Fabienne Comte

Claire Lacour

Anisotropic adaptive kernel deconvolution

Keywords: Adaptive kernel estimator, Anisotropic estimation, Deconvolution, Density estimation, Measurement errors, Multidimensional, Primary subjects, 62G07

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Introduction

There have been a lot of studies dedicated to the problem of recovering the distribution f of a signal when it is measured with an additive noise with known density. Several strategies have been proposed since [START_REF] Fan | On the optimal rates of convergence for nonparametric deconvolution problems[END_REF] in order to provide adaptive strategies for kernel [START_REF] Delaigle | Bootstrap bandwidth selection in kernel density estimation from a contaminated sample[END_REF]) or projection [START_REF] Pensky | Adaptive wavelet estimator for nonparametric density deconvolution[END_REF], [START_REF] Comte | Penalized contrast estimator for adaptive density deconvolution[END_REF]) estimators. The question of the optimality of the rates revealed real difficulties, after the somehow classical cases studied by [START_REF] Fan | On the optimal rates of convergence for nonparametric deconvolution problems[END_REF]: the case of super smooth noise (i.e. with exponential decay of its characteristic function) in presence of possibly also super smooth density implies non standard 1 bias variance compromises that require new methods for proving lower bounds. These problems have been studied by [START_REF] Butucea | Deconvolution of supersmooth densities with smooth noise[END_REF], Butucea and Tsybakov [2008a,b] and by [START_REF] Butucea | Adaptive estimation of linear functionals in the convolution model and applications[END_REF].

Then new directions lead researchers to release the assumption that the characteristic function of the noise never vanishes, see [START_REF] Hall | Deconvolution with unknown error distribution[END_REF], [START_REF] Meister | Deconvolution from Fourier-oscillating error densities under decay and smoothness restrictions[END_REF]. Others released the assumption that the density of the noise is known. In physical contexts, where it is possible to obtain samples of noise alone, a solution has been proposed by [START_REF] Neumann | On the effect of estimating the error density in nonparametric deconvolution[END_REF], extended to the adaptive setting by [START_REF] Comte | Data-driven density estimation in the presence of additive noise with unknown distribution[END_REF], another idea is developed in Johannes [2009]. Other authors assumed repeated measurements of the same signal, and proposed estimation strategy without noise sample, see [START_REF] Delaigle | On deconvolution with repeated measurements[END_REF].

All these works are in one dimensional setting. Our aim here is to study the multidimensional setting, and to propose adaptive strategies that would take into account possible anisotropy for both the function to estimate and the noise structure. As already explained in [START_REF] Kerkyacharian | Nonlinear estimation in anisotropic multi-index denoising[END_REF], adaptive procedures are delicate in a multidimensional setting because of the lack of natural ordering. For instance, the model selection method is difficult to apply here since it requires to bound terms on sums of anisotropic models. In this paper, we use a unified setting where all estimators can be seen as kernel estimators, and we use the method recently developed in Goldenshluger andLepski [2010, 2011] to face anisotropy problems. The originality of our work is to use Talagrand inequality as the key of the deviation in the mean squared error case. This idea is also exploited in a different context by [START_REF] Doumic | Nonparametric estimation of the division rate of a size-structured population[END_REF]. And indeed, we succeed in building adaptive kernel estimators in many contexts. The bandwidth is automatically selected. We provide risk bounds for these estimators, for both pointwise risk when local bandwidth selection is proposed and for the integrated mean square risk (MISE) when the global selection is studied. We also consider both anisotropic Hölder and Sobolev classes for f , the Fourier-domaindefinition of the last ones allowing to also deal with the case of super smooth functions. Few papers study the multidimensional deconvolution problem; we can only mention [START_REF] Masry | Multivariate probability density deconvolution for stationary random processes[END_REF] who considers mainly the problem of dependency between the variables without anisotropy nor adaptation, and [START_REF] Youndjé | Optimal bandwidth selection for multivariate kernel deconvolution density estimation[END_REF] who consider a cross-validation method for bandwidth selection in an isotropic and ordinary smooth setting. Our paper considerably generalizes their results with a different method, and provides new results and new rates in both pointwise and global setting.

We want here to emphasize that our setting is indeed very general. We consider all possible cases: the noise can have both ordinary smooth (O.S.) components (i.e. a characteristic function with polynomial rate of decay in the corresponding directions) and super smooth (S.S.) components (exponential rate of decay), and the signal density also. In particular, we obtain surprising results in the mixed cases: if one component only of the noise is S.S. (all the others being O.S.), in presence of an O.S. signal, then the rate of convergence of the estimator is logarithmic. On the contrary, if the signal has k out of d components S.S. in presence of an O.S. noise, then the rate of the estimator is almost as good as if the dimension of the problem was dk instead of d. We obtain also natural extensions of the univariate rates, and in particular the important fact that the rates can be logarithmic if the noise is S.S. (for instance in the Gaussian case) but are much improved if the signal is also S.S.: for instance, if the signal is also Gaussian, then polynomial rates are recovered.

In spite of the difficulty of the problem, in particular because of the large number of parameters required to formalize the regularity indexes of the functions, we exhibit very synthetic penalties than can be used in all cases. We also provide more precise but more technical results. It is certainly worth mentioning that the adaptive strategy we propose in the pointwise setting is not only a generalization of the one-dimensional results obtained in [START_REF] Butucea | Adaptive estimation of linear functionals in the convolution model and applications[END_REF], but is also a different procedure. Lastly, we prove original lower bounds for both pointwise and global setting, and this requires specific constructions.

The plan of the paper is the following. In Section 2, we describe the model and the assumptions: the functional classes and the kernels used in the following. We both give the conditions required in the following for the kernels and provide concrete examples of kernels fulfilling them. We define the general estimator by generalization of the one-dimensional kernel to multidimensional setting. In Section 3, we study the pointwise risk and we discuss the rates. We also assert the optimality of most rates by proving lower bounds. Then we propose a pointwise bandwidth selection strategy and prove risk bounds for the estimator in the case of Hölder classes and for Sobolev classes. As in the univariate case, adaptation costs a logarithmic loss in the rates. In Section 4, we provide global (upper and lower) MISE bounds and describe an adaptive estimator, which is studied both on Nilkols'kii (see [START_REF] Nikol'skiȋ | Approximation of functions of several variables and imbedding theorems[END_REF] and [START_REF] Kerkyacharian | Nonlinear estimation in anisotropic multi-index denoising[END_REF]) classes and for Sobolev densities. Here, it is possible that adaptation has no price and that the rate corresponds exactly to the optimal one found without adaptation. We provide in Section 5 illustrations and examples in dimension 2, for models having possibly very different behavior in the two directions. We give results of a small Monte-Carlo study, obtained by clever use of IFFT to speed the programs. Up to our knowledge, these effective experiments are the first ones in such a general setting. In a concluding Section 6, we pave the way for a generalization of the method to the case where the known noise density is replaced by an estimation based on a preliminary sample. To finish, all proofs are gathered in Section 7.

2. Model, estimator and assumptions.

2.1. Model and notations. We consider the following d-dimensional convolution model ( 1)

Y i =    Y i,1 . . . Y i,d    = X i + ε i =    X i,1 . . . X i,d    +    ε i,1 . . . ε i,d    , i = 1, . . . , n.
We assume that the ε i and the X i are i.i.d. and the two sequences are independent. Only the Y i 's are observed and our aim is to estimate the density f of X 1 when the density f ε of ε is known.

As far as possible, we shall denote by x variables in the time domain and by t or u variables in the frequency domain. We denote by g * the Fourier transform of an integrable function g, g * (t) = e i t,x g(x)dx where t, x = d j=1 t j x j is the standard scalar product in R d . Moreover the convolution product of two functions g 1 and g 2 is denoted by

g 1 ⋆ g 2 (x) = g 1 (x -u)g 2 (u)du. We recall that (g 1 ⋆ g 2 ) * = g * 1 g * 2 .
As usual, we define

g 1 = |g(x)|dx and g = g 2 = |g(x)| 2 dx 1/2 . The notation x + means max(x, 0), and a ≤ b for a, b ∈ R d means a 1 ≤ b 1 , . . . , a d ≤ b d . For two functions u, v, we denote u(x) v(x) if there exists a positive constant C not depending on x such that u(x) ≤ Cv(x) and u(x) ≈ v(x) if u(x) v(x) and v(x) u(x).
2.2. The estimator. Let us now define our collection of estimators. Let K be a kernel in L 2 (R d ) such that K * exists. Then we define, for h

∈ (R * + ) d , K h (x) = 1 h 1 . . . h d K x 1 h 1 , . . . , x d h d and L * (h) (t) = K * h (t) f * ε (t)
.

The kernel K is such that Fourier inversion can be applied:

L (h) (x) = (2π) -d e -i t,x K * h (t)/f * ε (t)dt, if f * ε (t) = 0. Considering that f Y = f ⋆ f ε and thus f * = f * Y /f * ε , a natural estimator of f is such that f * h (t) = f * Y (t)L * (h) (t) = K * h (t) f * Y (t) f * ε (t)
, where f *

Y (t) = 1 n n k=1 e i t,Y k ,
provided that f * ε does not vanish, and thus, by Fourier inversion,

fh (x) = 1 n n k=1 L (h) (x -Y k ).
Note that our estimator here is the same, in multivariate context, as the one proposed in onedimensional setting by [START_REF] Fan | On the optimal rates of convergence for nonparametric deconvolution problems[END_REF]. It verifies

E( f * h (t)) = K * h (t) f * Y (t) f * ε (t) = K * h (t)f * (t) so that E( fh ) = K h ⋆ f =: f h .
To construct an adaptive estimator, we also introduce auxiliary estimators involving two kernels. This idea, already used in [START_REF] Devroye | The double kernel method in density estimation[END_REF], allows us in the following to automatically select the bandwidth h (see section 3.4), following a method described in [START_REF] Goldenshluger | Bandwidth selection in kernel density estimation: oracle inequalities and adaptive minimax optimality[END_REF]. We consider fh,h

′ (x) = K h ′ ⋆ fh (x),
which implies that

f * h,h ′ (t) = K * h ′ (t)K * h (t) f * Y (t) f * ε (t)
.

Note that, for all x ∈ R d , we have fh,h ′ (x) = fh ′ ,h (x). The estimator which is finally studied is fĥ where ĥ is defined by using the collection ( fh,h ′ ).

2.3. Noise assumptions. We assume that the characteristic function of the noise has a polynomial or exponential decrease:

(H ε ) ∃α ∈ (R + ) d , ρ ∈ (R + ) d , β ∈ R d (β j > 0 if ρ j = 0) s. t. ∀t ∈ R d , |f * ε (t)| ≈ d j=1 (t 2 j + 1) -β j /2 exp(-α j |t j | ρ j ).
Note that this assumption implies f * ε (t) = 0, ∀t ∈ R d . A component j of the noise is said to be ordinary smooth (OS) if α j = 0 or ρ j = 0 and super smooth (SS) otherwise. We take the convention that α j = 0 if ρ j = 0 and ρ j = 0 if α j = 0.

Let us recall that exponential or gamma type densities are ordinary smooth, and that Cauchy or Gaussian densities are super smooth. The Gaussian case is considered in many problems and enhances the interest of super smooth contexts. But exponential-type densities keep a great interest in physical contexts, see for instance the fluorescence model studied in [START_REF] Comte | Adaptive density estimation in the pile-up model involving measurement errors[END_REF] where the measurement error density is fitted as an exponential type distribution, belonging to the ordinary smooth class.

To be more precise, we introduce the following notation. We denote by OS the set of directions j with ordinary smooth regularity (α j = ρ j = 0), and by SS the set of directions j with super smooth regularity (ρ j > 0) so that under (H ε ),

|f * ε (t)| ≈ j∈OS (t 2 j + 1) -β j /2 k∈SS (t 2 k + 1) -β k /2 exp(-α k |t k | ρ k ).
2.4. Regularity assumptions. We consider in the sequel several types of regularity for the target function f , associated with slightly different definition of the estimator: the choice of the kernel depends on the type of regularity space. We used Greek letters for the noise regularity, and now, we use Latin letters for the function f regularity indexes. First, for pointwise estimation purpose, we consider functions f belonging to Hölder classes denoted by H(b, L), b = (b 1 , . . . , b d ) such that: the function f admits derivatives with respect to x j up to order ⌊b j ⌋ (where ⌊b j ⌋ denotes the largest integer less than b j ) and

∂ ⌊b j ⌋ f (∂x j ) ⌊b j ⌋ (x 1 , . . . , x j-1 , x ′ j , x j+1 , . . . , x d ) - ∂ ⌊b j ⌋ f (∂x j ) ⌊b j ⌋ (x) ≤ L|x ′ j -x j | b j -⌊b j ⌋ .
Next for global estimation purpose, the functional spaces associated with standard kernel estimators are the anisotropic Nikol'skii class of functions, as in [START_REF] Goldenshluger | Uniform bounds for norms of sums of independent random functions[END_REF], see also [START_REF] Nikol'skiȋ | Approximation of functions of several variables and imbedding theorems[END_REF], [START_REF] Kerkyacharian | Nonlinear estimation in anisotropic multi-index denoising[END_REF]. We consider the class N (b, L) which is the set of functions f : R d → R such that f admits derivatives with respect to x j up to order ⌊b j ⌋, and

(i)
∂ ⌊b j ⌋ f (∂x j ) ⌊b j ⌋ ≤ L, for all j = 1, . . . , d, where . denotes the L 2 (R d )-norm. (ii) For all j = 1, . . . , d, for all t ∈ R,

∂ ⌊b j ⌋ f (∂x j ) ⌊b j ⌋ (x 1 , . . . , x j-1 , x j + y, x j+1 , . . . , x d ) - ∂ ⌊b j ⌋ f (∂x j ) ⌊b j ⌋ (x) 2 dx ≤ L 2 |y| 2(b j -⌊b j ⌋) .
Lastly, and for both pointwise and global estimation, we shall consider general anisotropic Sobolev spaces S(b, a, r, L) defined as the class of integrable functions f :

R d → R satisfying d j=1 |f * (t 1 , . . . , t d )| 2 (1 + t 2 j ) b j exp(2a j |t j | r j )dt 1 . . . dt d ≤ L 2 ,
for a j ≥ 0, r j ≥ 0, b j ∈ R, when j = 1, . . . , d. We set a j = 0 if r j = 0, and reciprocally, and in this case, b j > 1/2 (otherwise b j ∈ R). If some a j are nonzero, the corresponding directions are associated with so-called "super smooth" regularities. To standardize notations, we set a j = r j = 0 when Hölder or Nikol'skii regularity is considered. We refer to [START_REF] Triebel | Theory of function spaces[END_REF] for definitions and comparison of these spaces with other type of anisotropic regularity spaces such as Besov spaces.

We can note that Sobolev spaces allow one to take into account a global regularity rather than a pointwise one. Nevertheless, they have a convenient Fourier-domain representation, in particular when one wants to consider super smooth or analytical functions, even in pointwise setting. If the noise density can have such property in the case of Gaussian measurement error, it is natural to think that the signal density may have the same behavior.

2.5. Assumptions on the kernel. For the estimators to be correctly defined, the kernel must be chosen sufficiently regular to recover integrability in spite of the noise density.

We assume that K(x) = K(x 1 , . . . , x d ) = d j=1 K j (x j ). This assumption is not necessary, but simplifies the proofs. Besides, the kernels used in practice verify this condition. Moreover, we recall that K belongs to L 2 (R d ) and admits a Fourier transform.

To ensure the finiteness of the estimators, we shall use the following assumption:

Kvar(β) For j ∈ OS: |K * j (u)| 2 (1 + u 2 ) β j du < ∞ and |K * j (u)|(1 + u 2 ) β j /2 du < ∞ For j ∈ SS: K * j (t) = 0 if |t| > 1 and sup |t|≤1 |K * j (t)| < ∞
Moreover, we may require a classical assumption to control the bias for functions in Hölder or Nikol'skii spaces described above.

Korder(ℓ) The kernel K is of order ℓ = (ℓ 1 , . . . , ℓ d ) ∈ R d + , i.e. * K(x)dx = 1 * ∀1 ≤ j ≤ d, ∀1 ≤ k ≤ ℓ j , x k j K(x)dx = 0 * ∀1 ≤ j ≤ d, (1 + |x j |) ℓ j |K(x)|dx < ∞
Note that this implies condition (A2) used in [START_REF] Fan | On the optimal rates of convergence for nonparametric deconvolution problems[END_REF] which is stated in the Fourier domain. Condition Korder(ℓ) is verified by the following kernels defined in [START_REF] Goldenshluger | Uniform bounds for norms of sums of independent random functions[END_REF]. We start by defining univariate functions u j (x) such that u j (x)dx = 1, |x| ℓ j |u j (x)|dx < +∞ and then

(2)

K j (x j ) = ℓ j k=1 ℓ j k (-1) k+1 1 k u j ( x j k ).
Then K j is a univariate kernel of order ℓ j . The multivariate kernel is defined by

(3) K(x) = K(x 1 , . . . , x d ) = d j=1 K j (x j ).
The resulting kernel is such that

d j=1 x k j j K(x)dx 1 . . . dx d = 0 if 1 ≤ k j ≤ ℓ j
for one j ∈ {1, . . . , d}, and thus satisfies Korder(ℓ). We can give an example of kernel satisfying Assumptions Kvar(β) and Korder(ℓ). We can use the construction above with u j (x j ) = v ℓ j +2 (x j ) where

v p (x) = c p sin(x/p) x/p p , v p (0) = c p , v * p (t) = 2πpc p 2 p 1 [-1,1] ⋆ • • • ⋆ 1 [-1,1] p times (pt),
and c p is such that v p (x)dx = 1. This is what can be done when the function under estimation is assumed to be in a Hölder or in a Nikol'skii space.

When considering Sobolev space, since Assumption Kvar(β) only is required, we simply use the sinus cardinal kernel denoted by K = sinc and defined by

K * j (t) = 1 [-1,1] (t) = v * 1 (t), K j (x j ) = sin(x j ) πx j , K j (0) = 1 π .
Remark. When only ordinary smooth noises are considered on Hölder or Nikol'skii spaces, we may also use other type of kernels. For instance, the construction of kernel of order ℓ based on u j (x j ) = c j x j -1 2

⌊β j ⌋+1
x j + 1 2

⌊β j ⌋+1 1 [-1 2 , 1 2 ] (x j )
would suit. Indeed, it can be proved that K * j (t j ) = O(|t j | -(⌊β j ⌋+2) ) when |t j | → +∞.

Pointwise estimation

3.1. Bias and variance. Let x 0 be a point in R d . We aim to study the risk of the estimator fh of f at point

x 0 : |f (x 0 ) -fh (x 0 )|. Recall that f h = E( fh ) = K h ⋆ f and that E|f (x 0 ) -fh (x 0 )| 2 = |f (x 0 ) -f h (x 0 )| 2 bias + E|f h (x 0 ) -fh (x 0 )| 2 variance .
We first control the bias. We define

B 0 (h) = f -f h ∞ if K 1 < ∞ f * -f * h 1 /(2π) d otherwise
We recall that, when considering all types of spaces (Hölder and Sobolev), we standardized notations by setting a j = r j = 0 when Hölder regularity is considered. The following proposition holds.

Proposition 1. The bias verifies |f (x 0 )f h (x 0 )| ≤ B 0 (h) and, under assumptions

• f belongs to Hölder class H(b, L) and the kernel verifies Korder(ℓ) with ℓ ≥ ⌊b⌋, or • f * ∈ L 1 (R), f belongs to Sobolev class S(b + 1/2, a, r, L) and K = sinc,

Then B 0 (h) L d j=1 h b j +r j /2 j exp(-a j h -r j j ).
Thus, we recover the classical order h b j j when a j = 0. Let us now study the variance of estimators fh .

Proposition 2. The variance verifies E|f

h (x 0 ) -fh (x 0 )| 2 ≤ V 0 (h) where (4) V 0 (h) = 1 (2π) 2d 1 n min f * ε 1 K * h f * ε 2 2 , K * h f * ε 2 1 . Moreover, under (H ε ) and Kvar(β), if h j ≤ 1 for all j, V 0 (h) 1 n d j=1 h (ρ j -1) + +ρ j -1-2β j j exp(2α j h -ρ j j ).
When f * ε = 1 (no noise), we obtain the classical order j 1/(nh j ). Eventually, the bound on the MSE is obtained by adding the squared bias bound and the variance bound.

Rates of convergence.

3.2.1. Homogeneous cases. We first give the bandwidth choices and rates of convergence which are obtained when all components of both f and f ε have the same type of smoothness (all OS or all SS). Recall that in dimension 1, the minimax rates are logarithmic when the noise is super smooth, unless the function f is super smooth too : see [START_REF] Fan | On the optimal rates of convergence for nonparametric deconvolution problems[END_REF], [START_REF] Pensky | Adaptive wavelet estimator for nonparametric density deconvolution[END_REF], [START_REF] Comte | Penalized contrast estimator for adaptive density deconvolution[END_REF].

First, consider that both the function f and the noise are ordinary smooth. We can compute the anisotropic rate that can be deduced from a "good" choice of h = (h 1 , . . . , h d ). Indeed, setting the gradient of

h 2b 1 1 + • • • + h 2b d d + n -1 d i=1 h -(2β i +1) i
w.r.t. h to zero, we easily obtain h 2b j j,opt = h 2b k k,opt . Therefore the optimal bandwidth choice to minimize the risk bound is

(5) h j,opt ∝ n -1/(2b j +b j d i=1 [(2β i +1)/b i ])
and the resulting rate is proportional to (6)

ψ n = n -1/(1+ 1 2 d i=1 2β i +1 b i
) .

Secondly, consider the case where the noise is super smooth (all (β j , ρ j ) nonzero) but the function is ordinary smooth. Then h j,opt = ((2α j + 1)/ log(n)) 1/ρ j and the rate is of order ( 7)

ψ n = [log(n)] -2 min 1≤j≤d (b j /ρ j ) .
We can remark two things in this case: the rates are logarithmic, and the bandwidth choice is known because it only depends on the parameters of the noise density, which is assumed to be known. This explains why no bandwidth selection procedure is required here, as long as only classical Hölder regularities are considered for f . Now consider the case where the noise is ordinary smooth (all ρ j 's are zeros) but the function is super smooth (with all (a j , r j ) nonzero). Then we take h j,opt = (a j / log(n)) 1/r j and the rate is (8)

ψ n = [log(n)] d j=1 (2β j +1)/r j /n.
We can see that here, the rates are very good. It is worth mentioning that the first paper considering super smooth function f is [START_REF] Pensky | Adaptive wavelet estimator for nonparametric density deconvolution[END_REF].

We do not give a general bandwidth choice in the case where both functions can be super smooth, because it is very intricate. General formula in dimension 1 are given in [START_REF] Lacour | Rates of convergence for nonparametric deconvolution[END_REF], see also Butucea and Tsybakov [2008a,b]. We can just emphasize that in such case the rates can be considerably improved, compared to the logarithmic issue above. We give an example below.

Super Smooth f /Super Smooth f ε example. For instance, it is easy to see that the compromise between a bias of order exp(-1/h 2 ) and a variance of order exp(1/h 2 )/n is obtained for h = 2/ log(n) and gives a rate of order 1/ √ n. To be even more precise, the optimal rate in dimension 1, if the signal is N (0, σ 2 ) and the noise N (0,

σ 2 ε ), is n -1/(1+θ 2 ) [log(n)] -(1+1/(1+θ 2 ))/2 , θ 2 = σ 2 ε /σ 2 , for 1/h opt = [log(n) + (1/2) log(log(n))]/(σ 2 + σ 2 ε ).
As the bandwidth choice is very difficult to describe in the general case, this enhances the interest of automatic adaptation which is proposed below, when Sobolev spaces are considered. Note that optimal choices of the bandwidth are of logarithmic orders in all those cases. 3.2.2. Discussion about mixed cases. Let us consider now the case where the function is still ordinary smooth, but components 1 to j 0 of the noise are ordinary smooth while components j 0 + 1 to d are super smooth, 1 ≤ j 0 < d. Then it is clear that exponential components must first be "killed" by choosing logarithmic bandwidths and as the bandwidths are involved additively in the bias term, the rate becomes logarithmic. More precisely, taking for j = 1, . . . , j 0 , h j,opt ∝ n -1/(2d(2β j +1)) and for j = j 0 + 1, . . . , d, h j,opt = [log(n)/(4dα j )] -1/ρ j gives a variance term of order

n -1 j 0 j=1 h -2β j -1 j,opt d j=j 0 +1 h (ρ j -1) + +(ρ j -1)-2β j j,opt exp(2α j h -ρ j j,opt ) ∝ n -1+j 0 /(2d)+(d-j 0 )/(2d) log ω (n) = n -1/2 log ω (n),
where ω = d j=j 0 +1 (2β j + 1ρ j -(ρ j -1) + )/ρ j . Therefore, the variance is negligible and the rate is determined by the bias terms and is proportional to

(9) ψ n = [log(n)] -2 min j 0 +1≤j≤d (b j /ρ j ) .
The conclusion is that the presence of one super smooth component of the noise implies a logarithmic rate, when the function to estimate is ordinary smooth (and bandwidth selection is not required).

The other case we can handle is when the noise has all its components ordinary smooth, but the function has its j 0 first components ordinary smooth and the dj 0 last ones super smooth. Let us take d = 2 and j 0 = 1 for simplicity. Clearly, we can choose h 2,opt = (log(n)/a 2 ) -1/r 2 , so that the MSE for (h 1,opt , h 2,opt ) is proportional to

h 2b 1 1,opt + h 2b 2 +r 2 2,opt exp(-2a 2 h -r 2 2,opt ) + n -1 h -2β 1 -1 1,opt h -2β 2 -1 2,opt ∝ h 2b 1 1,opt + n -2 [log(n)] -(2b 2 +r 2 )/r 2 + n -1 h -2β 1 -1 1,opt [log(n)] (2β 2 +1)/r 2 .
Therefore, the optimal choice of h 1 is obtained as in dimension 1 with respect to a sample size n/[log(n)] (2β 2 +1)/r 2 and we find ) . This is the rate corresponding to the one-dimensional problem, up to a logarithmic factor. In the general case, we obtain a rate proportional to (10)

h 1,opt ∝ (n/[log(n)] (2β 2 +1)/r 2 ) -1/(2β 1 +2b 1 +1) . The final rate is proportional to (n/[log(n)] (2β 2 +1)/r 2 ) -2b 1 /(2β 1 +2b 1 +1
ψ n =   n/ d j=j 0 +1 (log n) 2β j +1 r j   -1/(1+ 1 2 j 0 i=1 2β i +1 b i )
In other words, we obtain in dimension d, the rate corresponding to dimension j 0 of the OS-OS problem, up to logarithmic factors.

Lower bounds.

To get a validation of our method, we need to prove lower bounds for the rates computed above, at least in part of the cases. In particular, we can extend the results of [START_REF] Fan | On the optimal rates of convergence for nonparametric deconvolution problems[END_REF] and of Butucea and Tsybakov [2008b] to the multivariate setting. Our next result is not straightforward and requires specific constructions, since it captures mixed cases which could not be encountered in univariate setting.

Theorem 1. We assume that the noise has its components independent. We also assume that, for j = 1, . . . , d, and for almost all u j in R, f * ε 1,j (u j ) admits a derivative and

(11) |u j | β ′ j exp(α j |u j | ρ j )|(f * ε 1,j ) ′ (u j )| is bounded, for a constant β ′ j such that β ′ j > β j if ε 1,j is OS.

Moreover, either

Case A: For j = 1, . . . , d, the components ε j are ordinary smooth, D = H(b, L) or D = S(b + 1/2, a, r, L) with r j < 2, and if 1 ≤ r j < 2, f * ε 1,j (u j ) admits in addition a second order derivative for almost all u j in R such that

|u j | β ′′ j exp(α j |u j | ρ j )|(f * ε 1,j ) ′′ (u j )| is bounded, with β ′′ j a positive constant.

or

Case B: There exists at least one component of ε which is super smooth and D = H(b, L) or D = S(b + 1/2, 0, 0, L).

Then for any estimator fn (x 0 ), and for n large enough,

sup f ∈D E f ( fn (x 0 ) -f (x 0 )) 2 ψ n
where ψ n is defined by ( 6) in Case A and D = H(b, L), by ( 10) in Case A and D = S(b + 1/2, a, r, L) with all r j 's less than 2, and by ( 9) in Case B.

Note that our condition on the noise improves [START_REF] Fan | On the optimal rates of convergence for nonparametric deconvolution problems[END_REF]'s conditions: in the OS case, Fan requires a second order derivative of f * ε and in the SS case, he gives a technical condition which is difficult to link with the functions at hand. The improvement took inspiration in the book of [START_REF] Meister | Deconvolution problems in nonparametric statistics[END_REF] who also had first order type conditions.

We therefore conclude that the rates reached by our estimators for estimating an ordinary smooth function or a super smooth function if the noise is ordinary smooth, are optimal. We also have optimality in the case of an ordinary smooth function f and super smooth noise.

3.4. Adaptive estimator. Now, our aim is to automatically select a bandwidth in a discrete set H 0 (described below) such that the corresponding estimator reaches the minimax rate, without knowing the regularity of f . We may also ignore if f is ordinary or super smooth, or partially both, depending on the direction.

3.4.1. General result. We have at our disposal estimators fh (x 0 ) and fh,h

′ (x 0 ) = K h ′ ⋆ fh (x 0 ), for x 0 = (x 0,1 , . . . , x 0,d ) ∈ R d such that fh,h ′ (x 0 ) = fh ′ ,h (x 0 ). We define (12) A 0 (h, x 0 ) = sup h ′ ∈H 0 | fh ′ (x 0 ) -fh,h ′ (x 0 )| -Ṽ0 (h ′ ) + , and
ĥ(x 0 ) = arg min h∈H 0 A 0 (h, x 0 ) + Ṽ0 (h) with (13) Ṽ0 (h) = c 0 log(n)V 0 (h)
and c 0 is a numerical constant to be specified later. The final estimator is f (x 0 ) = fĥ (x 0 ) (x 0 ).

The term Ṽ0 (h) corresponds to the variance of the estimate fh (x 0 ) multiplied by log(n). Now, we can state the result concerning the adaptive estimator. Define

N (K) = K 1 if K 1 < ∞ K * ∞ otherwise Theorem 2. Assume that N (K) < ∞ and let H 0 = {h (k) s.t. h (k) j ≤ 1, for j = 1, . . . , d, V 0 (h (k) ) ≤ 1, K * h (k) f * ε 2 2 K * h (k) f * ε -2 1 ≥ log(n) n for k = 1, . . . , ⌊n ǫ ⌋}. (14)
Let q be a real larger than 1. Assume that c 0 ≥ (4

(1 + K * ∞ )(2ǫ + q)) 2 / min ( f * ε 1 , 1) . Then, with probability larger than 1 -4n -q , (15) | f (x 0 ) -f (x 0 )| ≤ inf h∈H 0 (1 + 2N (K))B 0 (h) + 3 Ṽ0 (h) .
We can make two comments about this result.

(1) Inequality ( 15) is a trajectorial oracle inequality, up to the log(n) factor in the term Ṽ0 (h) which appears in place of V 0 (h).

(2) Condition ( 14) is typically verified if

K * h /f * ε 2 2 ≥ log(n) and max( K * h /f * ε 2 2 , K * h /f * ε 2 1 ) ≤ n.
It is just slightly stronger than assuming the variance V 0 (h) bounded. It is also important to see that we can deduce from Theorem 2 a mean oracle inequality. More precisely, we have

| f (x 0 ) -f (x 0 )| ≤ ( K * h /f * ε 1 + |f (x 0 )|). Then, for h ∈ H 0 , K * h /f * ε 2 1 ≤ (n/ log(n)) K * h /f * ε 2 2 and V 0 (h) ≤ 1 imply K * h /f * ε 2 1 ≤ n. Thus | f (x 0 )-f (x 0 )| 2 n. Therefore, Theorem 2 implies that, ∀h ∈ H 0 , (16) E(| f (x 0 ) -f (x 0 )| 2 ) ≤ (1 + 2N (K))B 0 (h) + 3 Ṽ0 (h) 2 + C n ,
provided that we choose q ≥ 2 in Theorem 2.

Study of Condition (14).

Let us define hopt = ( h1,opt , . . . , hd,opt ) the minimizer of the right hand side of equation ( 16):

hopt = arg min h∈R d + B 2 0 (h) + Ṽ0 (h) .
Note that hopt here corresponds to the value of h opt computed in Section 3.2 where n is replaced by n/ log(n). We need to check that hopt belongs to H 0 to ensure that the infimum in ( 15) is reached. This is what is stated in the following Corollary.

Corollary 1. Assume that (H ε ) holds and either 1. f belongs to Hölder class H(b, L), the noise has all its components OS and the kernel verifies Korder(ℓ) with ℓ ≥ ⌊b⌋, Kvar(β), and is such that K * j is lower bounded on [-q j , q j ] for q j > 0, and j

= 1 . . . , d, or 2. f * ∈ L 1 (R), f belongs to Sobolev class S(b + 1/2, a, r, L) and K = sinc.
Then hopt ∈ H 0 defined by ( 14) and thus the infimum in Inequality ( 15) is reached.

In particular in case 1., we have

(17) E(| f (x 0 ) -f (x 0 )| 2 ) = O((n/ log(n)) -1/(1+ 1 2 d i=1 2β i +1 b i
) ).

We can notice that the proof of Corollary 1 relies on the intermediate result stating that Condition ( 14) is equivalent to the following one:

(18) d j=1 h (ρ j -1) j n/ log(n).
The consequence of Corollary 1 is that the right hand side of (15) always leads to the best compromise between the squared bias B 2 0 (h) and Ṽ0 (h), that is the optimal rates of section 3.2 with respect to a sample size n/ log(n).

Remark 1. As we already mentioned, we have an extra log(n) factor in Inequality (15). In case 1. above, we can concretely see the loss in the rate by comparing the right-hand-side of ( 17) to the optimal rate (6). This logarithmic loss, due to adaptation, is known to be nevertheless adaptive optimal for d = 1, see Butucea and Tsybakov [2008a,b] and [START_REF] Butucea | Adaptive estimation of linear functionals in the convolution model and applications[END_REF], and we can conjecture that it is also the case for larger dimension.

Remark 2. In the case of a noise having super smooth components and of a function f known to belong to an Hölder space, we already mentioned that no bandwidth selection is required. Indeed, we just have to take h j = (log(n)/2α j ) -1/ρ j for the super smooth components and h j = n -1/(2d(2β j +1)) for ordinary smooth components, and the rate has a logarithmic order determined by the bias term, see ( 9). This is the reason why general adaptation is studied only on Sobolev spaces. The rates can be then considerably improved compared to the rate (9).

Global estimation

Here, we study the procedure for global estimation. In this section we assume that f belongs to L 2 (R d ).

Bias and variance.

We study now the MISE E f -fh 2 , made up of a bias term plus a variance term. We can prove the following bound for the bias.

then f -f h L d j=1 h b j j exp(-a j h -r j j ).
Let us now bound the variance of the estimator.

Proposition 4. We have

E f h -fh 2 ≤ V (h) where V (h) = 1 (2π) d n K * h f * ε 2 .
Moreover, under (H ε ) and Kvar(β)

V (h) 1 n d j=1 h -1-2β j +ρ j j exp(2α j h -ρ j j ).
We emphasize that the rates of convergence ( 6), ( 7) and ( 8) are formally preserved here, for the same optimal bandwidth choices, but with a definition of the parameters b j which is different (in case 2. here, f belongs to S(b, a, r, L) while in the pointwise setting it was chosen in S(b + 1/2, a, r, L)). Therefore, we refer to section 3.2 for all remarks concerning the quality of the rates and to the cases where part of the components of f or f ε are ordinary smooth and others are super smooth.

Lower bounds corresponding to the integrated risk can be obtained, through non straightforward extensions of the pointwise case. Thus, we get the following result.

Theorem 3. Consider either Case A with D = S(b, a, r, L) and all r j 's less than 2 or Case B with D = S(b, 0, 0, L) as described in Theorem 1, still under the general assumption that the noise has its components independent and fulfill (11). Then for any estimator fn , and for n large enough,

sup f ∈D E f fn -f 2 ψ n
where ψ n is defined by ( 6) in Case A when r = a = 0, by ( 10) in general Case A where D = S(b, a, r, L) with all r j 's less than 2, and by ( 9) in Case B.

Next, we study when these rates can be reached adaptively.

4.2. The global adaptive estimator. Here, we describe the adaptive estimation. As previously, we define

A(h) = sup h ′ ∈H fh ′ -fh,h ′ -Ṽ (h ′ ) + ,
and ĥ = arg min

h∈H A(h) + Ṽ (h) with Ṽ (h) defined by (19) Ṽ (h) = (1 + K * ∞ ) 2 (1 + 2η) 2 V (h)C(h)
where η is a numerical constant and C(h) ≥ 1 is a correcting term discussed below. Ideally, this term would be a constant but in super smooth cases, this may not be possible. The final estimator is f = fĥ .

We give first an adaptive trajectorial result in term of a general constraint on C(h).

Theorem 4. Assume that K * ∞ < ∞ and let

H = {h (k) s.t. h (k) j ≤ 1, for j = 1, . . . , d, V (h (k) ) ≤ 1, C(h) max(1, K * h /f * ε 2 2 / K * h /f * ε 2 ∞ ) ≥ (log n) 2 for k = 1, . . . , ⌊n ǫ ⌋}. (20)
Then, with probability larger than 1n ǫ e -[min(η,1)η/46](log n) 2

(21) f -f ≤ inf h∈H (1 + 2 K * ∞ ) f -f h + 3 Ṽ (h) .
Remark 3. Clearly, asymptotically when n gets large, ∀ǫ > 0, n ǫ e -[min(η,1)η/46](log n) 2 = O(1/n -q ) for any integer q. But in practice, the cardinality ⌊n ǫ ⌋ of H should not be too large.

Note that, as in the pointwise setting, we can write

f -f ≤ f + f ≤ f + nV ( ĥ) ≤ f + √ n
as ĥ is chosen in H. Therefore, Inequality (21) implies that

(22) E( f -f 2 ) ≤ inf h∈H (1 + 2 K * ∞ ) f -f h + 3 Ṽ (h) 2 + C 2 (η) n .
Now we can study condition (20) in our usual specific settings. Let us define ȟopt as the optimal bandwidth choice: ȟopt = arg min

h∈R d + { f -f h 2 + Ṽ (h)}.
As in the pointwise setting, the optimal compromise is automatically reached by the estimator if ȟopt belongs to H; but contrary to the pointwise setting, we may preserve a rate without loss if C(h) can be taken equal to a constant. We can prove the following result.

Corollary 2. Assume that (H ε ) holds, that the noise has all its components OS and either 1. f belongs to Nikol'skii class N (b, L), and

K verifies Kvar(β), 0 < sup u j ∈R |K * j (u)|u 2β j < ∞ for j = 1, . . . , d, Korder(ℓ) with ℓ ≥ ⌊b⌋, or 2. f * ∈ L 1 (R), f belongs to a Sobolev class S(b, 0, 0, L) and K = sinc.
Then, we can take C(h) = 1 and we have ȟopt ∈ H (where H as defined in Theorem 4). Thus, the infimum in Inequalities ( 21) and ( 22) are reached. That is, we have

(23) E( f -f 2 ) = O(n -1/(1+ 1 2 d i=1 2β i +1 b i ) ).
Clearly in the case of ordinary smooth noise and function f , the estimator automatically reaches the optimal rate, without requiring the knowledge of the regularity of f , which is nevertheless involved in the resulting rate.

If we want to use constraint (20) in the general setting, we have to choose C(h) = log 2 (n) and then, a systematic loss occurs:

Corollary 3. Assume that (H ε ) holds, that f * ∈ L 1 (R), f belongs to Sobolev class S(b, a, r, L) and K = sinc. Take C(h) = log 2 (n).
Then ȟopt ∈ H and the infimum in Inequalities ( 21) and ( 22) are reached.

Nevertheless, if H is more precisely specified, we can prove a better result in expectation:

Theorem 5. Assume that (H ε ) holds, that f * ∈ L 1 (R), f belongs to Sobolev class S(b, a, r, L) and K = sinc. Define now for M given, M ≤ n,

H M = {h (k) s.t. h (k) j = 1 k , j = 1, . . . , d, k = 1, . . . , M, with V (h (k) ) ≤ 1}. Choose (24) C(h) = 1 + d j=1 h -2ρ j 1 ρ j ≥1/2 j 1 ρ j ≥1/2 .
Then choose M such that ȟopt ∈ H M (M = n always suits). Then we have

(25) E( f -f ) ≤ 3 inf h∈H M f -f h + Ṽ (h) + C 2 √ n . Remark 4. By ȟopt ∈ H M , we mean that 1/[1/ ȟopt ] ∈ H where [x]
denotes the integer part of x. In the formulation above, the infimum in ( 25) is necessarily reached.

The exact choice instead of ( 24) is the following

(26) C(h) = d j=1 ω j h -(2ρ j -1) + +(ρ j -1) + j
for constants ω j depending on α j , β j , ρ j that can be specified (see Section 7.11 in Appendix).

Let us discuss the possible loss in the rate of convergence of the estimator resulting from the choice (24) of C(h) and Inequality ( 25).

(1) If f ε is ordinary smooth, equation ( 24) says that C(h) = 1 and therefore, as ȟopt belongs to H, the optimal rate (n

-1/(1+ 1 2 d i=1 2β i +1 b i ) or [log(n)] d j=1 (2β j +1)/r j /n
) is automatically reached by the estimator.

(2) If f ε is super smooth, equation ( 24) says that the variance term has to be slightly increased.

(a) Nevertheless, if the function f is ordinary smooth, the minimization in ( 22) still yields to the optimal rate. Indeed, in that case the variance is made negligible with respect to the bias by the optimal bandwidth choice (see the computations in Section 3.2). (b) When f is also super smooth, if all ρ j 's are less than 1/2, then there is no loss.

Otherwise, the optimal bandwidth choice is such that, in part of the cases, the bias is dominating, and then there is still no loss. When some of the ρ j 's are larger than 1/2 and the variance is dominating, there is a loss. But as the selected bandwidths have logarithmic orders in the concerned cases, the rates are deteriorated in a negligible way and less than if they were computed with respect to a sample size n/[log(n)] 2 max j ρ j instead of n. In other words, the loss is always negligible with respect to the rate.

Numerical illustration

5.1. Implementation. The theoretical study shows the advantages of the kernel sinc. It has also good properties for practical purposes, since it allows to use Fast Fourier Transform. Thus we consider in this section, in the case d = 2, the kernel K(x, y) = sinc(x)sinc(y)/π 2 . Let us denote ϕ h,j (x

) = π/ √ h 1 h 2 K(x 1 /h 1 -πj 1 , x 2 /h 2 -πj 2 ).
The main trick used here follows from model selection works on deconvolution (see [START_REF] Comte | Penalized contrast estimator for adaptive density deconvolution[END_REF] and [START_REF] Comte | Data-driven density estimation in the presence of additive noise with unknown distribution[END_REF]). It is shown therein that (ϕ h,j ) j∈Z 2 is an orthonormal basis of the space of integrable functions having a Fourier transform with compact support included into

[-1/h 1 , 1/h 1 ] × [-1/h 2 , 1/h 2 ].
Then fh can be written in this basis: fh = j âh j ϕ h,j with

âh j = 1 4π 2 f * h ϕ * h,j = √ h 1 h 2 4π 1/h 1 -1/h 1 1/h 2 -1/h 2 f * Y f * ε (u 1 , u 2 )e iπ(u 1 h 1 j 1 +u 2 h 2 j 2 ) du 1 du 2 .
The interesting point is here that such coefficients can be computed via Fast Fourier Transform. So we implement our estimator in the following way

fh = |j 1 |≤M |j 2 |≤M âh j ϕ h,j
with M = 64. Moreover, we use that with cardinal sine kernel, we have

f h,h ′ = f h∨h ′ , by denoting h ∨ h ′ = (max(h 1 , h ′ 1 ), max(h 2 , h ′ 2 )
). Then in the pointwise setting, we compute A 0 (h, x 0 ) as given by (12) with Ṽ0 (h) given by ( 13) and c 0 = 0.01. Thus, the plots of the selected estimators fĥ (x 0 ) (x 0 ) are given on a grid of points x 0 in a domain which is specified in each example.

In the global setting, we can exploit additional useful properties of the representation. Indeed, for all h ′ , h ′′ , with

fh ′ -fh ′′ 2 = 1 4π 2 f * h ′ -f * h ′′ 2 = 1 4π 2 f * Y f * ε 1 D h ′ - f * Y f * ε 1 D h ′′ 2 -2 0 2 4 -4 -2 0 2 4 0 0.1 0.2 -2 0 2 4 -4 -2 0 2 4 0 0.1 0.2 -2 0 2 4 -
D h = [-1/h 1 , 1/h 1 ] × [-1/h 2 , 1/h 2 ]. Then, if D h ′′ ⊂ D h ′ , fh ′ -fh ′′ 2 = 1 4π 2 D h ′ \D h ′′ f * Y f * ε 2 = 1 4π 2 D h ′ f * Y f * ε 2 - 1 4π 2 D h ′′ f * Y f * ε 2 = fh ′ 2 -fh ′′ 2 ,
where we have fh 2 = j |â h j | 2 . Then the computation of 19) is taken equal to log 2 (n) as recommended by Corollary 3. Once the bandwidth is selected in the global setting, we have the coefficients âĥ j and thus, we can plot fĥ (x, y) at any point (x, y). We take H and H 0 included in {4/m, 1 ≤ m ≤ 3n 1/4 }.

A(h) = sup h ′ ∈H fh ′ 2 -fh∨h ′ 2 -Ṽ (h ′ ) + is considerably accelerated. We choose Ṽ (h) = 0.05 log 2 (n)V (h), that is C(h) in formula (

5.2.

Examples. Now we compute estimators for different signal densities and different noises.

Let λ = 6, µ = 1/4. Example 1 Cauchy distribution: f (x, y) = (π 2 (1 + x 2 )(1 + y 2 )) -1 on [-4, 4] 2 with a Laplace/Laplace noise, i.e. f ε (x, y) = λ 2 4 e -λ|x| e -λ|y| ; f * ε (x, y) = λ 2 λ 2 + x 2 λ 2 λ 2 + y 2 The smoothness parameters are b 1 = b 2 = 0, r 1 = r 2 = 1, β 1 = β 2 = 2 and ρ 1 = ρ 2 = 0.
For this example, we can compute that the rate is of order (log(n)) 10 /n. Example 2 Mixed Gaussian distribution: X i,1 = W/ √ 7 with W ∼ 0.4N (0, 1) + 0.6N (5, 1), and X i,2 independent with distribution N (0, 1). We estimate the density on [-2, 4] 2 . We consider that the noise follows a Laplace/Gaussian distribution, i.e.

f ε (x, y) = λ 2 e -λ|x| 1 µ √ 2π e -y 2 /(2µ 2 ) ; f * ε (x, y) = λ 2 λ 2 + x 2 e -µ 2 y 2 /2
The smoothness parameters are b

1 = b 2 = 0, r 1 = r 2 = 2, β 1 = 2, β 2 = 0 and ρ 1 = 0, α 2 = µ 2 /2, ρ 2 = 2.
Here the rate of convergence is n -16/17 [log(n)] 63/34 in the global setting and n -16/17 [log(n)] 23/17 for the bandwidths h -1 Example 3 Gamma distribution: X i,1 ∼ Γ(5, 1/ √ 5) and X i,2 ∼ Γ(5, 1/ √ 5). We estimate the density on [0, 8] 2 . The noise follow a Gaussian/Gaussian distribution, i.e. For these examples, we apply both global and pointwise estimation procedure, and we compute the Mean Integrated Squared Error on a grid of 50 × 50 points. The MISE (multiplied by 100, averaged over 100 samples) is given in Table 1. For each path, we also compare the MISE for the global procedure with the minimum risk for all bandwidths of the collection. Table 2 gives the empirical version of the oracle constant defined by

1 = 7 log(n) and h -1 2 = a log(n) -b log(log(n)) for a =
f ε (x, y) = 1 2πµ 2 e -(x 2 +y 2 )/(2µ 2 ) ; f * ε (x, y) = e -µ 2 (x 2 +y 2 )/2 So b 1 = b 2 = 5, r 1 = r 2 = 0, β 1 = β 2 = 0, α 1 = α 2 = µ 2 /2 and ρ 1 = ρ 2 = 2.
C oracle = E f -f 2 inf h∈H fh -f 2 .
It shows that the adaptation is performing, since the risk for the chosen ĥ is very close to the best possible in the collection (the nearest of one C oracle , the better the algorithm).

We also illustrate the results with some figures. Figure 1 shows the surface z = f (x, y) for Example 2 and the estimated surface z = f (x, y) obtained by global bandwidth selection. For more visibility, sections of the previous surface are drawn. We can see the curves z = f (x, -0.3) versus z = f (x, -0.3) and the curves z = f (-0.3, y) versus z = f (-0.3, y). For this figure, the selected bandwidth is ĥ = (0.29, 0.57). Thus, the bandwidth in the first direction is twice smaller, to recover the two modes: this shows that our procedure takes really anisotropy into account. Figure 2 is an analogous illustration of Example 3, but with a pointwise bandwidth selection, as described in Section 3. We obtain a slightly more angular figure. Nevertheless, we can notice by observing Table 1 that the MISE is almost always smaller for this kind of estimation.

To conclude this section, we would like to mention that we can keep good results even in case of dependent components of both the noise and the signal. More precisely, we can take X ∼ N (0, Σ) and ε ∼ N (0, Σ ε ) with Σ = 1 -0.7 -0.7 2 and Σ ε = 10 -2 1 0.25 0.25 1.0625 , with X and ε independent. We present in Figure 3 an illustration of the results for the global method.

Concluding remarks: the case of unknown noise density

The assumption of the knowledge of the error distribution is often disputed. Relaxing this assumption requires conditions for obvious reasons of identifiability. Here is a quick description of what can be done in case of additional observations of the noise ε -1 , . . . , ε -N (think of a

1 f * ε (x) = 1 {| f * ε (x)|≥N -1/2 } f * ε (x) =    1 f * ε (x) if | f * ε (x)| ≥ N -1/2 , 0 otherwise, where f * ε (x) = N -1 N j=1 e -i x,ε -j is the natural estimator of f * ε . Then it is sufficient to write f * h (t) = K * h (t) f * Y (t) f * ε (t)
to define new estimators of f in this context. Adapting all the previous results in this framework is beyond the scope of this paper, but we can observe the effect of this modification on the integrated squared error, for instance. The bias is unchanged, but an additional term appears in the variance:

Proposition 5. We have E f h -fh 2 V (h) + W (h) where W (h) = 1 (2π) d N K * h f * f * ε 2 .
It is possible to give a bound of W (h) in term of the smoothness indices of f ε and f but we skip this tedious formula, which is just a generalization of Lemma 2 in [START_REF] Comte | Data-driven density estimation in the presence of additive noise with unknown distribution[END_REF]. In the case of an ordinary smooth function and a fully ordinary smooth noise, we obtain

W (h) N -1 d j=1 h -2(β j -b j ) + j
.

Thus, we get new rates of convergence in terms of n and N . If N > n, W (h) is always smaller than V (h). In this case, an adaptive procedure is conceivable, replacing Ṽ (h) by

V (h) = C(h) K * h / f * ε 2
/n and modifying H in the same way. The efficiency of this strategy can be proved by controlling terms of the form [ fhfh 2 -V (h)] + . This was successfully established in [START_REF] Comte | Data-driven density estimation in the presence of additive noise with unknown distribution[END_REF] in dimension 1, but such a study in dimension d would be much too long here.

Proofs

We start with three useful lemmas.

Lemma 1. Consider c, s nonnegative real numbers, and γ a real such that 2γ > -1 if c = 0 or s = 0. Then, for all m > 0,

• m -m (x 2 + 1) γ exp(c|x| j s)dx ≈ m 2γ+1-s e cm s , and if in addition 2γ

> 1 if c = 0 or s = 0, • ∞ m (x 2 + 1) -γ exp(-c|x| s )dx ≈ m -2γ+1
-s e -cm s . Proof of this lemma is based on integration by parts and is omitted. See also Lemma 2 p. 35 in Butucea and Tsybakov [2008a].

Lemma 2. [Bernstein inequality] Let T 1 , . . . , T n be independent random variables and

S n (T ) = n i=1 [T i -E(T i )]. Then, for η > 0, P(|S n (T ) -E(S n (T ))| ≥ nη) ≤ 2 max exp - nη 2 4v , exp - nη 4b ,
where Var(T 1 ) ≤ v and |T 1 | ≤ b.

It is proved in [START_REF] Birgé | Minimum contrast estimators on sieves: exponential bounds and rates of convergence[END_REF]

E sup t∈ B |ν n (t)| -(1 + 2η)H + ≤ 3π 2 v n e -η 2 6 nH 2 v + 21 η ∧ 1 M n e -(η∧1)η 21 nH M , with sup t∈ B ψ t ∞ ≤ M, E sup t∈ B |ν n (t)| ≤ H, sup t∈ B 1 n n k=1 Var(ψ t (Y k )) ≤ v.
Proof of Lemma 3: We apply the Talagrand concentration inequality given in [START_REF] Klein | Concentration around the mean for maxima of empirical processes[END_REF] to the functions s i (x) = t(x) -E(t(Y i )) and we obtain

P(sup t∈ B |ν n (t)| ≥ H + λ) ≤ exp - nλ 2 2(v + 4HM ) + 6M λ .
Then we modify this inequality following [START_REF] Birgé | Minimum contrast estimators on sieves: exponential bounds and rates of convergence[END_REF] Corollary 2 p.354. It gives ( 29)

P(sup t∈ B |ν n (t)| ≥ (1 + η)H + λ) ≤ exp - n 3 min λ 2 2v , min(η, 1)λ 7M .
To conclude for (27), we set λ = ηH. For (28), we take λ = ηH + u and write

E sup t∈ B(0,1) |ν n (t)| -(1 + 2η)H + ≤ +∞ 0 P sup t∈ B(0,1) |ν n (t)| ≥ (1 + η)H + ηH + u du ≤ +∞ 0 e -nη 2 H 2 6v e -nu 2 6v du + +∞ 0 e -nη(η∧1)H 21M e -n(η∧1)u 21M du = 3π 2 v n e -nη 2 H 2 6v + 21M n(η ∧ 1) e -nη(η∧1)H 21M
which is the result of (28).

7.1. Proof of Proposition 1. In the first case, the bias term is the same as in density estimation (see [START_REF] Tsybakov | Introduction to nonparametric estimation[END_REF]) and the use of Taylor formula to partial functions t

→ f (x 1 - v 1 h 1 , . . . , x i-1 -v i-1 h i-1 , t, x i+1 , . . . , x d ) yield |f h (x 0 ) -f (x 0 )| ≤ L d j=1 |x j | b j |K(x)|dx ⌊b j ⌋! h b j j .
In the second case, since f * , f * h ∈ L 1 (R), we can write

f (x 0 ) -f h (x 0 ) = 1 (2π) d e -i x 0 ,u 1 ( d j=1 [-1/h j ,1/h j ]) c (u j )f * (u 1 , . . . , u d )du 1 . . . du d
Then, for f ∈ S(b, a, r, L), the bias term is

|f (x 0 ) -f h (x 0 )| ≤ 1 (2π) d d j=1 1 |u j |≥1/h j |f * (u 1 , . . . , u d )|du 1 . . . du d ≤ 1 (2π) d d j=1 1 |u j |≥1/h j | d k=1 (1 + u 2 k ) -b k /2 exp(-a k |u k | r k ) × |f * (u 1 , . . . , u d )| d k=1 (1 + u 2 k ) b k /2 exp(a k |u k | r k ) du 1 . . . du d ≤ L (2π) d d j=1 |u|≥1/h j (1 + u 2 ) -b j exp(-2a j |u| r j )du 1/2 since k =j (1 + u 2 k ) -b k /2 exp(-a k |u k | r k ) ≤ 1. Then, using Lemma 1, |f (x 0 ) -f h (x 0 )| L d j=1 h b j +r j /2-1/2 j
exp(-a j h -r j ).

Proof of Proposition 2. The independence of the observations gives

Var( fh (x 0 )) = 1 n Var 1 (2π) d e -i u,x 0 K * h (u) e i u,Y 1 f * ε (u) du .
A simple bound of the variance by the expectation of the square yields Var( fh

(x 0 )) ≤ (n(2π) 2d ) -1 K * h /f * ε 2
1 . But we can also write

Var( fh (x 0 ))n(2π) 2d = e -i u-v,x 0 K * h (u)K * h (-v) f * ε (u)f * ε (-v) (f * Y (u -v) -f * Y (u)f * Y (-v))dudv ≤ K * h (u)K * h (-v) f * ε (u)f * ε (-v) |f * Y (u -v)|dudv ≤ K * h (u) f * ε (u) 2 du |f * Y (t)|dt ≤ K * h f * ε 2 2 f * ε 1 using Schwarz inequality. Now, under (H ε ), (2π) 2d nV 0 (h) is bounded by the minimum of f * ε 1 d j=1 K * j (u j h j ) (u 2 j + 1) -β j /2 exp(-α j |u j | ρ j ) 2 du j and   d j=1 K * j (u j h j ) (u 2 j + 1) -β j /2 exp(-α j |u j | ρ j ) du j   2 . If j ∈ SS, i.e. ρ j > 0 then K * j (t) = 0 if |t| ≥ 1. Consequently, using Lemma 1, K * j (uh j ) (u 2 + 1) -β j /2 exp(-α j |u| ρ j ) 2 du = 1/h j -1/h j |K * j (uh)| 2 (u 2 + 1) β j exp(2α j |u| ρ j )du ≤ K * j 2 ∞ 1/h j -1/h j (u 2 + 1) β j exp(2α j |u| ρ j )du h -2β j -1+ρ j j exp(2α j h -ρ j j
). In the same way

K * j (uh j ) (u 2 + 1) -β j /2 exp(-α j |u| ρ j ) du = 1/h j -1/h j |K * j (uh)|(u 2 + 1) β j /2 exp(α j |u| ρ j )du ≤ K * j ∞ 1/h j -1/h j (u 2 + 1) β j /2 exp(α j |u| ρ j )du h -β j -1+ρ j j exp(α j h -ρ j j ). Now, if j ∈ OS, i.e. α j = ρ j = 0, then K * j (uh j ) (u 2 + 1) -β j /2 2 du = h -1 j |K * j (u)| 2 ((uh -1 j ) 2 + 1) β j du h -1-2β j j |K * j (u)| 2 (u 2 + 1) β j du and K * j (uh j ) (u 2 + 1) -β j /2 du h -1-β j j
|K * j (u)|(u 2 + 1) β j /2 du. Finally, using that h j ≤ 1, we obtain the following bound for nV 0 (h)

j∈SS min(1, h -1+ρ j j )h -2β j -1+ρ j j exp(2α j h -ρ j j ) j∈0S h -1-2β j j = d j=1 h (ρ j -1) + j h -2β j -1+ρ j j exp(2α j h -ρ j j ).

Proof of Theorem 1. We shall consider two cases:

• Case A: the noise is OS and f belongs to D = H(b, L) or D = S(b + 1/2, a, r, L), with 0 ≤ r j < 2 for all j = 1, . . . , d.

In this case we set h n such that h n,j = n -1/(2b j +b j j 0 i=1 [(2β i +1)/b i ]) if r j = 0 (ordinary smooth components of f ) and h n,j = (log(n)/(2a j )) -1/r j when r j > 0 (super smooth components). Moreover ψ n is defined by ( 10) (recall that we standardized notations by setting r j = 0 when Hölder smoothness is considered, thus in the case of none super smooth components, we retrieve optimal h n given by ( 5) and rate ( 6)).

• Case B: the noise has at least one SS component and f belongs to D = H(b, L) or D = S(b + 1/2, 0, 0, L).

Then we set h n,j = n -1/(2β j +2b j +1) for j ∈ OS, and for j ∈ SS, h n,j = (2 ρ j log(n)/α j ) -1/ρ j . We recall that in this case ψ n = [log(n)] -2 min j∈SS (b j /ρ j ) . Before starting with the proof, we need to define preliminary material.

Let H be the kernel function defined in [START_REF] Fan | On the optimal rates of convergence for nonparametric deconvolution problems[END_REF], which is such that:

H = 0, H(0) = 0, H ∈ H(b i , L) ∩ S(b i , 0, 0, L), for i = 1, . . . , d, |H(x)| = O(x -δ ) as |x| → ∞ with δ > 3, and H * (t) = 0, (and thus also (H * ) ′ (t) = 0, (H * ) ′′ (t) = 0) when |t| is outside [1, 2].
We also use g s the symmetric stable law with characteristic function g * s (u) = exp(-|u| s ) where 0 < s < 2. An interest of this function relies on the Lemma: Lemma 4. The density g s satisfies the following properties:

• g -1 s (x) = O(|x| s+1 ) • If s < 1,
for all b > 0, there exists L ′ such that g s belongs to the Hölder space of dimension one H(b, L ′ ).

Proof of Lemma 4: Since the density is symmetric, we only consider positive x. [START_REF] Devroye | Nonuniform random variate generation[END_REF] shows that, for x > 0, g s (x) = x -1 ∞ j=1 b j (x -s ) j with b j = (-1) j-1 Γ(sj + 1) sin(sjπ/2) πj! First, we can write g s (x) = b 1 x -s-1 + o(x -s-1 ) as x → ∞, which proves the first point for s < 1. The case s ≥ 1 can be found in Butucea and Tsybakov [2008b]. The power series b j u j converges for all x (as pointed by Devroye, Stirling formula allows to show a geometric convergence -in fact of order j j(s-1) ). So it is differentiable with differentiate jb j u j-1 . Then, an easy computation leads to

g ′ s (x) = ∞ j=1 b j (-1 -sj)x -sj-2 = b 1 (-1 -s)x -s-2 + o(x -s-2 )
in some neighbourhood of infinity. In the same way, for all k ≥ 0,

g (k+1) s (x) = cx -s-k-2 + o(x -s-k-2 ).
But Hölder inequality provides |g

(k) s (x ′ ) -g (k) s (x)| ≤ ( |g (k+1) s | p ) 1/p |x -x ′ | b-k where 1/p = 1 + k -b. Since s + k + 2 > k + 1 -b, p(s + k + 2) > 1 and g (k+1) s is in L p . Thus g s ∈ H(b, L ′ ) with L ′ = g (k+1) s p .
Now, we define two couples (f 0 , f 1,A ) and (f 0 , f 1,B ). From now on, we assume that x 0 = (0, . . . , 0) since it is sufficient to translate functions at point x 0 in the other cases.

Case A. Let f 0 (x) = d j=1 1 c j g s j x j c j
with c j positive constants large enough (they will be made precise later). Here s j = s < 1 for j = 1, . . . , d if D = H(b, L), and if D = S(b + 1/2, a, r, L), for r j < 1, r j < s j < 1 and for 1 ≤ r j < 2, r j < s j < 2. We also define

f 1,A (x) = f 0 (x) + c V 0 (h n ) d j=1 H x j 2h n,j
.

Case B. Here we consider f 0 with s j = s < 1 for all j, and

f 1,B (x) = f 0 (x) + c d j=1 h b j n,j H x j h n,j 1≤i≤d, i =j g s (x i ),
where c is a constant to be specified later.

In the sequel we show that, for Z = A, B, 1) f 0 and f 1,Z are density functions and belong to D.

2) χ 2 (P n f 1,Z , P n f 0 ) n -1 where P n f 1,Z (resp. P n f 0 ) is the probability associated with the distribution of a sample Y 1 , . . . , Y n for density of Y 1 given by f 1,Z (resp f 0 ) and χ 2 (P, Q) = (dP/dQ -1) 2 dQ.

3) |f 1,Z (x 0 ) -f 0 (x 0 )| ≥ Cψ n .
Then it is sufficient to use Theorem 2.2 (see also p.80) in [START_REF] Tsybakov | Introduction to nonparametric estimation[END_REF] to obtain Theorem 1.

Proof of 1).

Hypothesis functions are densities First, f 0 are densities by construction. Second, the definition of H guarantees that, for Z = A, B, f 1,Z = 1. To ensure the positivity of

f 1,Z , it is sufficient to prove that |f 1,Z -f 0 | ≤ f 0 . But, as |x| → ∞, f -1 0 (x)|f 1,A (x) -f 0 (x)| c V 0 (h n ) d j=1 h δ n,j d j=1 x s j +1-δ j ≤ 1
for c small enough, since δ > 3 > max(s j ) + 1. In the same way, for case B, as |x| → ∞,

f -1 0 (x)|f 1,B (x) -f 0 (x)| c d j=1 h b j +δ n,j x s j +1-δ j ≤ 1
for c small enough.

Belonging to the Hölder space

Recall that we take s < 1 when D is an Hölder space. Since g s is in Hölder space of dimension one for any smoothness (Lemma 4), f 0 ∈ H(b, L ′ ) for some L ′ , and it is sufficient to choose c j to have

L ′ ≤ L/2. Now let G A (.) = (f 1,A -f 0 )(x 1 , . . . , x j-1 , . , x j+1 , . . . , x d ). Since H ∈ H(b j , L), |G (k) A (x ′ ) -G (k) A (x)| = c V 0 (h n ) j =i H x j 2h n,j (2h n,i ) -k H (k) x ′ 2h n,i -H (k) x 2h n,i ≤ c H d-1 ∞ L V 0 (h n )(2h n,i ) -b i |x ′ -x| b i -k . Then f 1,A -f 0 ∈ H(b, L/2) as soon as c H d-1 ∞ V 0 (h n )(2h n,i ) -b i ≤ 1/2
, which holds for our selected h n and suitable c. Thus f 0 and f 1,A belong to H(b, L).

Now let G B (.) = (f 1,B -f 0 )(x 1 , . . . , x j-1 , . , x j+1 , . . . , x d ). Since H ∈ H(b j , L), |G (k) B (x ′ ) -G (k) B (x)| ≤ ch b j n,j h -k n,j H (k) x ′ h n,j -H (k) x j h n,j g s d-1 ∞ +cL g s d-2 ∞ H ∞ p =j h bp n,p |x -x ′ | b j -k ≤ ch b j n,j h -k n,j L x -x ′ h n,j b j -k g s d-1 ∞ + cdL g s d-2 ∞ H ∞ |x -x ′ | b j -k Then f 1,B -f 0 ∈ H(b, L/2) if c is chosen small enough, so that f 1,B belongs to H(b, L).
Belonging to the Sobolev space By construction and because s j > r j , for c j large enough, f 0 ∈ S(b + 1/2, a, r, L/2) for r j < 2, j = 1, . . . , d. The computation of the Fourier transform of f 1,Af 0 gives

|(f 1,A -f 0 ) * (t)| = c V 0 (h n ) d j=1 2h n,j |H * (2t j h n,j ) |. Therefore |(f 1,A -f 0 ) * (t)| 2 d j=1 (1 + t 2 j ) b j +1/2 exp(2a j |t j | r j )dt ≤ c 2 V 0 (h n ) d j=1 h 2 n,j |H * (2t j h n,j ) | 2 (1 + t 2 j ) b j +1/2 exp(2a j |t j | r j )dt j k =j h n,k |H * (2t k h n,k ) | 2 dt k ≤ C(H)c 2 V 0 (h n ) d j=1 h -2b j n,j exp(2a j h -r j n,j ), using that H * (t) = 0 when |t| is outside [1, 2]. Then f 1,A -f 0 ∈ S(b + 1/2, a, r, L/2) as soon as C(H)c 2 V 0 (h n )h -2b i n,i exp(2a i h -r i n,i ) ≤ L 2 /(4d)
. This is verified for h n as chosen (the variance dominates the bias). The computation of the Fourier transform of f 1,Bf 0 gives [START_REF] Tsybakov | Introduction to nonparametric estimation[END_REF] p.86). Thus, it is sufficient to prove that χ 2 (P f 1,Z , P f 0 ) = O(n -1 ) where

(f 1,B -f 0 ) * (t) = c d k=1 h b k +1 n,k H * (t k h n,k ) d ℓ=1,ℓ =k g * s (t ℓ ). Therefore d j=1 |(f 1,B -f 0 ) * (t)| 2 (1 + t 2 j ) b j +1/2 dt ≤ c 2 d d j=1 h 2b j +2 n,j |H * (t j h n,j ) | 2 (1 + t 2 j ) b j +1/2 dt j d i=1,i =j |g * s (t i )| 2 dt i +c 2 d 1≤j,k≤d,j =k h 2b k +2 n,k |H * (t k h n,k )| 2 dt k (1 + t 2 j ) b j +1/2 |g * s (t j )| 2 dt j ℓ =k, ℓ =j |g * s (t ℓ )| 2 dt ℓ which is bounded since |H * (t j h n,j ) | 2 (1 + t 2 j ) b j +1/2 dt j = O(h -2b j -2 n,j ), (1 + t 2 j ) b j +1/2 |g * s (t j )| 2 dt j is a finite constant and h 2b k +2 n,k |g * s (t k h n,k )| 2 dt k = O(h 2b k +1 n,k ) = o(1) . Then there is some c such that f 1,B -f 0 ∈ S(b + 1/2, 0, 0, L/2). Proof of 2). Chi-square divergence For Z = A, B, since the observations are i.i.d., χ 2 (P n f 1,Z , P n f 0 ) = (1 + χ 2 (P f 1,Z , P f 0 )) n -1 (see e.g.
χ 2 (P f 1,Z , P f 0 ) = (f 1,Z * f ε -f 0 * f ε ) 2 (f 0 * f ε ) -1 .
Recall that we assume the independence of the noise components. Let us denote

q j (x j ) = 1 c j g s j x j -y c j f ε 1,j (y)dy = 1 c j g s j . c j * f ε 1,j (x j ) so that j q j (x j ) = (f 0 * f ε )(x). Then χ 2 (P f 1,A , P f 0 ) = c 2 V 0 (h n ) d j=1 H x j -y 2h n,j f ε 1,j (y)dy 2 q -1 j (x j )dx j and χ 2 (P f 1,B , P f 0 ) ≤ c 2 d d j=1 h 2b j n,j H x j -y h n,j f ε 1,j (y)dy 2 q -1 j (x j )dx j × j =i q i (x)dx and q i (x)dx = 1,
Now it follows from our Lemma 4 and Fan [1991]'s Lemma 5.1 that q j (x j ) ≥ C|x j | -(s j +1) for |x j | large enough, say |x j | ≥ A ≥ 1. Indeed Fan [1991]'s proof of his Lemma 5.1 works in our case because of the heavy tail property in Lemma 4. Using this property, we prove that

H x j -y h n,j f ε 1,j (y)dy 2 q -1 j (x j )dx j = O(h 2β j +1 n,j exp(-2 1-ρ j α j h -ρ j n,j )). (30)
Let us bound the term H ( . /2h n,j ) * f ε 1,j 2 (x j )q -1 j (x j )dx j = I 1 + I 2 where I 1 is the integral for |x j | < A and I 2 for |x j | ≥ A. Since q j (x j ) ≥ C|x j | -s j -1 for |x j | large enough

I 2 = |x j |≥A H ( . /2h n,j ) * f ε 1,j (x j ) 2 q j (x j ) dx j |x j |≥A H ( . /2h n,j ) * f ε 1,j (x j ) 2 |x j | s j +1 dx j |x j |≥A H ( . /2h n,j ) * f ε 1,j (x j )x j 2 dx j (31) if r j < s j < 1.
Then, with Parseval equality

I 2 ≤ H ( . /2h n,j ) * f ε 1, (y)y 2 dy = |(2h n,j H * (2h n2 u)f * ε 1,j (u)) ′ | 2 du ≤ 2 |4h 2 n,j (H * ) ′ (2h n2 u)f * ε 1,j (u)| 2 du + 2 |(2h n,j H * (2h n,j u)(f * ε 1,j ) ′ (u))| 2 du h 4 n,j |(H * ) ′ (2h n,j u)| 2 |u| -2β j exp(-2α j |u| ρ j )du + h 2 n,j |(H * (2h n,j u)| 2 u -2β ′ j exp(-2α j |u| ρ j )du h 2β j +3 n,j exp(-2 1-ρ j α j h -ρ j j ) 1≤|v|≤2 |(H * ) ′ (v)| 2 |v| -2β j dv +h 2β ′ j +1 n,j exp(-2 1-ρ j α j h -ρ j j ) 1≤|v|≤2 |(H * (v)| 2 v -2β ′ j dv (h 2β j +3 n,j + h 2β ′ j +1 n,j ) exp(-2 1-ρ j α j h -ρ j j ). (32)
If 1 ≤ r j < s j < 2, then x 2 j in (31) must be replaced by x 4 j and the same computations can be done with derivatives of order 2. The other term can be bounded in the same way, using that q j ≥ C for |x j | small

I 1 = |x j |<A H ( . /2h n,j ) * f ε 1,j (x j ) 2 q -1 j (x j )dx j ≤ C -1 |x j |<A H ( . /2h n,j ) * f ε 1,j (x j ) 2 dx j 1≤|2h n,j u|≤2 |2h n,j H * (2h n,j u)f * ε 1,j (u)| 2 du h 2β j +1 n,j exp(-2 1-ρ j α j h -ρ j j ) |H * (v)| 2 v -2β j dv (33)
Finally, by gathering ( 33) and (32), we obtain (30). Thus, in the OS case, we get

χ 2 (P f 1,A , P f 0 ) V 0 (h n ) d j=1 h 2β j +1 n,j n -1 .
For the other case, we get

χ 2 (P f 1,B , P f 0 ) j∈OS h 2b j n,j h 2β j +1 n,j + j∈SS h 2b j +2β j +1 n,j exp(-2 1-ρ j α j h -ρ j n,j ) n -1
by using the choices of the h n,j 's.

Proof of 3). Rate We can see that |f

1,A (0) -f 0 (0)| = c V 0 (h n )|H(0)| d , and |f 1,B (0) -f 0 (0)| = c( d j=1 h b j n,j )|H(0)||g s (0)| d-1
, which all have the announced order of the rate ψ n for the selected h n . This ends the proof of Theorem 1. 7.4. Proof of Theorem 2. 7.4.1. Proof of Theorem 2. We want to bound | f (x 0 )f (x 0 )|. Let h ∈ H 0 be fixed. The following decomposition holds:

| f (x 0 ) -f (x 0 )| ≤ | fĥ (x 0 ) (x 0 ) -fh, ĥ(x 0 ) (x 0 )| D 1 + | fh, ĥ(x 0 ) (x 0 ) -fh (x 0 )| D 2 + | fh (x 0 ) -f (x 0 )|.

By definition of

A(h, x 0 ), D 1 ≤ A 0 (h, x 0 ) + Ṽ0 ( ĥ(x 0 )).
And by definition of A 0 ( ĥ(x 0 ), x 0 ),

D 2 ≤ A 0 ( ĥ(x 0 ), x 0 ) + Ṽ0 (h). Therefore D 1 + D 2 ≤ A 0 (h, x 0 ) + Ṽ0 ( ĥ(x 0 )) + A 0 ( ĥ(x 0 ), x 0 ) + Ṽ0 (h) ≤ 2 A 0 (h, x 0 ) + Ṽ0 (h) ,
by using the definition of ĥ(x 0 ).

Thus (34) | f (x 0 ) -f (x 0 )| ≤ 2A 0 (h, x 0 ) + 2 Ṽ0 (h) + | fh (x 0 ) -f (x 0 )|.
To study A 0 (h, x 0 ), we can write

fh ′ (x 0 ) -fh,h ′ (x 0 ) = fh ′ (x 0 ) -f h ′ (x 0 ) -( fh,h ′ (x 0 ) -f h,h ′ (x 0 )) + f h ′ (x 0 ) -f h,h ′ (x 0 ),
where

f h (x 0 ) = E( fh (x 0 )) = K h ⋆ f (x 0 ) f h,h ′ (x 0 ) = E( fh,h ′ (x 0 )) = K h ′ ⋆ K h ⋆ f (x 0 ). For any h ′ , |f h ′ (x 0 ) -f h,h ′ (x 0 )| = |K h ′ ⋆ (f -K h ⋆ f )(x 0 )| ≤ N (K)B 0 (h).
We get back to the definition of A 0 (h, x 0 )

A 0 (h, x 0 ) = sup h ′ ∈H 0 | fh ′ (x 0 ) -fh,h ′ (x 0 )| -Ṽ0 (h ′ ) + ≤ sup h ′ ∈H 0 | fh ′ (x 0 ) -f h ′ (x 0 )| -Ṽ0 (h ′ )/(1 + K * ∞ ) + (35)
+ sup

h ′ ∈H 0 | fh,h ′ (x 0 ) -f h,h ′ (x 0 )| -K * ∞ Ṽ0 (h ′ )/(1 + K * ∞ ) + + N (K)B 0 (h)
We can prove the following concentration result: Proposition 6. Under the Assumptions of Theorem 2, for all h, h ′ ∈ H 2 0 , for all p ≥ 1,

P | fh (x 0 ) -f h (x 0 )| > c 1 (p) Ṽ0 (h) ≤ 2/n p , (36)
P | fh,h ′ (x 0 ) -f h,h ′ (x 0 )| > c 1 (p) K * ∞ Ṽ0 (h ′ ) ≤ 2/n p , ( 37
)
as soon as c 1 (p) 2 c 0 ≥ 16p 2 / min ( f * ε 1 , 1). The proposition is proved below. It implies that if c 1 (p) = 1/(1 + K * ∞ ) and c 0 ≥ 16p 2 (1 + K * ∞ ) 2 / min ( f * ε 1 , 1), P sup h∈H 0 | fh (x 0 ) -f h (x 0 )| -Ṽ0 (h)/(1 + K * 2 ∞ ) + > 0 ≤ 2 h∈H 0 n -p ≤ 2n ǫ-p
as Card(H 0 ) ≤ n ǫ . In the same way, for all h ∈ H 0 ,

P sup h ′ ∈H 0 | fh,h ′ (x 0 ) -f h,h ′ (x 0 )| -K * ∞ Ṽ0 (h)/(1 + K * ∞ ) + > 0 ≤ 2n ǫ-p
Thus, the following set

Ω = sup h ′ ∈H 0 | fh ′ (x 0 ) -f h ′ (x 0 )| -Ṽ0 (h ′ )/(1 + K * ∞ ) + = 0 ∩ ∀h ∈ H 0 , sup h ′ ∈H 0 | fh,h ′ (x 0 ) -f h,h ′ (x 0 )| -K * ∞ Ṽ0 (h ′ )/(1 + K * ∞ ) + = 0
has probability larger than 1 -4n 2ǫ-p . Now we choose p = 2ǫ + q and then c 0 ≥ 16(1 + K * ∞ ) 2 (2ǫ + q) 2 / min ( f * ε 1 , 1). Thus P(Ω) > 1 -4n -q . By gathering inequalities (34) and ( 35), we have on

Ω | f (x 0 ) -f (x 0 )| ≤ 2A 0 (h, x 0 ) + 2 Ṽ0 (h) + | fh (x 0 ) -f (x 0 )| ≤ 2N (K)B 0 (h) + 2 Ṽ0 (h) + | fh (x 0 ) -f (x 0 )| But, still on Ω | fh (x 0 ) -f (x 0 )| ≤ B 0 (h) + | fh (x 0 ) -f h (x 0 )| -Ṽ0 (h)/(1 + K * ∞ ) + Ṽ0 (h)/(1 + K * ∞ ) ≤ B 0 (h) + Ṽ0 (h) Then, on Ω, | f (x 0 ) -f (x 0 )| ≤ (1 + 2N (K))B 0 (h) + 3 Ṽ0 (h),
which ends the proof of Theorem 2.

7.4.2. Proof of Proposition 6. Let us define the independent random variables

Z k (x 0 ) = 1 (2π) d e -i u,x 0 K * h (u) e i u,Y k f * ε (u) du.
Clearly,

fh (x 0 ) -f h (x 0 ) = 1 n n k=1 [Z k (x 0 ) -E(Z k (x 0 ))].
We apply Bernstein Inequality recalled in Lemma 2 to the Z k (x 0 )'s, with η = c 1 (p) Ṽ0 (h). We find

|Z 1 (x 0 )| ≤ (2π) -d K * h (u) f * ε (u) du =: b
and Var(Z 1 (x 0 )) ≤ nV 0 (h). We obtain

P |( fh (x 0 ) -f h (x 0 )| > c 1 (p) Ṽ0 (h) ≤ P |S n (Z(x 0 )) -E(S n (Z(x 0 )))| ≥ c 1 (p) Ṽ0 (h) ≤ 2 max   exp   - n(c 1 (p) Ṽ0 (h)) 2 4nV 0 (h)   , exp   - n(c 1 (p) Ṽ0 (h)) 4b     , (38)
where c 1 (p) is chosen such that

(39) nc 1 (p) 2 Ṽ0 (h) 4nV 0 (h) ≥ p log(n) that is c 1 (p) 2 c 0 ≥ 4p (c 0 is the constant in the definition of Ṽ0 (h)). Moreover, n c 1 (p) 2 Ṽ0 (h) 4b = c 1 (p) 2 c 0 4 n log(n) nV 0 (h) b 2 . But for h ∈ H 0 , nV 0 (h)/b 2 = min f * ε 1 K * h f * ε 2 2 K * h f * ε -2 1 , 1 ≥ c 3 log(n) n with c 3 = min ( f * ε 1 , 1). Thus (40) n c 1 (p) 2 Ṽ0 (h) 4b ≥ p log(n)
provided that c 3 c 2 1 (p)c 0 ≥ 4p. Note now that this last condition also ensures the first constraint c 1 (p) 2 c 0 ≥ 4p. Therefore, inserting (39) and ( 40) in (38) implies the first inequality (36) of Proposition 6.

To prove (37), we follow the same line. For the study of

fh,h ′ (x 0 ) -f h,h ′ (x 0 ) = K h ⋆ ( fh ′ -f h ′ )(x 0 ), we can simply replace K * h (u) by K * h (u)K * h ′ (u), with |K * h (u)| ≤ K * ∞
so that it adds a term K * ∞ in the previous computations. Thus we get (37) and this end the proof of Proposition 6. 7.5. Proof of Corollary 1. Let us denote |f * ε,j (t)| the j-th component of the order of the noise characteristic function, i.e. |f * ε,j (t)| = (1 + t 2 ) -β j /2 exp(-α j |t| ρ j ). First, we write

K * h /f * ε 2 1 K * h /f * ε 2 2 d j=1 |K * j (t j h j )||f * ε,j (t j )| -1 dt j 2 |K * j (t j h j )| 2 |f * ε,j (t j )| -2 dt j   d j=1 1 h j   d j=1 |K * j (u j )||f * ε,j (u j /h j )| -1 du j 2 |K * j (u j )| 2 |f * ε,j (u j /h j )| -2 du j .
Consider now case 1. Under (H ε ), in the OS case, we get

K * h /f * ε 2 1 K * h /f * ε 2 2   d j=1 1 h j   d j=1 |K * j (u j )|(1 + (u j /h j ) 2 ) β j /2 du j 2 |K * j (u j )| 2 (1 + (u j /h j ) 2 ) β j du j   d j=1 1 h j   d j=1 |K * j (u j )|(h 2 j + u 2 j ) β j /2 du j 2 |K * j (u j )| 2 (h 2 j + u 2 j ) β j du j   d j=1 1 h j   d j=1 |K * j (u j )|(1 + u 2 j ) β j /2 du j 2 |K * j (u j )| 2 u 2β j j du j := C(ε, K) d j=1 1 h j .
because 0 < h j ≤ 1 and the assumptions make all integrals finite.

Consider case 2., where K j = sinc, and use the equivalence Lemma 1. Then we get straightforwardly

K * h /f * ε 2 1 K * h /f * ε 2 2 d j=1 h ρ j -1 j .
Therefore hopt belongs to H 0 if condition (18) is satisfied. Let us explain why constraint (18) is fulfilled in the two cases of Corollary 1.

First, in case 1., it follows from (5) that hj,opt are such that

( d i=1 1/ hi,opt ) ≤ ( d i=1 ( hP i,opt ) -2β i -1 )( d i=1,i =j h-2b i j,opt ) ∝ n/ log(n)
for j = 1, . . . , d which implies clearly that they satisfy the constraint d j=1 (1/h j ) ≤ n/ log(n). This is the reason why (18) and thus (17) hold.

Second, in case 2., the general constraint is also satisfied by the optimal bandwidths because the negative powers on the h j 's get smaller when ρ j increases, and each time a ρ j is nonzero, it is associated to a logarithmic order for the h j 's. Condition (18) can also easily be checked for mixed cases. Therefore, hopt also belongs to H 0 and Corollary 1 is proved. 7.6. Proof of Proposition 3. In the first case, standard methods (see [START_REF] Tsybakov | Introduction to nonparametric estimation[END_REF] or [START_REF] Kerkyacharian | Nonlinear estimation in anisotropic multi-index denoising[END_REF]) yield

f h -f ≤ C(K, d, b)L d j=1 h b j j .
In the Sobolev case, Parseval formula gives

f h -f 2 = (2π) -d f * h -f * 2 and f * h -f * 2 = ( d j=1 [-1/h j ,1/h j ]) c |f * (u)| 2 du ≤ d j=1 (1 + u 2 j ) -b j exp(-2a j |u j | r j )1 |u j |≥1/h j |f * (u 1 , . . . , u d )| 2 (1 + u 2 j ) b j exp(2a j |u j | r j )du 1 . . . du j L d j=1 h 2b j j exp(-2a j h -r j j ).
7.7. Proof of Proposition 4. The first bound is obtained by writing

E fh -f h 2 = 1 (2π) d Var K * h f * ε f * Y ≤ 1 (2π) d n K * h f * ε e i u,Y 1 2 .

Now we use the bound on K

* h /f * ε 2 proved for Proposition 2: nV (h) j∈SS h -2β j -1+ρ j j exp(2α j h -ρ j j ) j∈0S h -1-2β j j = d j=1 h -2β j -1+ρ j j exp(2α j h -ρ j j ).
7.8. Proof of Theorem 3. The proof uses all the tools given in the proof of Theorem 1 and we refer to it. We consider almost the same two cases.

• Case A: the noise is OS and f belongs to D = S(b, a, r, L), with 0 ≤ r j < 2 for all j = 1, . . . , d. • Case B: the noise has at least one SS component and f belongs to D = S(b, 0, 0, L).

In both cases, the bandwidth h n and the rate ψ n are as in the proof of Theorem 1. We also keep the same functions H, g s , f 0 . Next, we define below a collection of alternatives (f θ ) θ in case A, and a single alternative in case B. We follow the same three steps as previously, with, in step 3), integral norms instead of pointwise distance. We will use an integral on a compact set [a, b] = d j=1 [a j , b j ] in case A (which nevertheless minorates the norm on R d ) and on R d in case B. Here, we consider the two cases separately, the extension of the first one being more complicated than the extension of the second one. Case A, OS-noise: We take

f θ (x) = f 0 (x) + c V (h n ) k∈K θ k d j=1 H x j -x nkj 2h n,j with K = {1, . . . , M 1 } × • • • × {1, . . . , M d }, θ ∈ {0, 1} M 1 ו••×M d , M j = ⌊h -1 j ⌋,
x n,k is a vector with j-th coordinate x nkj = a j + k j (b ja j )h n,j . Moreover, we assume that b ja j ≥ 1/(2π) for j = 1, . . . d. Let θ a sequence in {0, 1} K . For i = 0, 1, we denote θ i k the sequence such that (θ i k ) k = i and for all l ∈ K different from k, (θ i k ) l = θ l . We now follow the three announced steps.

1) Hypothesis functions are densities

We already know that f 0 is a density and the definition of H guarantees that f θ = 1. To ensure the positivity of f θ , it is sufficient to prove that |f θf 0 | ≤ f 0 . But, as |x| → ∞,

(41) f -1 0 (x)|f θ (x) -f 0 (x)| c V (h n ) d j=1 h -1+δ n,j d j=1 x s j +1 j (x j -x nkj ) -δ ≤ 1 2
for c small enough, since δ > 3 > s j + 1.

Belonging to the Sobolev space S(b, a, r, L/2), r j < 2.

The computation of the Fourier transform of f θf 0 gives

|(f θ -f 0 ) * (t)| 2 = c 2 V (h n ) d j=1 2h 2 n,j |H * (2t j h n,j ) | 2 |v(t 1 (b 1 -a 1 )h n,1 , . . . , t d (b d -a d )h n,d )| 2 ,
where

v(t) = k∈K θ k e i<t,k> .
Therefore, using that

H * (t) = 0 when |t| is outside [1, 2], |(f θ -f 0 ) * (t)| 2 d j=1 (1 + t 2 j ) b j exp(2a j |t j | r j )dt ≤ c 2 V (h n ) d j=1 h 1-2b j n,j exp(2a j h -r j j ) |H * (2t j ) | 2 (1 + t 2 j ) b j l =j h n,l |H * (2t l ) | 2 |v(t 1 (b 1 -a 1 ), . . . , t d (b d -a d ))| 2 dt ≤ c 2 H * 2d ∞ V (h n ) d j=1 h 1-2b j n,j exp(2a j h -r j j ) l =j h n,l [1/2,1] d |v(t 1 (b 1 -a 1 ), . . . , t d (b d -a d ))| 2 dt But v is 2π-periodic so that (42) 1 (2π) d [0,2π] d |v(u)| 2 du = k θ 2 k ≤ Card(K) = d j=1 M j ≤ d j=1 h -1 j and, using that 1/(b j -a j ) ≤ 2π, [1/2,1] d |v(t 1 (b 1 -a 1 ), . . . , t d (b d -a d ))| 2 dt = d j=1 (b j -a j ) d j=1 [ 1 2(b j -a j ) , 1 b j -a j ] |v(u 1 , . . . , u d )| 2 du 1 . . . du d ≤ d j=1 (b j -a j ) [0,2π] d |v(u)| 2 du ≤ d j=1 [2π(b j -a j )] k∈K θ 2 k .
Then using (42), we get

|(f θ -f 0 ) * (t)| 2 d j=1 (1 + t 2 j ) b j exp(2a j |t| r j )dt c 2 V (h n ) d j=1 h 1-2b j n,j exp(2a j h -r j j ) l =j h n,l k θ 2 k c 2 V (h n ) d j=1 h -2b j n,j exp(2a j h -r j j ) ≤ (L/2) 2
for c small enough.

2) Chi-square divergence Let some k ∈ K. We shall first compare f 0 and f 0 (.x n,k ). Since g s j is symmetric, we only study the case of x j ≥ 0. First remark that for x j large enough,

x s j +1 j ≤ 2 s j (x j -x nkj ) s j +1 + 2 s j x s j +1 nkj (x j -x nkj ) s j +1 .
Then, according to Lemma 4, for x j large enough,

g s j x j c j c s j +1 j x 1-s j j c s j +1 j (x j -x nkj ) 1-s j g s j x j -x nkj c j .
Moreover, for x j small, i.e. in an interval I, since g s j is continuous and g s j (0) > 0,

g s j x j -x nkj c j ≤ g s j ∞ ≤ ( inf I/c j g s j ) -1 g s j ∞ g s j x j c j .
Thus, for all k ∈ K, f 0 f 0 (.x n,k ). In addition, we use (41) to conclude

f θ * f ε ≥ 1 2 f 0 * f ε ≥ max k∈K f 0 * f ε (. -x n,k ).
This implies

χ 2 (P f θ 0 k , P f θ 1 k ) = (f θ 1 k * f ε -f θ 0 k * f ε ) 2 f θ 0 k * f ε (f θ 1 k * f ε -f θ 0 k * f ε ) 2 f 0 * f ε (. -x n,k ) = χ 2 (P f 1,A , P f 0 ),
where f 0 and f 1,A are defined in the proof of Theorem 1. Hence, using the corresponding part of the proof of Theorem 1, we get χ 2 (P f θ 0

k , P f θ 1 k ) = O(n -1 ), uniformly in θ.
3) Rate.

For some estimator fn , let us denote the quadratic risk by

R = sup f E f [a,b] ( fn (x) -f (x)) 2 dx,
and by

A k (x) = |f θ 0 k -f θ 1 k (x)|/2 = | V (h n ) d j=1 H x j -x nkj 2h n,j
|/2. Using a Bernoulli distribution for θ and Markov inequality, we can prove as in [START_REF] Fan | Adaptively local one-dimensional subproblems with application to a deconvolution problem[END_REF] that

sup f E f ( fn (x) -f (x)) 2 ≥ max k∈K A k (x) 2 2 E θ (S n,k (θ)) where S n,k (θ) = 1 i=0 P θ i k (| fn (x) -f θ i k (x)| ≥ A k (x)
). Now, given our bound on the chi-square divergence, Theorem 2.2 (iii) in [START_REF] Tsybakov | Introduction to nonparametric estimation[END_REF] shows the existence of a constant s c such that

S n,k (θ) ≥ s c . Thus R ≥ s c 2 [a,b] max k A k (x) 2 dx = C [a,b] max k V (h n ) d j=1 H x j -x nkj 2h n,j 2 dx = CV (h n ) l∈K D l max k d j=1 H x j -x nkj 2h n,j 2 dx ≥ CV (h n ) l∈K D l d j=1 H x j -x nlj 2h n,j 2 dx with D l = [a 1 + (l 1 -1)(b 1 -a 1 )h n,1 , a 1 + l 1 (b 1 -a 1 )h n,1 ] × • • • × [a d + (l d -1)(b d -a d )h n,d , a d + l d (b d -a d )h n,d ]. But a j +l j (b j -a j )h n,j a j +(l j -1)(b j -a j )h n,j H x j -x nlj 2h n,j 2 dx j = h n,j b j -a j 0 |H(-x j /2)| 2 dx j Thus R ≥ CV (h n ) l∈K d j=1 h n,j b j -a j 0 |H(-x j /2)| 2 dx j ≥ CV (h n ) d j=1 b j -a j 0 |H(-x j /2)| 2 dx j ≥ C ′ V (h n ) Since V (h n ) ≈ ψ n ,
this ends the proof of the lower bound in case A for the integrated risk.

Case B. Noise with at least one SS component. Here, we can extend the proof of Case B of the pointwise setting more directly. We take f 0 as previously and define

f 1 (x) = f 0 (x) + c d j=1 h b j -1/2 n,j H x j h n,j 1≤i≤d, i =j H(x i ).
Let us follow again the three steps of the proof of Theorem 1 and Theorem 2.2 in Tsybakov [2009]. 1) Clearly, with the previous computations, f 0 and f 1 are densities (for c chosen small enough), and belong to the Sobolev space S(b, 0, 0, L).

2) Let us study the χ 2 -divergence.

χ 2 (P f 1 , P f 0 ) ≤ c 2 d d j=1 h 2b j -1 n,j H x j -y h n,j f ε 1,j (y)dy 2 q -1 j (x j )dx j × j =i (H * f ε,i (x)) 2 q i (x) dx, = c 2 d d j=1 h 2b j -1 n,j H ( . /2h n,j ) * f ε 1,j 2 (x j )q -1 j (x j )dx j × O(1), since replacing h nj by 1 in equation (30) implies (H * f ε,i ) 2 q -1 i = O(1). Thus we get χ 2 (P f 1 , P f 0 ) j∈OS h 2b j +2β j n,j + j∈SS h 2b j +2β j n,j exp(-2 1-ρ j α j h -ρ j n,j )
and with h n,j = n -1/(2β j +2b j +1) , for j ∈ OS and h n,j = (2 ρ j log(n)/α j ) -1/ρ j for j ∈ SS, we get an order less than 1/n.

3) Rate.

f 0 -f 1 2 = d j=1 h 2b j -1 n,j H 2 x j h n,j dx j i =j H 2 (x i )dx i +2 j<k h b j -1/2 n,j h b k -1/2 n,k . . . H x j h n,j H x k h n,k i =j H(x i ) ℓ =k H(x ℓ )dx 1 . . . dx d = d j=1 h 2b j n,j H 2 ( H 2 ) d-1 +2 j<k h b j -1/2 n,j h b k -1/2 n,k H x j h n,j H(x j )dx j H x k h n,k H(x k )dx k ( H 2 ) d-2 = d j=1 h 2b j n,j H 2d since H x j h n,j H(x j )dx j = 0 for h n,j < 1/2. Indeed H x j h n,j H(x j )dx j = h n,j 2π H * (th n,j )H * (t)dt = 0
since H * is supported by [1, 2] and H * (th n,j ) has support [1/h n,j , 2/h n,j ] ⊂]2, +∞) for h n,j < 1/2. This ends the proof of the lower bound in case B and thus of Theorem 3.

7.9. Proof of Theorem 4.

7.9.1. Proof of Theorem 4. The beginning of the proof is the same as the one of Theorem 2. Let h ∈ H be fixed. The following decomposition holds:

(43) ff ≤ fĥfh, ĥ D 3 + fh, ĥ -fh

D 4 + fh -f . By definition of A(h), D 3 ≤ A(h) + Ṽ ( ĥ).
And by definition of A( ĥ),

D 4 ≤ A( ĥ) + Ṽ (h). Therefore (44) D 3 + D 4 ≤ A(h) + Ṽ ( ĥ) + A( ĥ) + Ṽ (h) ≤ 2 A(h) + Ṽ (h) ,
by using the definition of ĥ. To study A(h), we can write fh

′ -fh,h ′ = fh ′ -f h ′ -( fh,h ′ -f h,h ′ ) + f h ′ -f h,h ′ . But f h ′ -f h,h ′ = K h ′ ⋆ (f -K h ⋆ f ) ≤ K * h ′ ∞ f -f ⋆ K h as u ⋆ v ≤ u * ∞ v , for functions u with Fourier transform and v ∈ L 2 (R d ). As K * h ′ ∞ = K * ∞ , we get f h ′ -f h,h ′ ≤ K * ∞ f -f h . In the same way, fh,h ′ -f h,h ′ ≤ K * ∞ fh ′ -f h ′ . Then fh ′ -fh,h ′ ≤ (1 + K * ∞ ) fh ′ -f h ′ + K * ∞ f -f h . We get back to the definition of A(h) A(h) = sup h ′ ∈H fh ′ -fh,h ′ -Ṽ (h ′ ) + ≤ (1 + K * ∞ ) sup h ′ ∈H fh ′ -f h ′ -Ṽ (h ′ )/(1 + K * ∞ ) + + K * ∞ f -f h . (45)
We can prove the following concentration result: Proposition 7. [Variance concentration] Under the assumptions of Theorem 4, for all h ′ in H,

P fh ′ -f h ′ ≥ Ṽ (h ′ )/(1 + K * ∞ ) ≤ exp - min(η, 1)η 46 (log n) 2
This proposition is proved below.

Then, if we define

Ω = {∀h ′ ∈ H fh ′ -f h ′ ≤ Ṽ (h ′ )/(1 + K * ∞ )}, then P(Ω c ) ≤ h ′ ∈H e -κ(log n) 2 ≤ card (
H)e -κ(log n) 2 with κ = min(η, 1)η/46. Now, gathering the terms yields, on Ω, ∀h ∈ H,

f -f ≤ 2 K * ∞ f -f h + 2 Ṽ (h) + fh -f ≤ (1 + 2 K * ∞ ) f -f h + 2 Ṽ (h) + fh -f h But, on Ω, fh -f h ≤ Ṽ (h)/(1 + K * ∞ ) ≤ Ṽ (h). Thus, on Ω, f -f ≤ (1 + 2 K * ∞ ) f -f h + 3 Ṽ (h) which ends the proof of Theorem 4. 7.9.2. Proof of Proposition 7. Let B(0, 1) = {t ∈ L 2 (R d ) ∩ L 1 (R d ), t = 1}. We can note that fh and f h belong to L 2 (R d ) ∩ L 1 (R d ), and fh -f h = sup t∈B(0,1) fh -f h , t = sup t∈ B(0,1) fh -f h , t
where B(0, 1) is a dense countable subset of B(0, 1) (thanks to the separability of L 2 (R)

K * h (-v) f * ε (-v) f * Y 1 (u -v)dudv
Clearly we can get first sup t∈ B(0,1) Var(ψ t (Y 1 )) ≤ nV (h). But we can also apply Cauchy-Schwarz Inequality with respect to the measure |f * Y 1 (uv)|dudv and we obtain thus sup t∈ B(0,1)

Var(ψ t (Y 1 )) ≤ sup t∈ B(0,1)

1 (2π) 2d |t * (u)| 2 K * h (u) f * ε (u) 2 |f * Y 1 (u -v)|dudv ≤ 1 (2π) 2d sup u∈R d K * h (u) f * ε (u) 2 sup t∈ B(0,1) t * 2 |f Y 1 (z)|dz ≤ 1 (2π) d K * h /f * ε 2 ∞ .
Therefore, To prove Corollary 2, consider case 1. Under (H ε ), in the OS case, we get because 0 < h j < 1 and the assumptions make all terms finite.

v := 1 (2π) d min( K * h /f * ε 2 ∞ , K * h /f * ε
K * h /f * ε 2 2 K * h /f * ε 2 ∞ ≈   d j=1 1 h j   d j=1 |K * j (u j )|(h 2 j + u 2 j ) β j du j sup u j ∈R |K * j (u j )| 2 (h 2 j + u 2 j ) β j
The result of Corollary 3 is obvious. Indeed, the choice C(h) = log 2 (n) ensures that condition (20) is fulfilled and thus ȟopt ∈ H.

To understand why it can not be improved, consider case 2. (in the general terminology of Corollary 1), where K j = sinc, and use the equivalence Lemma 1. Then we get straightforwardly ( 46) max(1,

K * h /f * ε 2 2 K * h /f * ε 2 ∞ ) ≈ d j=1 h -(1-ρ j ) + j
.

Then we obtain the same order as in case 1. above if the ρ j 's are all zero, thus the same conclusion holds for K taken as sinc and f ordinary smooth. It also follows from ( 46) that condition (20) in the definition of H is equivalent to (47)

d j=1 h -(1-ρ j ) + j C(h) log 2 (n).
In the case of ordinary smooth f * ε , consider the case where the function f is super smooth. Then the condition (47) can be written j (1/h j )C(h log 2 (n). This is not necessarily satisfied by the optimal bandwidths which have logarithmic orders, if we only set C(h) = 1. But as the powers of log(n) involved in ȟopt depend on the regularity of f , which is unknown, the quantity missing to reach log 2 (n) is unknown. In the case of super smooth f * ε , it is clear that if all ρ j 's are larger than one, C(h) = log 2 (n) is the only possible choice for condition (47) to be fulfilled.

7.11. Proof of Theorem 5. The proof starts like the proof of Theorem 4 but we replace Proposition 7 by a bound in expectation obtained in an analogous way, but by using equation ( 28) instead of equation ( 27). As all bounds M, v, H have been computed in the proof of Proposition 7, we easily obtain that

E fh ′ -f h ′ -Ṽ (h)/(1 + K * ∞ ) + ≤ C v n e -η 2 6 n Ṽ (h) v + V (h) n e -(η∧1)η 21 √ n .
To obtain the result, we need to prove that, in case 2. of the above terminology and with our new definition of H, we have

h∈H √ ve -η 2 6 n Ṽ (h) v < +∞.
Now we use the previous evaluations and in particular (46). We write C(h) = d j=1 C j (k j ). The following inequalities hold. k β j -(ρ j -1) + /2 e α j k ρ j -κC j (k)k (1-ρ j ) +   := Σ, where κ can be specified in function of η 2 /6 and the constants involved in Lemma 1. This explains why we choose C j (k) = 1 if 0 ≤ ρ j < 1/2 which corresponds to the case where k ρ j < k (1-ρ j ) + = k 1-ρ j . We choose C j (k) = (2α j /κ)k 2ρ j -1 if 1/2 ≤ ρ j < 1 because then α j k ρ j -κC j (k)k (1-ρ j ) + = -α j k ρ j . In the same way, we take C j (k) = (2α j /κ)k ρ j if ρ j > 1. Then the sums over k are bounded and Σ < +∞. These values give formula (26) which is overestimated by the proposal (24) in order to avoid the specification of tedious constants. Thus, we have

h ′ ∈H M E fh ′ -f h ′ -Ṽ (h)/(1 + K * ∞ ) + ≤ C Σ √ n + card(H M )e -(η∧1)η 21 √ n ≤ C ′ √ n since card(H M ) ≤ n d .
Therefore, it follows from (45) that, as K * ∞ = 1 for K =sinc, then

E(A(h)) ≤ 2C ′ √ n + 2 f -f h
and inserting this in ( 43) and ( 44) yields

E( f -f ) ≤ 3 f -f h + 3 Ṽ (h) + 2C ′ √ n ,
which is (25). This ends the proof of Theorem 5.

7.12. Proof of Proposition 5. First, note that Neumann's Lemma 2.1 (see [START_REF] Neumann | On the effect of estimating the error density in nonparametric deconvolution[END_REF], and in particular the proof of the Lemma 2.1 page 323) can be straightforwardly extended to the multivariate setting. Define

R(t) = 1 f * ε (t) - 1 f * ε (t)
.

The result can be written

E |R(t)| 2 ≤ C 1 |f * ε (t)| 2 ∧ N -1 |f * ε (t)| 4 .
Then the following decomposition holds:

f h -fh = 1 (2π) d/2 K * h f * Y -f * Y f * ε + ( f * Y -f * Y )R + f * Y R
and thus

E( f h -fh 2 ) |K * h (t)| 2 E[|f Y (t) -f * Y (t)| 2 ] |f * ε (t)| 2 dt + |K * h (t)| 2 E[|f Y (t) -f * Y (t)| 2 ]E(|R(t)| 2 )dt + |K * h (t)| 2 |f * Y (t)| 2 E[|R(t)| 2 ]dt 1 n K * h f * ε 2 + 1 n K * h f * ε 2 + N -1 K * h f * f * ε 2
where the second term is obtained by bounding R(t) by 1/|f * ε (t)| 2 and the last one uses the second bound of R(t) and the fact that f * Y = f * f * ε . The first two terms are V (h) and the last one is W (h). Thus, we obtain E( f h -fh

2 ) V (h) + W (h).

  Proposition 3. Under assumptions • f belongs to Nikol'skii class N (b, L) and the kernel verifies Korder(ℓ) with ℓ ≥ ⌊b⌋, or • f belongs to Sobolev class S(b, a, r, L) and K = sinc,

Figure 1 .

 1 Figure 1. Example 2, global bandwidth selection, with n = 500. Top right: true density f , top left: estimator f , bottom: sections, dark line for f and light line for the estimator

Figure 2 .

 2 Figure 2. Example 3, pointwise bandwidth selection,with n = 500. Top right: true density f , top left: estimator f , bottom: sections, dark line for f and light line for the estimator

Figure 3 .

 3 Figure 3. Dependent case, global bandwidth selection,with n = 500. Top right: true density f , top left: estimator f , bottom: sections, dark line for f and light line for the estimator

  j )|(1 + u 2 j ) β j du j sup u j ∈R |K * j (u j )| 2 u

Table 1 .

 1 16/17 and b = 40/17 in both cases. We use that µ 2 = 1/16. MISE ×100 averaged over 100 samples

		n = 100 n = 300	n = 500	n = 750	n = 1000
	Ex 1 Global	0.419	0.289	0.215	0.161	0.137
	Ex 1 Pointwise	0.269	0.140	0.101	0.083	0.068
	Ex 2 Global	3.615	1.699	0.761	0.473	0.367
	Ex 2 Pointwise	3.477	1.714	0.799	0.526	0.363
	Ex 3 Global	0.800	0.402	0.303	0.248	0.212
	Ex 3 Pointwise	0.622	0.293	0.212	0.167	0.138

Table 2 .

 2 This is an example with pointwise rate [log(n)] -4 and global rate (log(n)) -9/2 (which is not so slow, for instance, for n = 1000, this term is smaller than 1/n). C oracle averaged over 100 samples

		n = 100 n = 300	n = 500	n = 750	n = 1000
	Ex 1	1.48	2.04	2.01	1.96	1.97
	Ex 2	1.08	1.03	1.05	1.07	1.25
	Ex 3	1.36	1.53	1.57	1.57	1.62

  ≤ 2 exp -nη 2 /(2v 2 + 2bη) . Lemma 2 follows.

	Lemma 3. [Talagrand Inequality] Let Y 1 , . . . , Y n be i.i.d. random variables and ν n (t) = 1 n E(ψ t (Y i ))] for t belonging to B a countable subset of functions. For any η > 0,	n i=1 |ψ t (Y i )-
	(27)	P(sup	η 2 6	nH 2 v	, exp -	min(η, 1)η 21	nH M	.
	and
	(28)

, p.366 that

P(|S n (T ) -E(S n (T ))| ≥ nη) t∈ B |ν n (t)| ≥ (1 + 2η)H) ≤ max exp -

  (t) is an empirical process, such that t → ν n (t) is continuous. We can apply Talagrand Inequality recalled in Lemma 3. To this aim, we compute H 2 , M and v.≤ E( fhf h 2 ) ≤ V (h) ≤ V (h)C(h) =: H 2 .

	where
						ψ t (y) =	1 (2π) d	e i u,y t * (-u)	K * h (u) f * ε (u)	du.
	then ν n First
	E	sup t∈ B(0,1)	ν 2 n (t)	= E	sup t∈ B(0,1)	fh -f h , t 2 ≤ E	sup t∈ B(0,1)	fh -f h	2 t 2
	Next,
	sup t∈ B(0,1)	ψ t ∞ =	sup t∈ B(0,1)	sup y∈R d	1 (2π) d	e i u,y t * (-u)	K * h (u) f * ε (u)	du
					≤	sup t∈ B(0,1)		1 (2π) d	t * 2	K * f * ε (u) h (u)	2	du	1/2
	Last,
	sup t∈ B(0,1)	Var(ψ t (Y 1 )) ≤	sup t∈ B(0,1)	E	1 (2π) d	e i u,Y 1 t * (u)	K * f * ε (u) h (u)	du	2
					≤	sup t∈ B(0,1)	1 (2π) 2d	t * (u)t * (-v)	K * h (u) f * ε (u)
										d , such
	a set exists).
	Now
					fh -f h , t =	1 n

n i=1 |ψ t (Y i ) -E(ψ t (Y i ))] =: ν n (t) ≤ nV (h) =: M.

  Now, it is sufficient to use assumption (20) to obtain nH 2 /v ≥ (log n) 2 . Moreover (1+2η)H = Ṽ (h)/(1 + K * ∞ ). Then Proof of Corollary 2 and Corollary 3. We proceed as in the proof of Corollary 1 and we get(t j h j )| 2 |f * ε,j (t j )| -2 dt j sup t j ∈R |K * j (t j h j )| 2 |f * ε,j (t j )| -2 (u j )| 2 |f * ε,j (u j /h j )| -2 du j sup u j ∈R |K * j (u j )| 2 |f * ε,j (u j /h j )| -2 .

	P(sup t∈ B |ν n (t)| ≥ Ṽ (h)/(1 + K *	∞ )) ≤ max exp -	η 2 6	(log n) 2 , exp -	min(η, 1)η 21	√	n
				≤ exp -	min(η, 1)η 46	(log n) 2 .
	7.10. K * h /f * ε K * h /f * ε	2 2 2 ∞	≈ j ≈ d j=1 |K *   d j=1 1 j=1 h j   d	|K * j
									2 ).
	Inequality (27) gives
	P(sup					η 2 6	nH 2 v	, exp -	min(η, 1)η 21	√	n	.

t∈ B |ν n (t)| ≥ (1 + 2η)H) ≤ max exp -

Acknowledgements.

The authors are grateful to Oleg Lepski and Vincent Rivoirard for helpful and decisive discussions.