END_REF] or the application of these results in [START_REF] Härdle | Nonparametric vector autoregression[END_REF]. The functions b and σ remain real valued and the errors ε i as well, which makes most extensions straightforward.

First step of the estimation procedure

To estimate b on a given compact set A, we consider the contrast

γ (1) n (t) = 1 n n i=1 [X i+1 -t(X i )] 2 (2.12)
based on the observations X 1 , X 2 , . . . , X n+1 . We consider a collection of linear subspaces of IL 2 (A, dx), (S

(1)

m ) m∈M (1) n of dimension D (1)
m , as described in section 2.1 and satisfying Assumption (H Φ 1 ). [START_REF] Baraud | Adaptive estimation in an autoregression or a β-mixing regression[END_REF] proved non asymptotic risk bounds for the estimate b m1 defined as follows, when the variance is a known constant denoted by σ 2 . Let bm be a minimizer of γ th (m) where pen

(1)

th (m) = κΦ 2 1 σ 2 µ D (1) m n ,
where µ is the stationary law of the X i 's and κ a universal constant. Then b m1 has the same properties as in the case of a known constant variance. As σ 2 µ is unknown, we complete the procedure by replacing this quantity by an estimate. Let

r2 n = 1 n n i=1 (X i+1 -bmn (X i )) 2
(2.13)

where S

(1)

mn is a space of the family for a given m n ∈ M

(1)

n with dimension D (1)
mn to be chosen in Theorem 3.1 (see also the comments herewith). Then we define the final estimate as b := b m1 where m1 = arg min

m∈M (1) n γ (1) n ( bm ) + pen (1) (m) (2.14) with pen (1) (m) = κΦ 2 1 r2 n D
(1) m n and r2 n given by (2.13).

(2.15)

Comment: It is now well-known that it is safer to take for κ too great than too small values. An empirical calibration study, similar to the one extensively done for density estimation by [START_REF] Birgé | How many bins must be put in a regular histogram[END_REF], can be lead in order to compute κ. When the collection of models is chosen, Φ 1 is known but it is probably a computational artefact rather than a structural constant of the penalty. Indeed, in an independent fixed design framework with constant volatility σ 2 2 , the optimal penalty is found by [START_REF] Baraud | Model selection for regression on a fixed design[END_REF] to be 2σ 2 2 D

(1) m /n.

Second step of the estimation procedure

We consider now the following procedure. Let S

(2)

m , m ∈ M (2)
n , be a collection of linear subspaces of IL 2 (A, dx), of dimension D

(2) m , as described in section 2.1 and satisfying assumption (H Φ 2 ). Let

γ (2) n (t) = 1 n n i=1 X 2 i+1 -b2 (X i ) -t(X i ) 2 (2.16)
and define σ 2 m as a minimizer of γ

(2)

n (t) over t ∈ S (2)
m . Then our estimate is

σ 2 = σ 2 m2 with m2 = arg min m∈M (2) n γ (2) n ( σ 2 m ) + pen (2) (m) (2.17)
where

pen (2) (m) = κΦ 2 2 ŝ2 n D (2) m n .
(2.18) and

ŝ2 n = 1 n n i=1 (X 2 i+1 -ĝmn (X i )) 2 and ĝmn = arg min t∈S (2) mn 1 n n i=1 X 2 i+1 -t(X i ) 2 (2.19)
on some well chosen S

(2)

m = S (2)
mn . The theoretical value of the penalty that pen (2) estimates is:

pen (2) th = κΦ 2 2 (m 4 IE µ (σ 4 ) + 4IE µ (b 2 σ 2 )) D (2) m n . Comment. The choice (1/n) n i=1 [(X i+1 -b(X i )) 2 -t(X i )]
2 for the contrast is more standard and is the one empirically used. Only technical reasons lead to our slightly different choice.

3 The theoretical results

Estimation of the mean

Recall that the empirical euclidian norm is u 2 n = (1/n) n i=1 u 2 (X i ) and that A is the given compact set on which we aim to estimate the functions. We denote by b m the IL 2 (A, dx)-orthogonal projection of b on S m . We have the following result:

Theorem 3.1 Let X 1 , . . . ,
) = D (1) mn ≤ n 1/2-2/p , (3.21) satisfies IE[ b1 I A -b 2 n ] ≤ C inf m∈M (1) n b1 I A -b m 2 µ + D (1) m n ( b 2 µ + σ 2 µ ) + R n where C is a universal constant and R is a constant depending on σ p , Φ 1 , b1 I A ∞ , σ1 I A ∞ , Σ 1 , C 1 .
Comments: 1. The estimate performs almost as well as the best estimator that could be chosen among the collection. We insist on the fact that the procedure automatically selects a model very close to the (unobservable) best model (called oracle) in the collection i.e. the most adequate dimension for the space of approximation. 2. For the families (W) and (DP), any a 1 , b 1 > 0 suit so that condition (3.20) reduces to p ≥ 8. For the family (T), as a 1 = 1 + ε for any ε > 0 and b 1 = 1/2, condition (3.20) becomes p > 8. For family (RP), if we want to consider the maximal number of possible models, we find p ≥ 10.

3. In the general case, under condition (3.20), the constraint (3.21) is fulfilled as soon as D

(1) mn ≤ n 1/4 . For Gaussian errors, we can take p → +∞ and we find D

(1)

mn ≤ √ n.
This kind of result is known to lead to results of adaptation to unknown smoothness. For further applications, see [START_REF] Barron | Risks bounds for model selection via penalization[END_REF]. We first recall that a function f belongs to the Besov space

B α,l,∞ ([0, 1]) if it satisfies |f | α,l = sup y>0 y -α w d (f, y) l < +∞, d = [α] + 1,
where w d (f, y) l denotes the modulus of smoothness. For a precise definition of those notions, we refer to DeVore and Lorentz (1993) Chapter 2, Section 7, where it also proved that

B α,l,∞ ([0, 1]) ⊂ B α,2,∞ ([0, 1]) for l ≥ 2.
This justifies that we now restrict our attention to B α,2,∞ (A).

Proposition 3.1 Assume that the Assumptions of Theorem 3.1 hold and consider the families (RP), (DP), (W) or (T). Let α be a real number greater than 1/2, α ≤ r for (RP), (DP) and (W) and assume that b belongs to some Besov space B α,2,∞ (A). Then

sup b∈IB α,2,∞ (L) IE b1 I A -b 2 n 1 2 ≤ C(α, L)n -α 2α+1 (3.22) where IB α,2,∞ (L) = {t ∈ B α,2,∞ (A), |t| α,2 ≤ L}.
Proof. The result is straightforward with Lemma 12 in [START_REF] Barron | Risks bounds for model selection via penalization[END_REF] which imply that b1

I A -b m is of order (D (1)
m ) -α on the specified collections of models. Moreover the norm µ on the compact A is bounded by h 1 times the Lebesgue-norm as mentioned in (2.11).

Remark. Since the optimal choice is D

(1) 2α+1) , it satisfies in particular D

m * = n 1/(
(1) m * ≤ √ n, ∀α > 1/2. This allows to reach the optimal rate even with the family (T), restricted to dimensions less than √ n.

Estimation of σ 2

The result for the variance function can be given in two steps. σ 2 m denotes the IL 2 projection of σ 2 on S

(2) m .

Theorem 3.2 Let X 1 , . . . , X n be a stationary sequence drawn from model ( 1

dim(S (2) mn ) = D (2) mn ≤ n 1/2-4/p , (3.24) satisfies IE[ σ 2 1 I A -σ 2 2 n ] ≤ C inf m∈M (2) n σ 2 1 I A -σ 2 m 2 µ + S 2 Φ 2 2 D (2) m n + R n + C ′ IE[ b 2 1 I A -b2 2 n ] (3.25)
where C and C ′ are universal constants,

S 2 = b 2 + σ 2 2 µ + m 4 σ 2 2 µ + 4 bσ 2 µ and R is a constant depending on σ 16 , Φ 2 , b1 I A ∞ , σ1 I A ∞ .
Condition (3.23) reduces to p ≥ 16 for families (DP) or (W), to p ≥ 20 for family (RP) and to p > 16 for the family (T).

Remark. Note that considering the contrast

γn (t) = 1 n n i=1 X 2 i+1 -t(X i ) 2
leads to an estimate f of f = b 2 + σ 2 . In particular, it is possible to provide in an analogous way a bound for f -f 2 n . This gives the rate n -γ/(2γ+1) where γ = min(α, β) if b belongs to a Besov space B α,2,∞ and σ 2 to a Besov space B β,2,∞ . But it does not allow to separate the smoothness α and β of b and σ, without avoiding the loss in the rate when coming back to the evaluation of the rate of convergence of the estimator of σ 2 given by f -( b) 2 .

The interest of (3.25) is to illustrate the dependency in the first step estimator, and to show where some loss in the rates of convergence can happen. Indeed as

b 2 1 I A -b2 2 n = (b1 I A -b)(b1 I A + b) 2 n = (b1 I A -b)(2b1 I A -(b1 I A -b)) 2 n ≤ 4 b1 I A ∞ b1 I A -b 2 n + 2 (b1 I A -b) 2 2 n ≤ 4 b1 I A ∞ b1 I A -b 2 n + 2n b1 I A -b 4 n ,
so that we can find as another extension of Theorem 3.1 and as a tool for completing (3.25) in Theorem 3.2, the following bound:

Proposition 3.2 Under the Assumptions of Theorem 3.1 and if p ≥ 2(a 1 + 2) and p > 4b 1 + 10, (3.26)

we have:

IE( b1 I A -b 4 n ) ≤ C inf m∈M (1) n b1 I A -b m 4 + b1 I A -b m 2 8 n + (D (1) m ) 2 n 2 + R ′ n 2 . (3.27)
where C depends now on C, h 1 , M , θ and b 2 µ + σ 2 µ and

f 8 8 = |f (x)| 8 dx. Therefore if b belongs to some Besov space B α,2,∞ for α > 1/2, then b1 I A -b m 8 is of order (D (1) m ) -(α-(1/2-1/8)) , b1 I A -b m is of order (D (1) m ) -α . Therefore, choosing D (1)
m of order n 1/(2α+1) ensures that the infimum in (3.27) is less than Cn -4α/(2α+1) [1+n -(2α-1/2)/(2α+1) ] and therefore less than 2Cn -4α/(2α+1) , ∀α > 1/2. The rate corresponding to the term depending on b via nIE( b1

I A -b 4 n ) is n × n -4α/(2α+1) = n -(2α-1)/(2α+1) .
Next if σ 2 is in some B β,2,∞ , then the first term of the right-hand-side of (3.25) is of order

n -2β 2β+1 .
Thus it is easy to see that the minimax rate is obtained for

σ 2 if α ≥ 2β + 1/2,
i.e. it requires the regularity of b to be significantly greater than that of σ 2 . Moreover for the part IE( b1 2α+1) , it is negligible with respect to n -2β/(2β+1) as soon as αβ. Therefore, we proved the following result: Proposition 3.3 Assume that the Assumptions of Theorems 3.1, 3.2 and (3.26) hold and consider the collections of models (DP), (RP), (W) or (T). Let α and β be real numbers greater than 1/2 and less than r for families (RP), (DP) or (W) and assume that b belongs to some Besov space B α,2,∞ (A) and that σ 2 belongs to some Besov space

I A -b 2 n ) which has rate n -2α/(
B β,2,∞ (A) with α ≥ 2β + 1/2. Then sup b∈IB α,2,∞ (R 1 ,R 2 ),σ 2 ∈IB β,2,∞ (L) IE σ 2 1 I A -σ 2 2 n ≤ C(α, L, R 1 , R 2 )n -2β 2β+1 (3.28) where IB α,2,∞ (R 1 , R 2 ) = {t ∈ B α,2,∞ (A), |t| α,2 ≤ R 1 , |t| ∞ ≤ R 2 } and IB β,2,∞ (L) = {t ∈ B β,2,∞ (A), |t| β,2 ≤ L}.
Comments: 1.

If the condition α ≥ 2β + 1/2 is not fulfilled, the rate becomes n -(2α-1/2)/(2α+1) and is clearly suboptimal. 2. It has already been mentioned in the introduction that [START_REF] Neumann | Fully data-driven nonparametric variance estimators[END_REF] reaches the optimal rate for the estimation of σ 2 under the simpler condition α > 1; but he works with a fixed design regressive model under moment condition of any order for ε. It is also worth comparing this result with Hoffmann (1999) who deals with a more general risk IL p ′ and with functions belonging to more general Besov spaces B s,p,q , with s = α or s = β. Taking p ′ = p = 2 and q = ∞ in his main result for comparison shows that his conditions reduce simply to α > 3/2 (even when estimating b alone) and β > 3/2. Moreover, he requires the finiteness of exponential moments of the noise and reaches the optimal rate up to ln(n) factors. Therefore, the result given in Proposition 3.1 is always better, and the result given by Proposition 3.3 is better if α ≥ 2β + 1/2.

Simulation results

We generate samples using several regressive and autoregressive models. All models are denoted by Y i = b(X i ) + σ(X i )ε i+1 , with possibly Y i = X i+1 in the autoregressive case. For all paths, to make sure that the process has reached stationarity in the autoregressive case, we forget the 200 first data. For each model, we consider S = 400 samples with length n = 500 which provides paths denoted by (Y (s) i , X (s) i ) 1≤i≤n for s = 1, . . . , S. We consider various couples of regression or autoregression functions (b, σ). The couples of functions are gathered in Table 1. The values of the parameters in the regressive and autoregressive cases are given in the appendix (section 6).

Model drift and volatility

M1 b(x) = 0.4x + 1, σ(x) ≡ σ M2 b(x) = (0.5 + 0.25x) exp(0.5 -0.25x), σ(x) ≡ σ M3 b(x) = 0.5(x + 2 exp(-16x 2 ), σ(x) ≡ σ M4 b(x) = sin(2x) + 2 exp(-16x 2 ), σ(x) ≡ σ M5, M6 & M7 b(x) = sin(2ωπx + π/3), σ(x) ≡ σ M8 b(x) = sin(2πx + π/3), σ(x) = σ |x| M9 b(x) = sin(2πx + π/3) σ(x) = σ(0.31 + 0.7 exp(-5x 2 )) M10 b(x) = a(x + 2 exp(-16x 2 )) σ(x) = σ(0.2 + 0.4 exp(-2x 2 )) M11 1 b(x) = 1/(1 + exp(-x)) σ(x) = σ(ϕ(x + 1.2) + 1.5ϕ(x -1.2)) M12 & M13 b(x) = ax σ(x) = 0.05 + 1/(1 + βx 2 ) M14 & M15 b(x) = ax, σ(x) = 0.05 + π/2 + arctan(βx) M16 b(x) = a|x| if x < x 0 , a(x -2x 0 ) else. σ(x) = σ
Table 1. Couples of functions used to generate the models Note that the regressive case corresponds to the independent framework and the autoregressive case corresponds to the dependent context. Moreover models M1 to M7 correspond to (auto-)regressive models with constant volatility. Model M8 studies the problem of possible nullity of the variance function, together with some regularity problems in the volatility function. The models M10 and M11 are the one studied by [START_REF] Fan | Efficient estimation of conditional variance functions in stochastic regression[END_REF] and [START_REF] Härdle | Local polynomial estimators of the volatility function in nonparametric regression[END_REF] respectively. Lastly, model M16 studies the effect of a discontinuity in the mean function.

In the regressive case, the parameters are chosen to give some fixed level of the signal to noise ratio, denoted in all the following by s2n. Since in the regressive case, the X i 's are taken uniform on [-2, 2], we have s2n(reg) = 2 -2 b 2 (x)dx/ 2 -2 σ 2 (x)dx. In the autoregressive case, the choice of the parameters is done both to ensure the stability of the models and to provide some given signal to noise ratio. Since the law of the X i 's is unknown in this case, we compute for a given long sample s2n(autoreg) = n i=1 b 2 (X i )/ n i=1 σ 2 (X i ) and choose the coefficients giving the desired value of s2n(autoreg) in this particular case.

The results for models M12, M14, M16 are not reported in that context because the adjustment of most s2n ratios generate unstable models.

The estimation procedure is done using for both b and σ 2 the collection of models (RP) with degree r ≤ 5. We have implemented several procedures: six procedures working with piecewise polynomials of given (fixed) degree from r = 0 to r = 5, and a seventh procedure that chooses among those six global degrees the best one in terms of a penalized contrast. The interest of fixed degree estimation is that we can compute oracles which provide a benschmark to evaluate the performances of our estimates. More precisely, for each model, each degree r, each given dimension D = 1, . . . , D max = [n/((r + 1) ln(n))], we compute The oracle is then given by

L 2 (b, r, D) = 1 S S s=1 1 n n i=1 [b(X (s) i ) - b(s) D (X (s) i )]
L 2 oracle (b) = min 0≤r≤5 L 2 opt (b, r).
We define and compute analogously the oracles for σ 2 , L 2 opt (σ 2 , r) by using ([Y

(s) i - b(X (s) i )] 2 , X (s)
i ) as new data set and keep L 2 oracle (σ 2 ) = min 0≤r≤5 L 2 opt (σ 2 , r). Note that the oracles for σ 2 are computed with assuming that b is known. The oracles gives the best reachable performance, and are in practice unknown since the choice is performed with respect to the true function. The computation of the oracles represents the (very) long part (in time) of the numerical procedure. 2.6 0.7 1.7 0.7 1.2 0.7 1.1 0.7 M2 1.5 0.7 2.1 0.7 1.4 0.7 1.5 0.7 M3 1.4 0.7 1.8 0.7 1.4 0.8 1.7 0.8 M4 1.6 0.7 1.4 0.8 1.7 0.8 1.4 0.8 M5 1.7 0.7 1.1 0.7 1.4 0.7 1.4 0.7 M6 1.3 0.7 1.3 0.7 1.5 0.8 1.5 0.3 M7 1.4 0.9 1.3 0.9 1.3 1.0 1.3 1.0 M8 1.3 1.4 1.2 1.3 1.6 1.4 1.2 1.5 M9 1.6 1.2 1.4 1.2 1.5 1.2 1.6 1.2 M10 1.8 1.4 2.1 1.4 1.5 1.5 1.6 1.4 M11 2.7 2.1 1.6 2.0 1.2 2.1 1.1 2.1 M12 1.7 1.0 1.2 1.0 1.1 1.0 1.1 1.0 M13 2.2 1.1 1.8 1.1 1.6 1.1 1.6 1.1 M14 1.9 1.7 1.3 1.7 1.1 1.7 1.1 1.7 M15 1.9 1.2 1.3 1.2 1.1 1.2 1.1 1.2 M16 1.2 0.9 0.8 1.8 0.3 4.2 0.3 12 Table 2. Ratio to the oracle of the L 2 risk for the first step estimator in the regressive case for gaussian errors Let S r,D be the space of piecewise polynomials of degree r on the partition [(d-1)/D, d/D[, d = 1, . . . , D. Let X, Y, sX be vectors of IR n with coordinates X i , Y i , sX i , where sX i will be defined later, and let be t a function in some S r,D . We define here the contrast and the penalty function as

g n (X, Y, sX; t) = 1 n n i=1 Y i -t(X i ) sX i 2
and pen(D) = D + ln 2 (D).

Then we consider the following general procedures: E r (X, Y, sX, f X) for r = 0, . . . , 5 and E(X, Y , sX, f X), with input the IR n vectors X, Y, sX previously described and f X with coordinates f (X i ) for some given function f . The procedure E r (X, Y, sX, f X) proceeds as follows.

• For D = 1, . . . , D max = [n/((r + 1) ln(n))], compute fr,D (in fact fr,D (X i ), i = 1, . . . , n) the piecewise polynomial of S r,D minimizing g n (X, Y, sX; t) over all t in S r,D .

• Compute Dr = arg min 1≤D≤Dmax

[g n (X, Y, sX; fr,D ) + 2σ 2 r pen(D(r + 1))] where

σ2 r = g n (X, Y, sX, fr,[min( √ n,n/((r+1) ln(n))] ) if sX i = 1, ∀i = 1, . . . , n 1
else .

• Keep σ2 r , ( fr, Dr (X 1 ), . . . , fr, Dr (X n )) and f -fr, Dr

2 n = (1/n) n i=1 (f (X i )-fr, Dr (X i )) 2 .
The procedure E(X, Y, sX, f X) follows then and selects r = arg min 0≤r≤5 g n (X, Y, sX, fr,Dr ) + 2σ 2 r pen((r + 1) Dr ) .

The output is therefore f = fr, Dr and the associated error f -

f 2 n . s2n=1 s2n=3 s2n=7 s2n=10 b σ b σ b σ b σ M1
1.4 0.8 1.4 0.8 1.4 0.8 1.4 0.8 M2 2.2 1.2 2.1 0.8 2.1 0.8 2.1 0.8 M3 1.2 0.8 1.5 1.0 1.4 0.9 1.2 0.8 M4

1.1 1.3 1.2 1.2 1.7 126 0.6 0.7 M5 1.3 0.9 1.4 1.2 1.5 1.1

1.4 0.8 M6 1.3 1.3 1.2 1.0 1.2 1.1 1.1 1.2 M7 1.6 2.6 1.4 2.9 1.4 1.1 1.3 1.4 M8 1.1 4.1 1.2 2.3 1.3 2.3 1.1 1.6 M9 1.5 1.6 1.1 1.7 1.3 1.7 1.5 1.6 M10 1.5 1.6 1.6 1.3 1.5 1.3 1.2 2.2 M11 1.5 2.2 1.6 1.7 1.4 1.2 1.4 1.2
Table 3. Ratio to the oracle of the L 2 risk for the first step estimator in the autoregressive case for gaussian errors.

It follows that, as an output of the procedure E r (X (s) , Y (s) , 1 I, bX (s) ), where (X (s) , Y (s) ) is the sth sample drawn from a given regressive model, we obtain b(s) r , and b(s) as an output of E(X (s) , Y (s) , 1 I, bX (s) ). We compute b -b(s) 2 n for each sample. This allows to compute the mean IL 2 -empirical error:

L 2 emp (b, b) = IE (S) b -br 2 n = 1 S S s=1 1 n n i=1 [b(X (s) i ) -b(s) r (X (s) i )] 2 .
Therefore, we are interested in the ratios L 2 emp (b, b)/L 2 oracle (b). They are generally greater than one. The nearer of one, the better our method.

We also computed the output of E(X (s) , Y (s) , σX (s) , bX (s) ), where σX (s) has coordinates σ(X (s) i ) for i = 1, . . . , n, which delivers an estimator denoted by b(s) σ of b if σ were known. We compared the associated ratio L 2 emp (b, bσ )/L 2 oracle (b) to the previous one. Analogously, E r (X (s)

, [Y (s) -b(s) (X (s) )] 2 , 1 I, σ 2 X (s) ) where [Y (s) -b(s) (X (s) )] 2 has co- ordinates [Y (s) i -b(s) (X (s) i )] 2 for i = 1, . . . , n, gives estimators σ 2 (s) r , and E(X (s) , [Y (s) - b(s) (X (s) )] 2 , 1 I, σ 2 X (s) ) gives σ 2 (s)
. When s is varying we compute the error:

L 2 emp (σ 2 , σ 2 (s) ) = 1 S S s=1 1 n n i=1 σ 2 (X (s) i ) -σ 2 (s) (X (s) i ) 2 .
It can be compared with the estimate, denoted by σ 2 (s)

b of σ 2 if b were known by using

E(X (s) , [Y (s) -b(X (s) )] 2 , 1 I, σ 2 X (s) ).
Moreover, we studied a second stage of the procedure by computing b(s) as the output of

E r (X (s) , Y (s) , σ 2 (s)
, bX (s) ) and σ 2 (s) s) ). But this procedure happened to be very unstable in spite of several attempts to stabilize it.

as the output of E(X (s) , [Y (s) - b(s) (X (s) )] 2 , 1 I, σ 2 X (
We need to make two remarks about our procedure. First, when we have to divide by some estimate of the variance, when computing b(s) for instance, we divide in fact by the supremum of the value of interest and the 2%-quantile of all the positive estimated values. Secondly, there may be some restrictions on the values of the degrees when too few observations lie in one bin. In the regressive case, when working with global degree r, we take in fact locally on the subinterval

[d -1/D, d/D[, the degree min r, X i ∈ d -1 D , d D -1 .
In the autoregressive case, we take min(r, Rd -1) where Rd = rank(V (d, D)) and V (d, D) = (X q-1 ip / σ(X ip )) 1≤p≤k,1≤q≤r+1 , for i 1 , . . . i k the indexes of the X i 's in [(d-1)/D, d/D[. This is required for the inversion of the local linear system associated to the local computation of the estimator.

Our results are gathered in the Tables 2 and3 in the case of gaussian errors. Table 4 and 5 give the results for uniform errors and since they are of the same type, they are deferred to the appendix in section 6. All tables give the error ratios L 2 emp /L 2 oracle for b and σ as in the models given in Table 1 and for different values of the signal-to-noise ratio s2n. We can give several comments about these tables and other unreported results.

1. We can see that most ratios are near of 1, and almost all are less than 2, which means that our estimates perform very well.

2. We give the results of the first step estimator for b and σ because the second step is often unstable.

3. The results for b are most of the time better as those obtained by working with known σ and the knowledge of b does not improve significantly the estimation of σ. 5. When the s2n ratio become higher, the results do not improve significantly because the oracle decreases considerably in the same time.

In order to give a visual illustration of the results, we give confidence intervals (tenth and nineteenth percentiles) for curve estimation of b and σ 2 in 3 cases: Model M9 in the gaussian autoregressive case for s2n=3 (Figure 1), Model M10 for s2n=7 in the gaussian regressive (Figure 2) and autoregressive (Figure 3) cases. We generated here S = 100 samples with length n = 500. It appears clearly that the estimation of the mean function b is always very good, whereas the estimation of σ 2 is generally better in the regressive context than in the autoregressive one.

Proofs

5.1 Proof of Theorem 3.1

For the sake of simplicity, we omit the superscript (1) for the spaces and the dimensions and write S m for S

(1)

m , D m for D (1) m , M n for M (1)
n . There is no ambiguity all along this proof. We follow the line of the proof of Theorem 1 in [START_REF] Baraud | Adaptive estimation in an autoregression or a β-mixing regression[END_REF] and we use the same notations. We only recall that Ω * is the event

Ω * = {(ε i+1 , X i ) = (ε * i+1 , X * i ), i = 1, . . . , n}
where the variables (ε * i+1 , X * i ) are associated to the (ε i+1 , X i ) as in Claim 2 of [START_REF] Baraud | Adaptive estimation in an autoregression or a β-mixing regression[END_REF] recalled below: Claim 2: Let q n , q n,1 be integers such that 0 ≤ q n,1 ≤ q n /2, q n ≥ 1. Set

u i = (ε i , X i ), i = 1, ..., n, then there exist random variables u * i = (ε * i , X * i ), i = 1, .
.., n satisfying the following properties:

• For ℓ = 1, ..., ℓ n = [n/q n ], the random vectors U ℓ,1 = u (ℓ-1)qn+1 , ..., u (ℓ-1)qn+q n,1 ′ and U * ℓ,1 = u * (ℓ-1)qn+1 , ..., u * (ℓ-1)qn+q n,1 ′
have the same distribution, and so have the random vectors • For ℓ = 1, ..., ℓ n , IP U ℓ,1 = U * ℓ,1 ≤ β (qn-q n,1 ) and IP U ℓ,2 = U * ℓ,2 ≤ β q n,1 .

U ℓ,2 = u (ℓ-1)qn+q n,1 +1 , ..., u ℓqn ′ and U * ℓ,2 = u * (ℓ-1)qn+q n,
(5.29)

• For each δ ∈ {1, 2}, the random vectors U * 1,δ , ..., U * ℓn,δ are independent. The variables u * i are built using Berbee's coupling Lemma as in [START_REF] Viennet | Inequalities for absolutely regular processes: application to density estimation[END_REF]. For sake of simplicity, we assume that n = q n ℓ n . For ρ ≥ 1 we also recall that Ω ρ is the event:

Ω ρ =    t 2 µ ≤ ρ t 2 n , ∀t ∈ m,m ′ ∈Mn S m + S m ′    ,
that is Ω ρ is the event where the norms . and . n can be compared. Lastly, we add, for some τ ∈]0, 1[, the definition of the following event: (5.30) where rn is defined by (2.13). We denote by Ω * τ,ρ := Ω τ ∩ Ω ρ ∩ Ω * , by B(m ′ , µ) = {t ∈ S m + S m ′ , t µ ≤ 1}, and by D(m ′ ) = dim(S m + S m ′ ). Since m is fixed, we do not mention the dependence on m of the previous terms. Then we write the decomposition

Ω τ = (1 -τ ) σ 2 µ ≤ r2 n ≤ 2(1 + τ )( b 2 µ + σ 2 µ ) ,
γ n (t) -γ n (s) = b A -t 2 n -b A -s 2 n + 2 s -t, σε n where t, σε n = (1/n) n i=1 t(X i )σ(X i )ε i+1 . Moreover the definition of b implies that ∀m ∈ M n γ n ( b) + pen (1) ( m) ≤ γ n (b m ) + pen (1) (m)
with pen (1) defined by (2.15). Therefore, using that 2ab ≤ xa 2 + x -1 b 2 and (a + b) 2 ≤ (1 + y)a 2 + (1 + y -1 )b 2 for all positive a, b, x and y, we find, analogously to (38) in [START_REF] Baraud | Adaptive estimation in an autoregression or a β-mixing regression[END_REF], Claim 3, that on Ω * τ,ρ

1 -ρ 1 + y x b1 I A -b 2 n ≤ 1 + ρ 1 + y -1 x b1 I A -b m 2 n + pen (1) (m) + x n 2 sup t∈B( m,µ) n i=1 ε * i+1 σ(X * i )t(X * i ) 2 -pen (1) ( m) ≤ 1 + ρ 1 + y -1 x b1 I A -b m 2 n + 2(1 + τ )κ( b 2 µ + σ 2 µ ) D m n Φ 2 1 + x n 2   sup t∈B( m,µ) n i=1 ε * i+1 σ(X * i )t(X * i ) 2 -x 2 nD( m)Φ 2 1 σ 2 µ   + + x 3 D( m) n Φ 2 1 σ 2 µ - κ(1 -τ ) σ 2 µ D mΦ 2 1 n
The real numbers x and y are positive constant numbers to be chosen. Then setting

Wn (m ′ ) =   sup t∈B(m ′ ,µ) n i=1 ε * i+1 σ(X * i )t(X * i ) 2 -x 2 nD(m ′ )Φ 2 1 σ 2 µ   +
we find that if x, ρ are numbers satisfying x > ρ > 1, y = (x -ρ)/(x + ρ), τ > 0, and if κ = x 3 /(1 -τ ) in pen (1) , then:

b1 I A -b 2 n 1 I Ω * τ,ρ ≤ K 1 (x, ρ) b1 I A -b m 2 n + 4 ( b 2 µ + σ 2 µ ) 1 -τ D m n Φ 2 1 + x n 2 Wn ( m). (5.31) with K 1 (x, ρ) = (x + ρ) 2 /(x -ρ) 2 .
In [START_REF] Baraud | Adaptive estimation in an autoregression or a β-mixing regression[END_REF], Wn (m ′ ) for σ ≡ σ 2 where σ 2 is a constant is denoted by W n (m ′ ) and Proposition 6 in [START_REF] Baraud | Adaptive estimation in an autoregression or a β-mixing regression[END_REF] states that, under assumptions analogous to the assumptions of Theorem 3.1, for any p < p/2,

m ′ ∈Mn IE(W n (m ′ ) p) ≤ K 2 n p   m ′ ∈Mn D -p/2+p m ′ + q p n |M n | n p(p-2)/[4(p-1)]-p  
(5.32)

where

K 2 = C(p, p, x)(Φ 1 h -1/2 0
) p σ 2p p , σ p p = IE(|ε 1 | p ) and C(p, p, x) is a constant depending on p, p and x. This result has a straightforward extension to non constant variance function with only a σ A ∞ replacing σ 2 and ( σ A ∞ σ p ) 2p replacing σ 2p p . The only point to check is indeed the bound for the analogous of IE sup ℓn ℓ=1 G ℓ (t) with now

Gℓ (t) = i∈I (1) ℓ ε * i+1 σ A (X * i )t(X * i ).
We still find that for t =

D(m ′ ) j=1 a j ϕ j where (ϕ j ) 1≤j≤D(m ′ ) is a µ-orthonormal basis of S m + S m ′ , IE sup t∈B(m ′ ,µ) ℓn ℓ=1 Gℓ (t) ≤   D(m ′ ) j=1 IE ℓn ℓ=1 Gℓ (ϕ j ) 2   1/2 =   D(m ′ ) j=1 ℓn ℓ=1 IE G2 ℓ (ϕ j )   1/2
since the blocks are independent and centered. The difference is that here

IE( G2 ℓ (ϕ j )) = i∈I (1) ℓ IE(ε 2 i+1 )IE(σ 2 A (X i )ϕ 2 j (X i )) = q n,1 σ A ϕ j 2 µ .
Then using consequence (2.10) of assumption (H Φ 1 ) 2., we have j ϕ 2 j ∞ ≤ Φ 2 1 D(m ′ ), and therefore IE sup

t∈B(m ′ ,µ) ℓn ℓ=1 Gℓ (t) ≤ Φ 1 σ µ ℓ n q n,1 D(m ′ ).
This gives the announced extension of Proposition 6 of [START_REF] Baraud | Adaptive estimation in an autoregression or a β-mixing regression[END_REF], namely, for p = 1: (5.33) where

m ′ ∈Mn IE Wn (m ′ ) n 2 ≤ K 3 n -1   m ′ ∈Mn D -p/2+1 m ′ + q p n |M n | n p(p-2)/[4(p-1)]-1   ,
K 3 = C(x, p) σ A 2 ∞ σ 2 p (Φ 2 1 h -1 0 ) p/2
. Thus, in view of (H (a 1 ,b 1 ) ), the last bracketed term in (5.33) is uniformly bounded if

-p/2 + 1 ≤ -a 1 and b 1 + 1 -{p(p -2)/[4(p -1)]} < 0.
(

IE( b1 I A -b 2 n 1 I Ω * τ,ρ ) ≤ K 1 (x, ρ) b1 I A -b m 2 n + 4 ( b 2 µ + σ 2 µ ) 1 -τ D m n Φ 2 1 + K 4 n .
(5.35)

where K 4 = K 4 (x, ρ, Φ 1 , h 0 , Σ 1 , T 1 ). It remains to bound the expectation on the complementary of Ω * τ,ρ . Let C denote a constant that may change from line to line. It follows from Claim 5 in [START_REF] Baraud | Adaptive estimation in an autoregression or a β-mixing regression[END_REF] that IP((Ω ρ ∩Ω * ) c ) ≤ C/n 2 under geometrical mixing condition and (H Φ 1 ) 3. and that

IE b1 I A -b 2 n 1 I Ω * c τ,ρ ≤ C(h 0 , h 1 , Φ 1 , ρ)n -1 as soon as IP((Ω * τ,ρ ) c ) ≤ C n 2 . (5.36) Since IP((Ω * τ,ρ ) c ) ≤ IP(Ω c τ ) + IP((Ω * ρ ) c
) with obvious notations, and since

IP(Ω c τ ) = IP(Ω c τ ∩ Ω ρ ∩ Ω * ) + IP((Ω ρ ∩ Ω * ) c ),
we find that (5.36) holds if we have

IP(Ω c τ ∩ Ω ρ ∩ Ω * ) ≤ C/n 2 .
This is ensured by the following Lemma:

Lemma 5.1 Under the Assumptions of Theorem 3.1 and if

D (1) mn ≤ n 1/2-k/p for k = 2 or k = 4, (5.37) and p ≥ 8 if k = 2 and p ≥ 16 if k = 4 (so that D (1)
mn can always be taken of order

n 1/4 ), then IP (Ω c τ ∩ Ω ρ ∩ Ω * ) ≤ Cn -k where C is a constant depending in particular on Φ 1 , p ρ, σ µ , σ A ∞ , b A ∞ .
Recall that p denotes the order of the moment of the ε i 's in model (1.1) and that D

(1) mn is defined by (2.13) and the line following. This ends the proof of Theorem 3.1. . The limit choices τ → 0 and x → 0 give κ → 1 but imply that the multiplicative constant tends to infinity. The choice τ = 1/2, x = 2 and ρ = 1 gives κ = 4 and reasonable orders for the multiplicative constants. The value of κ must be investigated by simulation experiments.

Proof of Lemma 5.1. Most elements of this proof (in the case of a simpler model) can be found in a first draft of [START_REF] Baraud | Adaptive estimation in an autoregression or a β-mixing regression[END_REF] but it disappeared of the final version. For the sake of simplicity, we work on a space S m with dimension D m (instead of D

(1) mn ). We shall denote by t X the transpose of a vector X. Let R = t (X 2 , . . . , X n+1 ) and let ε = t (ε 2 , . . . , ε n+1 ). All along this section we abusively denote the same way a function g mapping IR into IR and the vector of IR n t (g(X 1 ), • • • , g(X n )). IR n is provided with the inner product u, v = n i=1 u i v i , we denote the corresponding norm by . and by . n the empirical norm:

u 2 n = (1/n) n i=1 u 2 i . From now on {ϕ λ , λ ∈ Λ m } denotes an orthonormal basis of S m relatively to µ and Φ m (X) is the D m × D m normalized Gram matrix defined by Φ m (X) = 1 n n i=1 ϕ λ (X i )ϕ λ ′ (X i ) λ,λ ′ ∈Λm . It follows from the definition of bm that bm = V m [nΦ m (X)] -1 t V m R = 1 n V m Φ -1 m (X) t V m R where V m denotes the n × D m matrix satisfying (V m ) (i,λ) = ϕ λ (X i ) for i = 1, • • • , n and λ ∈ Λ m . We denote by Π m (X) the projection matrix V m Φ -1 m (X) t V m . Note that Π m (X) t Π m (X) = V m Φ -1 m (X)( t V m V m )Φ -1 m (X) t V m = nV m Φ -1 m (X) t V m = nΠ m (X). Since r2 n = R -bm 2 n for R = b + σε, we have r2 n = R -Π m (X)R 2 n = b -Π m (X)b + σε -Π m (X)σε 2 n = 1 n b -Π m (X)b 2 + σε 2 -Π m (X)σε 2 + 2 b -Π m (X)b, σε .
We define the measure IP * ,ρ by IP * ,ρ (B) = IP(B ∩ Ω * ∩ Ω ρ ), and we take τ = 4η.

IP * ,ρ (r 2 n ≤ (1 -4η) σ 2 µ ) ≤ IP * ,ρ ( σε 2 n ≤ (1 -η) σ 2 µ ) +IP( Π m (X)σ A ε 2 n ≥ η σ 2 µ ) +IP * ,ρ (2| b -Π m (X)b, σε | ≥ 2nη σ 2 µ ).
We denote by σ A = σ1 I A . Note that σ can be replaced by σ A each time it is multiplicated by A-supported functions. The same holds for b and b

A = b1 I A . Let us bound first IP * ,ρ ( Π m (X)σ A ε 2 n ≥ 2η σ 2 µ ). Π m (X)σ A ε 2 = 1 n 2 t (σ A ε)Π m (X)Π m (X)(σ A ε) = 1 n t ( t V m σ A ε)Φ -1 m (X)( t V m σ A ε) = 1 n t V m σ A ε, Φ -1 m (X)( t V m σ A ε) ≤ ρ(Φ -1 m (X)) n t V m σ A ε 2 ,
where ρ(M ) denotes the spectral radius of the matrix M . We know from [START_REF] Baraud | Model selection for regression on a fixed design[END_REF], Lemma 3.1 p. 475, that

ρ(Φ -1 m (X)) = sup t∈Sm/{0} t 2 µ t 2 n . Therefore on Ω ρ , we have ρ(Φ -1 m (X)) ≤ ρ. This implies that on Ω * ∩ Ω ρ , Π m (X)σ A ε 2 n ≤ ρ n 2 t V m σ A ε 2 = ρ λ∈Λm 1 n n i=1 ε i+1 ϕ λ (X i )σ A (X i ) 2 . Therefore IP * ,ρ ( Π m (X)σε 2 n ≥ η σ 2 µ ) ≤ IP * ,ρ   λ∈Λm 1 n n i=1 ε i+1 ϕ λ (X i )σ A (X i ) 2 ≥ η ρ σ 2 µ   ≤ ρ η σ 2 µ p 2 IE   λ∈Λm 1 n n i=1 ε * i+1 ϕ λ (X * i )σ A (X * i ) 2   p 2 ≤ ρ η p 2 D p/2-1 m n p σ p µ λ∈Λm IE n i=1 ε * i+1 ϕ λ (X * i )σ A (X * i ) p .
This term is handled by using a Rosenthal moment inequality (see [START_REF] Petrov | Limit theorems of probability theory. Sequences of independent random variables[END_REF] or a recall in [START_REF] Baraud | Model selection for regression on a fixed design[END_REF], Theorem 8.1) applied to centered and block-independent variables admitting moments of order p (I

(1,2) ℓ is set for successively I

(1) ℓ and I

(2) ℓ ): there exists a constant c(p) such that (5.38) where

IE n i=1 Z i p ≤ 2c(p)        ℓn ℓ=1 IE i∈I (1,2) ℓ Z i p +    ℓn ℓ=1 IE    i∈I (1,2) ℓ Z i    2    p/2       
Z i = ε * i+1 ϕ λ (X * i )σ A (X * i ).
Next we bound both terms separately. (5.39) where C = C(p, ρ, τ, Φ 1 , σ p , σ A ∞ ). Therefore, under (5.37), and for p ≥ 8, we have

IE ℓn ℓ=1 i∈I (1,2) ℓ Z i p ≤ ℓn ℓ=1 IE    i∈I (1,2) ℓ |ε i+1 ϕ λ (X i )σ A (X i )|    p ≤ (Φ D m ) p σ A p ∞ q p-1 n ℓn ℓ=1 IE    i∈I (1,2) ℓ |ε i+1 | p    ≤ (2Φ σ A ∞ σ p ) p q p-
IP * ,ρ ( Π m (X)ε 2 n ≥ η σ 2 µ ) ≤ C(τ, p, Φ 1 , ρ)n -k .
A bound for IP( σε 2 n ≤ (1 -η) σ 2 µ ) is obtained by applying a Rosenthal type inequality as well.

IP * ,ρ ( σε 2 n ≤ (1 -η) σ 2 µ ) = IP * ,ρ 1 n n i=1 σ 2 (X i )(ε 2 i+1 -1) + 1 n n i=1 [σ 2 (X i ) -IE(σ 2 (X i )] ≤ -η σ 2 µ ≤ IP * ,ρ 1 n n i=1 σ 2 (X i )(ε 2 i+1 -1) ≥ η 2 σ 2 µ +IP * ,ρ 1 n n i=1 [σ 2 (X i ) -IE(σ 2 (X i )] ≥ η 2 σ 2 µ
where u i = ε 2 i -1 are i.i.d. centered variables, with u i independent of X i-1 and B 2 (m ′ , µ) = {t ∈ S We recall that S 2 = s 2 + IE µ [(b 2 + σ 2 ) 2 ]. Therefore, we find:

7 8 σ 2 -σ 2 A 2 n 1 I Ω * τ,ρ ≤ 9 8 σ 2 m -σ 2 A 2 n + pen (2) (m) + 8Φ 2 2 x 2 s 2 D (2) m n +32 W (1) n ( m2 ) + 8 W (2) n ( m2 ) + ρ 4 σ 2 -σ 2 m 2 n +16 b 2 A -b2 2 n + 8Φ 2 2 x 2 s 2 D m2 n -pen (2) ( m2 ).
(5.41)

For simplicity, we choose ρ = 3/2 and this yields (2) mn ≤ n 1/2-4/p (5.42) and p ≥ 16 (so that D

(2)

m can be of order n 1/4 ), then

IP Ωc τ ∩ Ω ρ ∩ Ω * ≤ Cn -2
where C is a constant depending in particular on Φ 2 , p ρ, σ µ , σ A ∞ , b A ∞ . 1.9 1.0 2.3 1.1 2.3 1.1 2.3 1.1 M3 1.5 1.4 1.6 1.3 1.2 1.2 1.3 1.3 M4 1.3 1.7 1.2 2.1 1.2 1.2 1.2 1.0 M5 1.2 1.6 1.3 1.4 1.5 1.2 1.2 1.5 M6 1.3 2.2 1.3 1.7 1.1 2.1 1.3 2.4 M7 1.6 3.1 1.3 2.7 1.3 2.5 1.3 2.4 M8 1.3 8.5 1.1 2.5 1.3 1.9 1.1 2.3 M9 1.5 1.6 1.1 1.7 1.3 1.5 1.3 1.6 M10 1.5 1.9 1.9 1.9 1.1 1.1 1.2 1.3 M11 1.8 2.6 1.5 2.3 1.6 1.3 1.6 1.1 Table 4. Ratio to the oracle of the L 2 risk for the first step estimator in the autoregressive case for uniform errors. 1.4 1.3 1.2 1.3 1.5 1.5 1.2 1.4 M9 1.6 1.2 1.4 1.2 1.5 1.2 1.6 1.2 M10 1.6 1.3 2.4 1.3 1.3 1.3 1.5 1.3 M11 2.4 1.3 1.6 1.3 1.3 1.3 1.3 1.3 M12 2.0 1.6 1.4 1.5 1.4 1.5 1.4 1.5 M13 2.7 1.5 2.2 1.5 2.2 1.5 2.2 1.5 M14 1.9 1.5 1.4 1.5 1.3 1.4 1.3 1.4 M15 2.0 1.1 1.5 1.1 1.4 1.1 1.4 1.1 M16 1.3 1.5 0.8 3.6 0.4 11 0.3 31 Table 5. Ratio to the oracle of the L 2 risk for the first step estimator in the regressive case for uniform errors.

  b m is the IL 2 (dx)-orthogonal projection of b on S(1)

( T )

 T Trigonometric polynomials : we consider spaces of dimension D (i) m generated by the functions ϕ 0 (x) = 1, ϕ 2j (x) = √ 2 cos(2πjx), ϕ 2j+1 (x

  (1) n (t), over t ∈ S(1) m . The bm 's define a collection of estimators of b. Then choose:m1 = arg min m∈M (1) n γ (1) n ( bm ) + pen(m) where pen(m) = κσ 2 2 D (1) m n ,and κ is a universal constant.[START_REF] Baraud | Adaptive estimation in an autoregression or a β-mixing regression[END_REF]'s results extend straightforwardly to a known varying variance function by considering the estimate b m1 with:

i

  ) 1≤i≤n , as an estimation of IE[ b -bD 2 n ]. Then we know L 2 opt (b, r) = min 1≤D≤Dmax L 2 (b, r, D) and D opt = arg min 1≤D≤Dmax L 2 (b, r, D).

Figure 1 :

 1 Figure 1: True functions (thick curves) b and σ 2 in the autoregressive model M9 with value of the parameters corresponding to s2n=3 and gaussian errors. Tenth and nineteenth percentiles (dotted curves) for S = 100 samples with length n = 500 of the estimation of b (right curves) and σ 2 (left curves) given by the algorithm.

Figure 2 :

 2 Figure 2: True functions (thick curves) b and σ 2 in the regressive model M10 with value of the parameters corresponding to s2n=7 and gaussian errors. Tenth and nineteenth percentiles (dotted curves) for S = 100 samples with length n = 500 of the estimation of b (right curves) and σ 2 (left curves) given by the algorithm.

.Figure 3 :

 3 Figure 3: True functions (thick curves) b and σ 2 in the autoregressive model M10 with value of the parameters corresponding to s2n=7 and gaussian errors. Tenth and nineteenth percentiles (dotted curves) for S = 100 samples with length n = 500 of the estimation of b (right curves) and σ 2 (left curves) given by the algorithm.

≤ε

  ( σ A ∞ σ 2 ) p n p/2 . i+1 ϕ λ (X i )σ(X i )

  m ′ , t µ ≤ 1}, D 2 (m ′ ) = dim(S the new Ω * τ,ρ := Ωτ ∩ Ω ρ ∩ Ω * where now Ωτ = {(1 -τ )s 2 ≤ ŝ2 n ≤ 2(1 + τ ) IE µ [(b 2 + σ 2 ) 2 ] + s 2 },(5.40)withs 2 = 4IE µ (b 2 σ 2 ) + m 4 IE µ (σ 4 ).

  κ in pen(2) is chosen in such a way that the last term in(5.41) is nonpositive i.e. κ = 8x 2 /(1 -τ ). The bound for m ′ ∈M (2) n IE[ W (1) n (m ′ )]is the same as the one given in (5.33) with only σ A ∞ replaced by b A σ A ∞ and the same conditions on p. To bound m ′ ∈Mn IE[ W (2) n (m ′ )] we must take into account that the u i 's admit moments of order p/2 only, thus (5.33) holds with σ A ∞ replaced by σ A 2 ∞ and p replaced by p/2. Then the conditions required now to bound the last term (see (5.33) with p replaced by p/2) are -p/4 + 1 ≤ -a 2 and b 2 + 1 -{(p/2)(p/2 -2)/[4(p/2 -1)]} < 0; those conditions are fulfilled under (3.23). Therefore, the end being the same as in the proof of Theorem 3.1, the result follows from the following Lemma: Lemma 5.2 Under the assumptions of Theorem 3.2 and if D

  M14 β = 4, a = s2n/0.6007; β = 0.25, a =(0.708, 0.951, 0.9919, 0.9961), β = 0.25, a = (0.707, 0.95, 0.9918, 0.9961), M15 β = 2, a = s2n/0.5675; no autoregressive counterpart M16 x 0 = 1/ √ 2, σ = 0.8616, a = s2n; a = 1.04, x 0 = √ 2, σ = (1.044, 0.314, 0.162, 0.125); a = 1.04, x 0 = √ 2, σ = (1.03, 0.314, 0.162, 0.1254).

  X n be a stationary sequence drawn from model (1.1) and consider a collection of models satisfying (H Φ 1 ) and (H (a 1 ,b 1 ) ).

	Assume that A1, A2(p), A3,
	A4 are fulfilled with
	p ≥ 8, p ≥ 2(1 + a 1 ) and p > 6 + 4b 1 , then b, defined by (2.14)-(2.15), with r2 n defined by (2.13) and such that	(3.20)
	dim(S (1) mn

  .1) and consider a collection of models satisfying (H Φ 2 ) and (H (a 2 ,b 2 ) ). Assume that A1, A2(p), A3, A4 are fulfilled with p ≥ 16, p ≥ 4(1 + a 2 ) and p > 8b 2 + 12, (3.23)

	then σ2 , defined by (2.17)-(2.18), with ŝ2 n defined by (2.19) and such that

ϕ is the normal density

We must also emphasize that the last step of the procedure which chooses the degree performs quite well and gives empirical errors of the same order as the error associated to the degree implying the lowest error.

IE|ε 2 1 -1| p/2 + n p/4 m p/4 4 + 2 p/2 nq p/2-1 n + 2 p/2 n p/4 .

Since p > 4, n 1-p/2 < n -p/4 and thus the order is n -p/4 which is less than n -2 if p ≥ 8 and less than n -4 if p ≥ 16.

To bound the last term IP * ,ρ (2| b -Π m (X)b, σε n | ≥ 2η σ 2 µ ), we consider the two terms IP * ,ρ (| b, σε n | ≥ η σ 2 µ /2) and IP * ,ρ (| Π m (X)b, σε n | ≥ η σ 2 µ /2).

Again, we clearly have

.

The moment of order p/2 is bounded by applying again the moment inequality (5.38) to the blocks of b(

Thus as previously and since q n is of order ln(n), p > 4, the order is n -p/4 so that:

A2(p) ensures that IE µ (|b| p ), IE µ (|σ| p ) are finite.

In the same way as previously,

This implies

This term is nearly the same as (5.39) except that there is a loss of D p/2 m in the final order. Ignoring the constants, we find for this probability an order D 3p/2 m n 1-p q p-1 n + D p m n -p/2 . The final order is D p m n -p/2 and is less than n -k as soon as

and the first right-hand-side term has already been studied; the second one gives the same order with a Rosenthal inequality again. This completes the proof of Lemma 5.1.

Proof of Theorem 3.2

We have

Since all functions s, t are A-supported, we can replace b and σ by b1 I A = b A and σ1 I A = σ A everywhere. Moreover, for any θ > 0,

The terms to control are sup

This ends the proof of Theorem 3.2.

Proof of Lemma 5.2. We follow the line and the notations of the proof of Lemma 5.1 and write, if X 2 has coordinates (X 2 i+1 ), i = 1, . . . , n:

Then all terms can be treated as previously. For instance

and all terms can be treated thanks to a Rosenthal inequality of order p/4. This implies an order n -p/8 , less than n -2 for p ≥ 16 as assumed in (3.23).

Analogously the term Π m (X)

and the scalar product term of order

They are less than n -2 if (D

(2) m ) p/2 ≤ n p/4-2 and p > 8 which explains condition (5.42).

Proof of Proposition 3.2

We start from (5.31) which only requires to be squared:

(5.43)

Choosing p = 2 in (5.32) and using the extended Proposition 6 of [START_REF] Baraud | Adaptive estimation in an autoregression or a β-mixing regression[END_REF], we can replace (5.33) by

The last bracketed term is bounded if -p/2 + 2 ≤ -a 1 and b 1 + 2 -{p(p -2)/[4(p -1)]} < 0. This gives the conditions p ≥ 2(2 + a 1 ) and p > 4b 1 + 10; these conditions are fulfilled under (3.26). Therefore under (3.26),

) is of order 1/n 2 . Taking the expectation of (5.43) gives

where K ′ depends on x, ρ, h 0 , σ A ∞ , Φ 1 , Σ 1 , T 1 . We write that

From Theorem 2.1 in [START_REF] Viennet | Inequalities for absolutely regular processes: application to density estimation[END_REF], we know that there exists a function

for a sequence (X i ) stationary with stationary law µ and absolutely regular with β-mixing coefficients β l . Therefore

Var

This yields

which is the first part of the right hand side of (3.27).

The last thing to check is the order of the expectation of b

≤ 2 b A ∞ + IE(ε 8 1 )IE µ (σ 8 ) IP((Ω * τ,ρ ) c ).

Thus, we need IP((Ω * τ,ρ ) c ) to be order n -2 i.e. that IP((Ω * τ,ρ ) c ) ≤ C/n 4 which is ensured by Lemma 5.1 (take k = 4) under the Assumptions of Theorem 3.2.

6 Appendix: Value of the parameters associated to the models and further numerical results.

The functions are given in Table 1 and the parameters are given first for the regressive case as a function of s2n, then in the autoregressive case, in the order corresponding to s2n values 1, 3, 7, 10, first for gaussian errors and next for uniform errors.