Yannick Baraud
email: baraud@unice.fr

ESTIMATOR SELECTION WITH RESPECT TO HELLINGER-TYPE RISKS

Keywords: May, 9 2009. 2000 Mathematics Subject Classification. Primary 62G05, Secondary 62N02, 62M05, 62M30, 62G07 Estimator selection, Model selection, Variable selection, T -estimator-Histogram, Estimator aggregation, Hellinger loss

Estimator selection with respect to Hellinger-type risks.

Introduction

1.1. The statistical setting. Let N 1 , . . . , N k be k independent random measures. Each N i is defined on an abstract probability space (Ω, T , P) and takes its values in the class of positive measures on a measured space (X i , A i , µ i ). Besides, we assume that (1)

E[N i (A)] =

A s i dµ i < +∞, for all A ∈ A i for some nonnegative and measurable function s i on X i . We shall call s i the intensity of N i . Equality (1) implies that N i is finite a.s. and that for all measurable and nonnegative functions f i on X i ,

(2)

E X i f i dN i = X i f i s i dµ i .
Our aim is to estimate s = (s 1 , . . . , s k ) from the observation of N = (N 1 , . . . , N k ).

Throughout, we shall set X = (X 1 , . . . , X k ), A = (A 1 , . . . , A k ), µ = (µ 1 , . . . , µ k ) and denote by L the cone of nonnegative and measurable functions t of the form (t 1 , . . . , t k ) where the t i are nonnegative and integrable functions on (X i , A i , µ i ). Moreover, for f = (f 1 , . . . , f k ) ∈ L we shall use the notations

X f dN = k i=1 X i f i dN i and X f dµ = k i=1 X i f i dµ i .
Finally, L 0 will denote a known subset of L containing the target function s.

This statistical framework allows to deal simultaneously with the more classical ones given below:

Example 1 (Density Estimation). Consider the problem of estimating a density s on (X , A, µ) from the observation of an n-sample X 1 , . . . , X n with distribution P s = s • µ. To handle this problem, we shall take k = 1, N = n -1 n i=1 δ X i and L 0 the set of densities on (X , A) with respect to µ.

Example 2 (Estimation of marginals). Let X 1 , . . . , X n be independent random variables with values in the measured spaces (X 1 , A 1 , µ 1 ), . . . , (X n , A n , µ n ) respectively. We assume that for all i, X i admits a density s i with respect to µ i and our aim is to estimate s = (s 1 , . . . , s n ) from the observation of X = (X 1 , . . . , X n ). We shall deal with this problem by taking k = n and N i = δ X i for i = 1, . . . , n. Note that this setting includes as a particular case that of the regression framework

(3)

X i = f i + ε i , i = 1, . . . , n
where the f i are unknown real numbers and the ε i = X i -f i are i.i.d. random variables with known distribution q. In this case s i (x) = q(x -f i ) for all i = 1, . . . , n and the problem of estimating the densities of the X i amounts to estimating the shift parameter f = (f 1 , . . . , f n ).

Example 3. Let X 1 , . . . , X n be n independent, nonnegative and integrable random variables. Our aim is to estimate the function s given by s(i) = E(X i ) < +∞ for i ∈ X = {1, . . . , n} on the basis of the observation X = (X 1 , . . . , X n ). This statistical setting is a particular case of our general one by taking k = 1, A = P(X ), µ the counting measure on (X , A), L 0 = L and N the measure defined for A ⊂ X by N (A) = i∈A X i .

Among the marginal distributions of X we have in mind, we mention the Binomial or Gamma among others.

Example 4 (Estimating the intensity of a Poisson process). Consider the problem of estimating the intensity s of a possibly inhomogeneous Poisson process N on a measurable space (X , A). We shall assume that s is integrable. This statistical setting is a particular case of our general one by taking k = 1 and L 0 = L.

Hereafter, we shall deal with estimators with values in L 0 and to measure their risks, endow L 0 with the distance H defined for t, t ′ in L 0 by

H 2 (t, t ′ ) = 1 2 X √ t - √ t ′ 2 dµ = 1 2
When k = 1 and t, t ′ are densities with respect to µ, H is merely the Hellinger distance h between the corresponding probabilities. Given an estimator ŝ of s, i.e. a measurable function of N with ŝ ∈ L 0 , we define its risk by E H 2 (s, ŝ) .

1.2. An account of the results. We start with an arbitrary collection E = {ŝ λ , λ ∈ Λ} of estimators based on N together with a family S of subsets of L 0 . The family E need not be countable even though we shall assume so in order to avoid measurability problems. In fact, the reader can check that the cardinality of E will play no role in our results. In contrast, the family S should be countable (we shall use the word countable for finite or countable) and its complexity is measured by means of a mapping ∆ from S into [1, +∞) satisfying (4) Σ = S∈S e -∆(S) < +∞.

When Σ = 1, e -∆ corresponds to a prior distribution on the family S and gives thus a Bayesian flavor to our statistical procedure. The fact that ∆ is assumed to be not smaller than 1 (although one usually assumes ∆ ≥ 0) is only here to simplify the presentation of the results. Hereafter, the elements S of S will be called models and assumed to have finite metric dimensions D(S) (in an appropriate sense).

In the present paper, the problem we consider is that of estimator selection. More precisely, our aim is to select some estimator ŝλ among the collection E from the same observation N in view of achieving the smallest risk bound over E. For appropriate choices of E and S, our approach allows us to deal simultaneously with the problems of model selection, (convex) aggregation and construction of T -estimators. As we shall see, very little assumptions on the estimators ŝλ will be required and in fact the way they depend on N need not even be known. Nevertheless, there should be some connections between the families E and S. Typically, each ŝλ should belong (or at least should be close enough) to S∈S S. More precisely, we associate to each estimator ŝλ a (possibly random) subfamily S λ ⊂ S of approximation models and introduce the accuracy index of ŝλ with respect to S λ as the (random) quantity ( 5)

A(ŝ λ , S λ ) = inf S∈S λ inf t∈S H 2 (ŝ λ , t) + pen λ (t) ,
where pen λ is a penalty function from S∈S λ S into R + . Typically, pen λ is of the form (6) pen λ (t) = c 0 τ inf

S∈S λ (t) (D(S) + ∆(S)) ,
where c 0 is a universal constant in (0, 1 -1/ √ 2), S λ (t) = {S ∈ S λ , t ∈ S} and τ is a scaling parameter depending on the statistical setting (τ is of order 1/n for Example 1 and of order a universal constant for Examples 2, 3 and 4). The quantity A(ŝ λ , S λ ) measures in some sense the complexity of the estimator ŝλ with respect to the collection S λ . For a choice of the penalty given by ( 6), the model S λ which minimizes (5) over S λ achieves the best trade-off between the approximation term inf t∈S H 2 (ŝ λ , t) and the complexity term τ (D(S) ∨ ∆(S)). The selection procedure we propose leads to an estimator s = ŝλ which satisfies the following inequality for some constant C ∈ (0, 1) which neither depends on τ nor s

(7) CE H 2 (s, s) ≤ inf λ∈Λ E H 2 (s, ŝλ ) + E [A(ŝ λ , S λ )] .
Inequality (7) leads to an oracle inequality as soon as the quantity

E [A(ŝ λ , S λ )] = E inf S∈S λ inf t∈S H 2 (ŝ λ , t) + pen λ (t)
is not larger that E H 2 (s, ŝλ ) up to a universal constant whatever s. Such a property depends on the choice of the subfamilies S λ . From a theoretical point of view, the choices S λ = S for all λ ∈ Λ lead to the smallest values of A(ŝ λ , S λ ). Nevertheless, for computational reasons it may be sometimes convenient to reduce the family S λ to a smaller number of models.

Selecting among estimators is an old problem in statistics. In density or regression, most of the statisticians use resampling techniques (cross-validation, V -fold,...). They seem to give satisfactory results in practice but little is known on the theoretical performances of the resulting choice. In the opposite, we provide a non-asymptotic risk bound for the estimator we select but more needs to be done to make our procedure practical. We shall point out the difficulties to be overcome in view of computing the final estimator s and also describe some situations for which these computations are indeed feasible.

1.3. Connections with Birgé's T -estimators. The starting point of this paper originates from a series of papers by Lucien [START_REF] Birgé | Model selection via testing: an alternative to (penalized) maximum likelihood estimators[END_REF]2007;2008) providing a new perspective on estimation theory. His approach relies on ideas borrowed from old papers by Le [START_REF] Cam | Convergence of estimates under dimensionality restrictions[END_REF][START_REF] Cam | On local and global properties in the theory of asymptotic normality of experiments[END_REF] and [START_REF] Birgé | Approximation dans les espaces métriques et théorie de l'estimation[END_REF][START_REF] Birgé | Sur un théorème de minimax et son application aux tests[END_REF]1984a), showing how to derive good estimators from families of robust tests between simple hypotheses, and also from more recent ones about complexity and model selection such as [START_REF] Barron | Minimum complexity density estimation[END_REF] and [START_REF] Barron | Risk bounds for model selection via penalization[END_REF]

C ′ E d 2 (s, ŝ m) ≤ inf m∈M E d 2 (s, ŝm ) + τ (D(S m ) ∨ ∆(S m )) .
There exist many different ways of designing a selection rule. Some are based on the minimization of a penalized criterion. For example, let us mention Castellan (2000a;2000b) and [START_REF] Massart | Concentration inequalities and model selection[END_REF] (Chapter 7) for the problem of estimating a density, [START_REF] Reynaud-Bouret | Adaptive estimation of the intensity of inhomogeneous Poisson processes via concentration inequalities[END_REF] for that of estimating the intensity of a Poisson process and in the regression setting [START_REF] Baraud | Model selection for regression on a fixed design[END_REF], [START_REF] Birgé | Gaussian model selection[END_REF] and [START_REF] Yang | Model selection for nonparametric regression[END_REF] among other references. Another way, usually called Lepski's method, appears in a series of papers by Lepski (1990;[START_REF] Lepskiȋ | Asymptotically minimax adaptive estimation. I. Upper bounds. Optimally adaptive estimates[END_REF]1992a;[START_REF] Lepskiȋ | On problems of adaptive estimation in white Gaussian noise[END_REF] and was originally designed to perform model selection among collections of nested models. In a more abstract way, [START_REF] Birgé | Model selection via testing: an alternative to (penalized) maximum likelihood estimators[END_REF] proposed a way of selecting among T -estimators and closer to ours, Baraud and [START_REF] Baraud | Estimating the intensity of a random measure by histogram type estimators[END_REF] suggested to compare pair by pair histogram-type estimators in the statistical frameworks described in Examples 1 and 4 (among others). Finally, we mention that other procedures based on resampling have interestingly emerged from the work of [START_REF] Arlot | Rééchantillonnage et Sélection de modèles[END_REF]2009) andCélisse (2008).

Our approach to estimator selection provides an alternative to solve the problem of model selection. By choosing d = H, Λ = M, S m = {S m } and pen m given by ( 6) for all m ∈ M, we have

A(ŝ m , S m ) ≤ 2c 0 τ (D(S m ) ∨ ∆(S m )) , ∀m ∈ M
and it is then straightforward to deduce from (7) an oracle-type inequality such as (8) for the estimator we select. Compared to the model selection procedures mentioned above, we shall see that ours possesses the advantage to apply in many statistical settings simultaneously and to require very few assumptions on the collection of models.

1.5. Organization of the paper. The paper is organized as follows. The basic ideas underlying our approach are described in Section 2. The main assumptions on the measure N and the family S are presented and discussed in Section 3 on the basis of Examples 1 to 4. The selection procedure and the main result can be found in Section 4. In Section 5 we consider models S which consist of piecewise constant functions on partitions of X . For the problem of estimating a density, we give a practical way of choosing the number of cells of a regular histogram (regular in the sense that each cell contains the same number of data, with a possible exception for the right-most). In Section 6, we deal with models with bounded metric dimensions and, as an application, handle the problems of (convex) aggregation and construction of T -estimators. The problem of estimating the means of nonnegative random variables as presented in Example 3 is tackled in Section 7. We establish there uniform rates of estimation over various classes of means and provide a lower bound on the minimax estimation rate over classes for which √ s = ( s(1), . . . , s(n)) belong to a given linear space. In Section 8, we consider the regression framework presented in Example 3 and deal with the problems of model selection, complete variable selection and that of selecting among linear estimators under weak integrability properties of the errors. Finally, Section 9 is devoted to the proofs.

Throughout, we use the following notations. The quantity |E| denotes the cardinality of a finite set E.

For x ∈ R + , ⌊x⌋ = sup {n ∈ N, n ≤ x}. The Euclidean norm of R n is denoted . We set N * = N \ {0}, R * + = R + \ {0} and for t ∈ R * n + , we denote by √ t the vector √ t 1 , . . . , √ t n . Given a closed convex subset A of an
Hilbert space, Π A denotes the projection operator onto A. For t ∈ L 0 and F ⊂ L 0 , we set

H(t, F) = inf f ∈F H(t, f ) and for y > 0, B(t, y) = {t ′ ∈ L 0 , H(t, t ′ ) ≤ y}. Throughout C, C ′ , C ′′ , .
.. denote constants that may vary from line to line.

Basic formulas and basic ideas

The aim of this section is to present the basic formulas and ideas underlying our approach. For the sake of simplicity, we assume that k = 1 until further notice. For t ∈ L 0 , we define ρ(s, t) = X √ st dµ. This quantity corresponds to the Hellinger affinity whenever s and t are densities. The squared distance H 2 (s, t) is related to ρ(s, t) by the formula 2H 2 (s, t) = X sdµ+ X tdµ-2ρ(s, t). Throughout, t, t ′ denote two elements of L 0 one should think of as estimators of s. One would prefer t

′ to t if H 2 (s, t ′ ) is smaller than H 2 (s, t) or equivalently if ρ(s, t ′ ) - 1 2 X t ′ dµ -ρ(s, t) - 1 2 X tdµ ≥ 0.
Since X tdµ and X t ′ dµ are both known, deciding whether t ′ is preferable to t amounts to estimating ρ(s, t) and ρ(s, t ′ ) in a suitable way. In the following sections, we present the material that enables us to estimate these quantities on the basis of the observation N .

2.1. An approximation of ρ(., .). For a measure ν on (X , A) and t, r ∈ L 0 , we set

(9) ρ r (ν, t) = 1 2 ρ(t, r) + X t r dν ≤ +∞
(using the conventions 0/0 = 0 and a/0 = +∞ for all a > 0). We start with the following result showing that ρ r (s • µ, t) over-approximates ρ(s, t).

Proposition 1. Let s, t, r ∈ L 0 . We have,

ρ r (s • µ, t) -ρ(s, t) = 1 2 X t r √ s - √ r 2 dµ ≥ 0. Besides, if r = (t + t ′ )/2 with t ′ ∈ L 0 then (10) 0 ≤ ρ r (s • µ, t) -ρ(s, t) ≤ 1 √ 2 H 2 (s, t) + H 2 (s, t ′ ) . Proof. It follows from the definition of ρ r that 2 [ρ r (s • µ, t) -ρ(s, t)] = X √ tr dµ + X t r sdµ -2 X √ st dµ = X t r √ s - √ r 2 dµ.
For the second part, note that (t/r)(x) ≤ 2 for all x ∈ X and therefore ρ r (s • µ, t) -ρ(s, t) ≤ √ 2 H 2 (s, r). It remains to bound H 2 (s, r) from above. The concavity of the map t → √ t implies that ρ(s, r) ≥ [ρ(s, t) + ρ(s, t ′ )] /2 and therefore 2H 2 (s, r) ≤ H 2 (s, t) + H 2 (s, t ′ ), which leads to the result.

The important point about Proposition 1 (more precisely inequality (10)) lies in the fact that the constant 1/ √ 2 is smaller than 1. This makes it possible to use the (sign of the) difference

T (s • µ, t, t ′ ) = ρ r (s • µ, t ′ ) - 1 2 X t ′ dµ -ρ r (s • µ, t) - 1 2 X tdµ
with r = (t + t ′ )/2 as an alternative benchmark to find which between t and t ′ is the closest element to s (up to a multiplicative constant). More precisely, we can deduce from Proposition 1 the following corollary.

Corollary 1. If T (s • µ, t, t ′ ) ≥ 0, then H 2 (s, t ′ ) ≤ √ 2 + 1 √ 2 -1 H 2 (s, t).
Proof. Using inequality (10) and the assumption, we have

H 2 s, t ′ -H 2 (s, t) = ρ(s, t) - 1 2 X tdµ -ρ(s, t ′ ) - 1 2 X t ′ dµ = ρ r (s • µ, t) - 1 2 X tdµ -ρ r (s • µ, t ′ ) - 1 2 X t ′ dµ + ρ (s, t) -ρ r (s • µ, t) + ρ r s • µ, t ′ -ρ s, t ′ ≤ 1 √ 2 H 2 (s, t) + H 2 s, t ′
which leads to the result.

2.2. An estimator of ρ r (., .). Throughout, given t, t ′ ∈ L 0 , we set

r = t + t ′ 2 ∈ L 0 .
The superiority of the quantity ρ r (s • µ, t) over ρ (s, t) lies in the fact that the former can easily be estimated by its empirical counterpart, namely

(11) ρ r (N, t) = 1 2 ρ(t, r) + t r dN .
Note that ρ r (N, t) is an unbiased estimator of ρ r (s • µ, t) because of (2). Consequently, a natural way of deciding which between t and t ′ is the closest to s is to consider the test statistic

T (N, t, t ′ ) = ρ r (N, t ′ ) - 1 2 X t ′ dµ -ρ r (N, t) - 1 2 X tdµ .
Replacing the "ideal" test statistic T (s • µ, t, t ′ ) by its empirical counterpart leads to an estimation error given by the process Z(N, ., .) defined on L 2 0 by

Z(N, t, t ′ ) = T (N, t, t ′ ) -T (s • µ, t, t ′ ) = ρ r N, t ′ -ρ r s • µ, t ′ -[ρ r (N, t) -ρ r (s • µ, t)] = X ψ(t, t ′ , x)dN - X ψ(t, t ′ , x)sdµ where ψ(t, t ′ , x) is the function on L 2 0 × X with values in [-1/ √ 2, 1/ √ 2] given by (12) ψ(t, t ′ , x) = 1 √ 2 1 1 + t(x)/t ′ (x) - 1 1 + t ′ (x)/t(x)
.

The study of the empirical process Z(N, ., .) over the product space S × S ′ is at the core of our techniques.

2.3. The multidimensional case k > 1. In the multidimensional case, the same results can be obtained by reasoning component by component. More precisely, the formulas of the above sections extend by using the convention that for all k-uplets ν = (ν 1 , . . . , ν k ) of measures on (X 1 , A 1 ), . . . , (X k , A k ) respectively,

X φ(s, t, t ′ , r)dν = k i=1 X i φ(s i , t i , t ′ i , r i )dν i ,
whatever the functions s, t, t ′ , r ∈ L 0 and mappings φ from R 4 + into R.

Assumptions on N and S

Let τ, γ be positive numbers. For (t, t ′ ) ∈ L 2 0 and y > 0, let us set

w 2 (t, t ′ , y) = H 2 (s, t) + H 2 s, t ′ ∨ y 2 .
We assume that the family S and the measure N satisfy the following.

Assumption 1. Let τ and γ be fixed positive numbers and c 0 ∈ (0, 1 -1/ √ 2). For all pairs (S, S ′ ) ∈ S 2 , there exist positive numbers D(S), D(S ′ ) such that for all ξ > 0 and y 2 ≥ τ (D(S) ∨ D(S ′ ) + ξ),

P sup (t,t ′ )∈S×S ′ Z(N, t, t ′ ) w 2 (t, t ′ , y) > c 0 ≤ γe -ξ .
This assumption means that for ξ large enough the random process Z(N, t, t ′ ) is uniformly controlled by w 2 (t, t ′ , y) over S × S ′ with probability close to 1. As we shall see on examples, such a property can be derived from concentration inequalities. Under suitable assumptions, the quantities D(S) measure (in some sense) the massiveness of the parameter sets S. Assumption 1 is met in the following typical examples.

3.1. Discrete models. When the collection S consists of discrete models S, Assumption 1 holds under mild conditions on N . The proof of the following proposition is postponed to Section 9.1.

Proposition 2. Let a, b, c and M be nonnegative numbers and c 0 ∈ (0, 1 -1/ √ 2). Assume that N satisfies for all y, ξ > 0 (13) sup t,t ′ ∈B(s,y)

P Z(N, t, t ′ ) > ξ ≤ b exp - aξ 2 y 2 + cξ .
Besides, assume that for all S ∈ S there exists η(S) ≥ 1/2 such that for all R ≥ 2η(S)

(14) S ∩ B(s, R √ τ ) ≤ M exp R 2 2 with τ = 4(2 + cc 0 ) ac 2 0 .
Then, Assumption 1 holds with γ = bM 2 and D(S) = 4η 2 (S) for all S ∈ S.

Inequality
(15) E e u(X i -s(i)) ≤ exp u 2 σs(i) 2(1 -|u|β) ∀u ∈ (-1/β, 1/β),
then ( 13) holds with a = 1/(12σ), b = 1 and c = β √ 2/(12σ).

The value of τ given in Proposition 2 is of order 1/n in the density case and is of order a constant in the other cases. The proof of Proposition 3 is postponed to Section 9.2.

3.2. Piecewise constant parameter sets. Assume that k = 1 and define for any finite partition m of X the set S m gathering the elements of L 0 which are piecewise constant on each element of the partition m, that is

S m = I∈m a I 1l I (a I ) I∈m ∈ R |m| L 0 .
Let M be a countable set consisting of such partitions. The family S = {S m , m ∈ M} and the measure N satisfy Assumption 1 provided that the following holds.

Proposition 4. Let a and δ be positive numbers. For any finite partition m of X , set

X 2 (m) = I∈m N (I) -E(N (I))
2 and assume that N satisfies for all ξ > 0 (16)

P X 2 (m) ≥ a (|m| + ξ) ≤ e -ξ .
Besides, assume that for all m, m ′ ∈ M,

(17) m ∨ m ′ ≤ δ |m| ∨ m ′ where m ∨ m ′ = {I ∩ I ′ , (I, I ′ ) ∈ m × m ′ }. Then Assumption 1 holds with γ = 1, τ = 20ac -2 0 and D(S m ) = δ|m| for all m ∈ M.
In this case, the parameter D(S m ) is proportional to the dimension of the linear space generated by S m . The assumptions given by ( 16) and ( 17) also appeared in [START_REF] Baraud | Estimating the intensity of a random measure by histogram type estimators[END_REF] as Assumptions H and H' in their Theorem 6. Inequality ( 16) can be obtained from concentration inequalities of suprema of empirical processes (based on N ) over classes of uniformly bounded functions. In particular, the following result is proved in [START_REF] Baraud | Estimating the intensity of a random measure by histogram type estimators[END_REF].

Proposition 5. Inequality ( 16) holds with a = 200/n in the case of Example 1, with a = 6 in the case of Example 4 and, in the case of Example 3, with

a = 3κ 1/ √ 2 + β κ - 1 2 +
provided that for some β ≥ 0 and κ > 0, the X i satisfy for i = 1, . . . , n

E e u(X i -s(i)) ≤ exp κ u 2 s(i) 2(1 -uβ) for all u ∈ 0, 1 β , with the (convention 1/β = +∞ if β = 0), and E e -u(X i -s(i)) ≤ exp κ u 2 s(i) 2 for all u ≥ 0.
One can check that the value of τ given in Proposition 4 is then of order 1/n in the density case and otherwise is of order a constant.

The selection procedure and the main result

Let {ŝ λ , λ ∈ Λ} be a collection of estimators of s with values in L 0 based on the observation N and let S be a countable family of subsets S of L 0 . We recall that together with S, we consider a nonnegative map ∆ on S satisfying (4) and in order to simplify the presentation of our results, we assume that ∆(S) ≥ 1 for all S ∈ S.

Throughout, c 0 is an arbitrary number in (0, 1 -1/ √ 2).

4.1. The estimation procedure. We associate to each λ ∈ Λ, both a family of (possibly random) subsets S λ of S and a penalty function pen λ from S∈S λ S into R + . The procedure includes three steps.

Step 1: Construction of intermediate estimators.

Let τ > 0. For each λ, define sλ as any element of S∈S λ S such that

(18) H 2 (ŝ λ , sλ ) + pen λ (s λ ) ≤ A(ŝ λ , S λ ) + c 0 τ
where A(ŝ λ , S λ ) is given by ( 5). When S λ reduces to a single model S λ containing ŝλ with probability one and when pen λ is constant over S λ , one may choose sλ = ŝλ , what we shall do for simplicity throughout this paper.

Step 2: Pairwise comparison of the estimators sλ . Given a pair (s λ , sλ ′ ) such that sλ = sλ ′ , we consider the test statistic

T(N, sλ , sλ ′ ) = ρ r (N, sλ ′ ) - 1 2 X sλ ′ dµ -pen λ ′ (s λ ′ ) (19) -ρ r (N, sλ ) - 1 2 X sλ dµ -pen λ (s λ )
where r = (s λ + sλ ′ )/2 and ρ r (N, .) is given by (11). We set

E(s λ ) = {s λ ′ , T(N, sλ , sλ ′ ) ≥ 0} and note that either sλ ∈ E(s λ ′ ) or sλ ′ ∈ E(s λ ) since T(N, sλ , sλ ′ ) = -T(N, sλ ′ , sλ ).
Then, we define

D(s λ ) = sup H 2 (s λ , sλ ′ ) sλ ′ ∈ E(s λ ) if E(s λ ) = ∅ and D(s λ ) = 0 otherwise.
Step 3: The final selection. Select λ among Λ as any element satisfying

D(s λ) ≤ D(s λ ) + c 0 τ, ∀λ ∈ Λ
and λ as any element of Λ such that

H 2 (ŝ λ, sλ ) ≤ inf λ∈Λ H 2 (ŝ λ , sλ ) + c 0 τ.
Our final estimator is s = ŝλ .

4.2. Discussion about the procedure. The choice of the value c 0 is arbitrary in (0, 1 -1/ √ 2) and can be fixed to ( √ 2 -1)/(2 √ 2). It seemed interesting to show how the constants we get in the risk bounds were depending upon the choice of c 0 , at least in the proofs. Even though an optimization with respect to c 0 looks theoretically untractable, the computations show that choices of c 0 too close to 0 or to 1 -1/ √ 2 lead to large constants and should therefore be avoided.

Let us now discuss the implementation issues. The computation of the estimator s requires the comparison pair by pair of the estimators sλ defined in Step 1. The whole procedure may therefore be performed in about |Λ| 2 steps once the sλ are available. When the cardinality of Λ is not too large, the main difficulty lies in the computations of the sλ . In the most favorable situations, one may take S λ as a (possibly random) singleton S λ for each λ ∈ Λ and the penalty function pen λ to be constant over S λ . In this case, sλ may be chosen as the best approximation of ŝλ in S λ . Whenever the model S λ is simple enough, this step can therefore be performed in a reasonable amount of time. Nevertheless, we sometimes use models S which result from an abstract discretization of manifolds and, in this least favorable case, the sλ are abstract as well.

4.3. The main result. We recall that for all t ∈ L 0 and λ ∈ Λ,

S λ (t) = {S ∈ S λ , t ∈ S} .
We obtain the following result the proof of which postponed to Section 9.4.

Theorem 1. Let c 0 ∈ (0, 1 -1/ √ 2).
Assume that N and S satisfy Assumption 1 for some positive constants τ and γ and let ∆ be some mapping from S into [1, +∞) satisfying ( 4). By applying the selection procedure of Section 4.1 with penalties pen λ satisfying for all λ ∈ Λ and t ∈ S∈S λ S,

(20) pen λ (t) ≥ c 0 τ inf {D(S) + ∆(S), S ∈ S λ (t)} ,
the estimator s = ŝλ satisfies for all ξ > 0,

P CH 2 (s, s) ≥ inf λ∈Λ H 2 (s, ŝλ ) + A(ŝ λ , S λ ) + τ ξ ≤ γΣ 2 e -ξ ∧ 1,
where C is a positive constant depending on c 0 only and

A(ŝ λ , S λ ) = inf S∈S λ inf t∈S H 2 (ŝ λ , t) + pen λ (t) , ∀λ ∈ Λ.
In particular, by integration with respect to ξ, for some C ′ depending on c 0 , γ and Σ only

C ′ E H 2 (s, s) ≤ E inf λ∈Λ H 2 (s, ŝλ ) + A(ŝ λ , S λ ) ≤ inf λ∈Λ E H 2 (s, ŝλ ) + E [A(ŝ λ , S λ )] . (21)
We deduce from Theorem 1 the following corollary.

Corollary 2. Under the assumptions of Theorem 1, if for all λ ∈ Λ, ŝλ ∈ S∈S λ S with probability 1 and if equality holds in ( 20), then

C ′ E H 2 (s, s) ≤ inf λ∈Λ E H 2 (s, ŝλ ) + 2c 0 E v 2 (ŝ λ ) (22)
where C ′ depends on c 0 , γ and Σ only and

(23) v 2 (ŝ λ ) = τ inf S∈S λ (ŝ λ ) D(S) ∨ ∆(S) for all λ ∈ Λ.
Inequality ( 22) compares the risk of the resulting estimator s to those of the ŝλ plus an additional term E v 2 (ŝ λ ) . If for some deterministic S ∈ S λ , the estimator ŝλ belongs to S with probability 1, we obtain that

(24) v 2 (ŝ λ ) ≤ τ [D(S) ∨ ∆(S)]
and hence E v 2 (ŝ λ ) is small compared to the risk of ŝλ as soon as for some universal constant C ′′ > 0,

C ′′ E H 2 (s, ŝλ ) ≥ τ D(S) for all s ∈ L 0 .
We emphasize that (24) does not depend on the cardinality of the collection of estimators {ŝ λ , λ ∈ Λ}. In particular, if with probability one all the estimators ŝλ belong to a same deterministic model S ∈ λ∈Λ S λ , by setting ∆(S) = 1, the resulting estimator s satisfies

C ′′′ E H 2 (s, s) ≤ inf λ∈Λ E H 2 (s, ŝλ ) + τ (D(S) ∨ 1)
no matter how large this collection is. For a choice of {ŝ λ , λ ∈ Λ} which is countable and dense in S, we therefore get

sup s∈S E H 2 (s, s) ≤ C ′ τ (D(S) ∨ 1) ,
showing thus that the quantity τ (D(S) ∨ 1) is an upper bound for the minimax rate over S.

Selecting among histogram-type estimators

Throughout this section, k = 1 and we consider a countable family M of finite partitions of X satisfying (17). We associate to M the family of models S = {S m , m ∈ M} described in Section 3.2. We assume that the measure N satisfies ( 16) and set τ = 20ac -2 0 . As already seen in Proposition 4, inequalities (17) together with ( 16) imply that Assumption 1 holds and we may therefore apply our main theorem (Theorem 1). The statistical settings we have in mind include Examples 1, 3 and 4 for which we already know from Proposition 5 that ( 16) holds true for a suitable value of a. We restrict ourselves to families of estimators ŝλ with values in m∈M S m which allows us to associate to each λ some (possibly random) partition m(λ) ∈ M such that ŝλ ∈ S m(λ) with probability 1. For all λ ∈ Λ, we choose S λ = S m(λ) and pen λ constant over S m(λ) for the sake of simplicity. We may therefore take sλ = ŝλ for all λ ∈ Λ in our selection procedure which only depends now on the choice of pen λ (ŝ λ ). We deduce from Theorem 1 the following result.

Theorem 2. Assume that N and S satisfy ( 16) and ( 17) respectively. Let {ŝ λ , λ ∈ Λ} be a collection of estimators of s with values in m∈M S m . If for all λ ∈ Λ,

(25) pen λ (ŝ λ ) ≥ c 0 τ δ| m(λ)| + ∆(S m(λ) ) ,
the estimator s satisfies for some positive constant C depending on c 0 and Σ only,

CE H 2 (s, s) ≤ E inf λ∈Λ H 2 (s, ŝλ ) + pen λ (ŝ λ ) ≤ inf λ∈Λ E H 2 (s, ŝλ ) + E [pen λ (ŝ λ )] .
In particular, if equality holds in ( 25)

C ′ E H 2 (s, s) ≤ inf λ∈Λ E H 2 (s, ŝλ ) + τ E | m(λ)| ∨ ∆(S m(λ) ) ,
for some C ′ depending on c 0 , Σ and δ only.

Let us now turn to examples.

5.1. Model selection. Theorem 2 holds for any choices of estimators {ŝ λ , λ ∈ Λ} with values in m∈M S m . However, the estimators ŝm defined by

(26) ŝm = I∈m N (I) µ(I) 1l I for m ∈ M
are of special interest. Note that when µ(I) = 0, E(N (I)) = I sdµ = 0 and N (I) = 0 a.s., the estimator ŝm is well-defined with the convention 0/0 = 0 and c/∞ = 0 for c > 0. Besides, in the context of Example 1, ŝm belongs to L 0 (that is, ŝm is a density on X ) as soon as µ is finite on X . For these estimators, one can prove (we refer to [START_REF] Baraud | Estimating the intensity of a random measure by histogram type estimators[END_REF]) that for all m ∈ M,

E H 2 (s, ŝm ) ≤ 4 H 2 (s, S m ) + τ |m| .
By applying Theorem 2 with Λ = M and for all m ∈ M, S m = {S m } and

pen m (t) = c 0 τ (δ|m| + ∆(S m )) , ∀t ∈ S m
we obtain the following result.

Corollary 3. Assume N and S satisfy ( 16) and ( 17) respectively. The estimator s satisfies for some constant C depending on c 0 , δ and Σ only

CE H 2 (s, s) ≤ inf m∈M E H 2 (s, ŝm ) + τ (|m| ∨ ∆ m ) ≤ inf m∈M H 2 (s, S m ) + τ (|m| ∨ ∆ m ) .
This corollary recovers the results of Theorem 6 in [START_REF] Baraud | Estimating the intensity of a random measure by histogram type estimators[END_REF] even though the selection procedure is different. The choice of a suitable family M of partitions is of course a crucial point in view of deducing nice statistical properties for s. This point has been discussed in [START_REF] Baraud | Estimating the intensity of a random measure by histogram type estimators[END_REF] (see their Section 3). However, the families M proposed there have large cardinalities and the computation of s becomes unfortunately NP-hard. 5.2. Selecting among histograms based on random partitions. In this section, we focus on the problem of estimating a density or the intensity of a Poisson process by an histogram(-type) estimator. More precisely, we consider the frameworks described in Examples 1 and 4 with X = [0, 1) and µ the Lebesgue measure. As already mentioned, the model selection approach developed in the previous section has the advantage to design an estimator possessing nice theoretical properties for suitable choices of families M but also has the drawback to be practically intractable for such choices. In view of designing a practical procedure for the problem of estimating a density, one may prefer families containing fewer but possibly datadependent partitions. For illustration, we consider a family of random partitions the elements of which contain a same number of data (with a possible exception for the rightmost). Our selection procedure gives a practical way of choosing the number of data that should be put in each bin of an histogram and provides thus an alternative to the cross-validation techniques which are quite popular in density for selecting such tuning parameters. This family of partitions can also be used for the problem of estimating the intensity of a Poisson process, the only difference being that the number n = N ([0, 1)) of observations becomes now randomly drawn from a Poisson distribution with parameter n = 1 0 s(x)dx (that we assume to be positive).

The procedure. For n ≥ 1, let M be the random variable defined as the smallest integer for which each interval [(i -1)/M, i/M [ with i = 1, . . . , M contains at most one datum. For λ ∈ {1, . . . , n}, define m(λ) as any partition of [0, 1) based on the grid {i/ M , i = 1, . . . , M } for which |I ∩ {X 1 , . . . , X n }| = λ for all intervals I in m(λ) (with a possible exception for the rightmost). Note that | m(λ)| ≤ nλ -1 + 1. Let us now define our collection of estimators. It would be natural to take Λ = {1, . . . , n} in order to index the family of estimators ŝ m(λ) defined by (26). However, this choice of Λ has the drawback to be random in the Poisson case and we rather take Λ = N * and define ŝλ as ŝ m(λ) when λ ≤ n and as n1l [0,1[ when λ > n. Note that both collections {ŝ λ , λ ∈ Λ} and ŝ m(λ) , λ ∈ {1, . . . , n} coincide on the event {n ≥ 1}. As to the family M, we introduce for each positive integer M the family of partitions M M consisting of intervals with end points belonging to {i/M, i = 0, . . . , M } and set M = M ≥1 M M . Finally, we define ∆ as follows. First, for M = 1 M 1 reduces to {[0, 1)} and we set ∆(S {[0,1)} ) = 1. Then we define recursively for M ≥ 2 and

m ∈ M M \ M -1 k=1 M k , ∆(S m ) = (|m| + 2) log(M ). Our choice of ∆ satisfies (4) since m∈M e -∆(Sm) ≤ M ≥1 M D=1 m∈M M ,|m|=D e -(D+2) log(M ) ≤ M ≥1 M -2 1 + M -1 M ≤ eπ 2 6 < +∞.
The computation of s requires n(n -1)/2 steps to compare the estimators ŝλ pair by pair plus n additional steps to minimize λ → D(ŝ λ ) for λ ∈ {1, . . . , n}. The whole selection procedure can therefore by implemented in about n 2 steps.

In view of facilitating the comparison of the risk bounds between the density and Poisson frameworks, we shall use the notation h for the Hellinger distance in the density case and the normalized Hellinger-type distance H/ √ n in the Poisson case.

Corollary 4. Assume that s Lq = 1 0 s q 1/q < +∞ for some q > 1. If for all λ ∈ Λ, pen λ (ŝ λ ) = 2c 0 τ (nλ -1 + 1) log(e + M )1l λ≤n + 1l λ>n , the estimator s = ŝλ satisfies for some C depending on q only

CE h 2 (s, s) ≤ inf 1≤λ≤n   E h 2 (s, ŝλ ) + log e + n 2 s Lq λ  
in the density case, and in the Poisson case

CE H 2 (s, s) n ≤ inf λ≥1   E H 2 (s, ŝλ ) n + log e + n2 s/n Lq λ ∧ n   .
Even though the estimators ŝλ are widely used in practice, at least for the purpose of estimating a density, little is known about their risks. In density estimation, the only result we are aware of is due to [START_REF] Lugosi | Consistency of data-driven histogram methods for density estimation and classification[END_REF] showing that if λ = λ(n) satisfies both λ(n) → +∞ and λ(n)/n → 0 as n tends to infinity, the L 1 -norm between s and ŝλ(n) tends to 0 a.s. (and therefore so does the Hellinger distance). The assumption that for some q > 1, s q is integrable is technical and ensures that the cardinality M of the random grid {i/ M , i = 1, . . . , M } keeps to a reasonable size as n increases.

Collections of models with bounded metric dimensions

Throughout this section, we consider a family S of subsets of L 0 with metric dimensions bounded by D(., .) (in the sense of Definition 6 in [START_REF] Birgé | Model selection via testing: an alternative to (penalized) maximum likelihood estimators[END_REF]). More precisely, we assume that for some universal constant M > 0, all S ∈ S and η > 0 there exist a number D(S, η) ∈ [1/2, +∞) and a discrete subset

S[η] ⊂ L 0 such that (27) H(t, S[η]) ≤ η √ τ , for all t ∈ S
and for all s ∈ L 0 and R ≥ 2η, ( 28)

S[η] ∩ B(s, R √ τ ) ≤ M exp D(S, η) R η 2 .
Furthermore, it is assumed that the mapping η → D(S, η) is right-continuous and, with no loss of generality, that is also non-increasing on (0, +∞). It follows from ( 27) and ( 28) that sets S with bounded metric dimensions may be approximated at any scale by discrete sets with controlled massiveness. Any compact set S has a bounded metric dimension: by definition for all η > 0 one can find a finite subset S[η] of S satisfying ( 27) and hence ( 28) holds with M = 1 and

D(S, η) = 4 -1 max log S[η] , 2 ≥ 1/2.
As to the measure N , we assume that it satisfies (13) and we take τ = 4(2 + cc 0 )/(ac 2 0 ) all along. As shown by Proposition 3, this assumption on N is met in all the examples of Section 1.1 and τ is then of order a constant (except in the density case where it is of order 1/n). To our knowledge, the results of this section are new for the problems presented in Examples 2 and 3 and we believe that they can also be solved by using the robust tests described in [START_REF] Birgé | Sur un théorème de minimax et son application aux tests[END_REF] and [START_REF] Birgé | Asymptotics: particles, processes and inverse problems[END_REF] in the contexts of Examples 1 and 4 respectively. 6.1. The selection procedure and the main result. We start with a collection of estimators {ŝ λ , λ ∈ Λ}, a family S of models S with bounded metric dimensions and a mapping ∆ on S with values in [1, +∞) satisfying (4). We associate to each model S ∈ S its discrete version

(29) S = S[η] with η = η(S) = inf η > 0, η 2 ≥ 2D(S, η) ≥ 1
and consider the family S of those S when S runs among S. Since the mappings D(S, .) are right-continuous and non-increasing, the sets η > 0, η 2 ≥ 2D(S, η) are non-void and contain their smallest elements η(S) for all S ∈ S. In order to select among the family {ŝ λ , λ ∈ Λ}, we use the selection procedure described in Section 4.1 with the choices S λ = S for all λ ∈ Λ, ∆(S) = ∆(S) for all S ∈ S and

pen λ (t) = pen(t) = c 0 τ inf S∈S(t)
4η 2 (S) + ∆(S) for all t ∈ S∈S S and λ ∈ Λ.

For the choice of η(S) given by (29), inequality ( 14) holds for all S ∈ S and therefore by Proposition 2, N and S satisfy Assumption 1 with D(S) = 4η 2 (S) and γ = bM 2 . In particular, pen satisfies (20). Finally, note that under ( 27) for all λ ∈ Λ and S ∈ S,

A (ŝ λ , S) ≤ inf t∈S H 2 (ŝ λ , t) + pen(t) ≤ 2H 2 ŝλ , S + 2τ η 2 S + c 0 τ 4η 2 (S) + ∆(S)
≤ 2H 2 ŝλ , S + τ 6η 2 (S) + ∆(S) .

By applying Theorem 1 we therefore deduce the following result.

Theorem 3. Let {ŝ λ , λ ∈ Λ} be a family of estimators based on a measure N satisfying ( 13). Let us assume that the family S consists of subsets S of L 0 with bounded metric dimensions D(S, η) and that ∆ is a mapping from S into [1, +∞) satisfying ( 4). By applying the selection procedure described above, the resulting estimator s satisfies

CE H 2 (s, s) ≤ inf λ∈Λ E H 2 (s, ŝλ ) + E inf S∈S H 2 (ŝ λ , S) + τ η 2 S ∨ ∆ S
where η(S) is given by ( 29) and C is a positive number depending on c 0 , b, M and Σ only. In particular, if for all λ ∈ Λ there exists some (possibly random) model S λ ∈ S such that ŝλ ∈ S λ with probability 1,

(30) CE H 2 (s, s) ≤ inf λ∈Λ E H 2 (s, ŝλ ) + τ E η 2 S λ ∨ ∆ S λ .
Let us now turn to examples.

6.2. Aggregation of arbitrary points. We assume here that the estimators ŝλ are deterministic and to emphasize the fact that they do not depend on N , denote them s λ hereafter. Typically, one should think of the s λ as estimators of s based on an independent copy N ′ of N in which case the result below should be understood as conditional on N ′ . In view of selecting among these points, we consider the family of models S given by S = {{s λ } , λ ∈ Λ}. Since each element S of S reduces to a single point, its metric dimension can be chosen as D(S, η) = 1/2 for all η > 0 and hence η(S) = 1. We deduce from Theorem 3 the following result.

Corollary 5. Let N be some random measure satisfying ( 13), {s λ , λ ∈ Λ} a countable collection of points in L 0 and ∆ a mapping from S into [1, +∞) satisfying (4). By applying the selection procedure described in Section 6.1 the estimator s satisfies

CE H 2 (s, s) ≤ inf λ∈Λ H 2 (s, s λ ) + τ ∆ (s λ )
for some positive number C depending on c 0 , b and Σ only. In particular, if Λ is finite, by choosing ∆(s λ ) = 1 + log |Λ| we obtain

E H 2 (s, s) ≤ C inf λ∈Λ H 2 (s, s λ ) + τ log |Λ| .

Construction of T -estimators.

Let us start with a family of models S with finite metric dimensions and consider the family of estimators (in fact points) obtained by gathering the elements of the sets S defined in Section 6.1 as they run among S. That is {ŝ λ , λ ∈ Λ} = S∈S S. Because of ( 27), this collection of estimators satisfies inf λ∈Λ H (s, ŝλ ) ≤ H(s, S) + η(S) √ τ for all s ∈ L 0 and S ∈ S.

By applying the selection procedure of Section 6.1 the resulting estimator s turns to be a T -estimator (according to Definition 2 in [START_REF] Birgé | Model selection via testing: an alternative to (penalized) maximum likelihood estimators[END_REF]). We deduce from Theorem 3 the following result for s.

Corollary 6. Let N be some measure satisfying ( 13), S a countable collection of subsets S of L 0 with bounded metric dimensions D(S, η) and ∆ a mapping on S satisfying (4). By applying the selection procedure described in Section 6.1 to the family of points S∈S S with S defined by ( 29), the resulting estimator s is a Testimator which satisfies

E H 2 (s, s) ≤ C inf S∈S H 2 (s, S) + τ η 2 (S) ∨ ∆(S)
for some C depending on c 0 , b, M and Σ only.

This result recovers Corollary 4 in [START_REF] Birgé | Model selection via testing: an alternative to (penalized) maximum likelihood estimators[END_REF] in the density case and Theorem 3 in [START_REF] Birgé | Asymptotics: particles, processes and inverse problems[END_REF] in the Poisson case. Example 2 was also considered in [START_REF] Birgé | Model selection via testing: an alternative to (penalized) maximum likelihood estimators[END_REF] (Proposition 6) but for a different loss function. The case of Example 3 is to our knowledge new.

6.4. Estimators with values in a simplex and convex aggregation. In this section, we assume that L 0 is a convex subset of L and consider a family {t 1 , . . . , t M } of M ≥ 2 distinct points of L 0 . We denote by M the class of nonempty subsets m of {1, . . . , M } and define S m as the convex hull of the t i for i ∈ m, namely

S m = i∈m q i t i (q i ) i∈m ∈ R |m| + , i∈m q i = 1 ⊂ L 0 .
Along the section, we assume that the estimators ŝλ take their values in the convex hull of {t 1 , . . . , t M }, that is, in

(31) C = m∈M S m = M i=1 q i t i q 1 , . . . , q M ∈ R + , M i=1 q i = 1 ⊂ L 0 .
As usual, our aim is to select some estimator s among the collection {ŝ λ , λ ∈ Λ} at best.

6.4.1. The example of convex aggregation. When the set {t 1 , . . . , t M } corresponds to M preliminary estimators of s (say obtained from an independent copy N ′ of N ), the problem of looking for some best convex combination of these is usually called convex aggregation. Given some integer D ∈ {1, . . . , M }, we tackle this problem by considering the (countable) collection of estimators (in fact points) given by ( 32)

s λ = M i=1 λ i t i , λ ∈ Λ with Λ = Λ D,M = λ ∈ Q M + , M i=1 λ i = 1, |{i, λ i = 0}| ≤ D .
The choices D = 1 and D = M correspond to the problems of estimator selection and convex aggregation respectively. These problems are particular cases of that of aggregation which aims at designing a suitable combination of given estimators in order to outperform each of these separately (and even the best combination of these) up to a remaining term. Aggregation techniques can be found in [START_REF] Juditsky | Functional aggregation for nonparametric regression[END_REF], [START_REF] Juditsky | Functional aggregation for nonparametric regression[END_REF], Yang (2000a), ( 2000b), ( 2001), [START_REF] Tsybakov | Optimal rates of aggregation[END_REF], [START_REF] Wegkamp | Model selection in nonparametric regression[END_REF], [START_REF] Birgé | Model selection via testing: an alternative to (penalized) maximum likelihood estimators[END_REF], [START_REF] Rigollet | Linear and convex aggregation of density estimators[END_REF], [START_REF] Bunea | Aggregation for Gaussian regression[END_REF], [START_REF] Goldenshluger | A universal procedure for aggregating estimators[END_REF] for L p -losses, and Catoni (2004) (we refer to his course of Saint Flour which takes back some mixing techniques he introduced earlier). Most of the aggregation procedures are based on a sample splitting and therefore usually requires that the data be i.i.d.. In a non-i.i.d. case, some nice results of aggregation can be found in [START_REF] Leung | Information theory and mixing least-squares regressions[END_REF] for the problem of mixing least-squares estimators of a mean of a Gaussian vector Y . In their paper, they assume that the components of Y are independent with a known common variance. [START_REF] Giraud | Mixing least-squares estimators when the variance is unknown[END_REF] extended their results to the case where it is unknown.

6.4.2. The selection procedure. In order to select among the estimators {ŝ λ , λ ∈ Λ}, we apply the selection procedure described in Section 6.1 with the family of models S = S m , m ∈ M . To do so, we need to find a mapping ∆ on S with values in [1, +∞) satisfying ( 4

q i j t i j +   1 - |m|-1 j=1 q i j   t i |m|
where the q i j vary among ℓε, ℓ = 0, . . . , ε -1 and satisfy |m|-1 j=1 q i j ≤ 1. The following result holds. Proposition 6. For all η > 0 and all non-void subsets m of {1, . . . , M }, the subset S m [η] defined above is an η √ τ -net for S m which satisfies

log |S m [η]| ≤ |m| log(1 + ε -1 )
where ε is given by ( 33) with the convention ε -1 = 0 if |m| = 1. In particular, the metric dimension D(S m , η) of S m may be chosen as

D(S m , η) = |m| log 1 + (|m| -1) t 1 η 2 τ ∨ 1 ≥ 1 2 .
Some simple calculations show that for all m ∈ M, the quantity η(S m ) given by ( 29) satisfies

η 2 (S m ) ≤ 2|m| log 1 + (|m| -1) t 1 |m|τ ∨ 1 for all m ∈ M.
6.4.3. The Main result. Hereafter, m(λ) denotes any (possibly random) element of M for which ŝλ ∈ S m(λ) . We deduce from Theorem 3 the following result.

Corollary 7. Let N be some measure satisfying (13) and {ŝ λ , λ ∈ Λ} a collection of arbitrary estimators with values in the simplex C given by ( 31). Let L be the mapping defined on N * by

L(k) = log(M ) if k = 1 1 + max log M k -1 ; log t 1 τ -1 otherwise.
By applying the selection procedure described above, the resulting estimator s satisfies

CE H 2 (s, s) ≤ inf λ∈Λ E H 2 (s, ŝλ ) + E inf m∈M H 2 (ŝ λ , S m ) + τ |m|L(|m|) ≤ inf λ∈Λ E H 2 (s, ŝλ ) + τ E [| m(λ)|L(| m(λ)|)]
where C depends on c 0 and b only.

For the problem of convex aggregation, that is, by considering the collection of points {s λ , λ ∈ Λ} defined by ( 32) with D = M , the estimator s satisfies

(34) CE H 2 (s, s) ≤ inf m∈M H 2 (s, S m ) + τ |m|L(|m|) .
Let us comment on this result in the density framework. In this case, t 1 = 1 and τ is of order 1/n. If one considers the collection of points {s λ , λ ∈ Λ} given by ( 32) with D = M , the right-hand side of ( 34) is of the form

inf m∈M H 2 (s, S m ) + |m|L(|m|) n = inf D=1,...,M inf λ∈Λ D,M H 2 (s, s λ ) + DL(D) n
where L(1) = log(M ) and L(D) is of order max {log(eM/D); log(n)} otherwise. In the density case, the problem of aggregating M densities has also been considered in [START_REF] Birgé | Model selection via testing: an alternative to (penalized) maximum likelihood estimators[END_REF] and [START_REF] Rigollet | Linear and convex aggregation of density estimators[END_REF]. In this latter paper, it is shown that the optimal rate of aggregation associated to the simplex Λ M,M is of order M/n for the L 2 -norm and leads thus to risk bounds which are similar to ours (up to constants and log(n) factors). The result we get is similar to [START_REF] Birgé | Model selection via testing: an alternative to (penalized) maximum likelihood estimators[END_REF] for the L 1 -norm (up to the logarithmic factor). We do not know whether this extra logarithmic factor is due to our discretization procedure or to the loss function we use.

Estimating the means of nonnegative random variables

In this section, we consider the statistical setting described in Example 3 and assume that (15) holds. We recall that it is satisfied for a large class of distributions including any random variables with values in [0, β] (then σ = β), the Binomial distribution (then σ = 1 = β), the Poisson distribution (for the same choice of parameters), or the Gamma distribution γ(p, q) (with mean p/q and β = 1/q = σ). By expanding (15) in a vicinity of 0, it is easy to see that (15) implies that Var(X i ) ≤ σE(X i ) for all i = 1, . . . , n. Throughout, we identify with the same notation the functions t on X = {1, . . . , n} with the vectors (t 1 , . . . , t n ) = (t(1), . . . , t(n)). The distance √ 2H between two elements t, t ′ ∈ L 0 corresponds to the Euclidean distance between the vectors √ t and √ t ′ and it seems natural to approximate the parameter √ s with respect to the Euclidean norm . For this purpose, we introduce a family V of linear subspaces V of R n with respective (linear) dimensions denoted D(V ). Together with this family, we associate, as usual, a mapping ∆ on V satisfying (4). The aim of this section is to design an estimator s satisfying the following risk bound (35)

C ′ E H 2 (s, s) ≤ inf V ∈V inf v∈V √ s -v 2 + D(V ) ∨ ∆(V ) for all s ∈ R n + .
To do so, we associate to each linear space V , some discrete set V ⊂ R n + obtained by the discretization device described in [START_REF] Birgé | Model selection via testing: an alternative to (penalized) maximum likelihood estimators[END_REF]. More precisely, the following result holds.

Proposition 7. Let τ > 0 and C be some closed convex subset of R n and V ⊂ R n a linear subspace with dimension D(V ). For all η > 0, there exists a discrete subset V (η) ⊂ C such that whatever f ∈ C, (36) inf

v∈V (η) f -v ≤ 4 inf v∈V f -v + η √ τ
and for all R ≥ 2η,

(37) v ∈ V (η), f -v ≤ R √ τ ≤ exp 5D(V ) R η 2 .
We apply the proposition with C = R n + and for V ∈ V, denote by V = V (η) the discrete set resulting from the choice η = η(V ) = 20D(V ) and set

S(V ) = (v 2 1 , . . . , v 2 n ), v ∈ V for all V ∈ V.
We shall now apply the selection procedure described in Section 4.1 with the collection of the estimators (in fact points) ŝλ = λ with λ ∈ Λ = V ∈V S(V ) together with the choices ∆(S(V )) = ∆(V ) and S λ = S for all V ∈ V and λ ∈ Λ. We know from Proposition 3 that N satisfies ( 13) with a = (12σ) -1 , b = 1 and c = β √ 2/(12σ) and since the mapping

t → √ t from (R n + , √ 2H) into (R n + ,
) is an isometry, it follows from (37) and our choice of η(V ) that all the models S(V ) of S satisfy ( 14) with M = 1 and τ depending on σ and β only. By Proposition 2, N and S satisfy thus Assumption 1 with D(S(V )) = 80D(V ) and τ given by ( 14). Finally, since (36

) implies 4 -1 √ 2H(s, S(V )) ≤ inf v∈V √ s -v + η √ τ for all V ∈ V, we deduce from
Theorem 1 that the selected estimator s satisfies the following:

Theorem 4. Let V be a countable family of linear spaces V with respective dimensions D(V ) and ∆ a mapping from V into [1, +∞[ satisfying (4). By applying the estimation procedure described above, one designs an estimator s satisfying ( 35) for some constant C ′ depending on c 0 , σ, β and Σ only.

7.1. Uniform convergence rates. In this section, we assume that √ s is of the

form √ s = √ s F = (F (x 1 ), . . . , F (x n ))
for some unknown nonnegative function F on [0, 1] and deterministic points 0 ≤ x 1 < . . . < x n ≤ 1. For a suitable choice of V, our aim is to deduce from Theorem 4 uniform rates of convergence over classes of means

S α p,∞ (R) of the form S α p,∞ (R) = s F , F ∈ B α p,∞ (R) ⊂ R n + where B α p,∞ (
R) is a Besov ball with radius R > 0 and parameters α > 0 and p ∈ [1, +∞]. For a precise definition of Besov spaces, we refer to [START_REF] Devore | Constructive approximation[END_REF]. The following result derives from Theorem 1 and Proposition 1 in [START_REF] Birgé | An adaptive compression algorithm in Besov spaces[END_REF].

Proposition 8. For all r ∈ N * and J ∈ N, there exist positive numbers C(r), C ′ (r), C ′′ (r) and a family V r,J of finite dimensional linear subspaces V of real-valued functions on [0, 1] with the following properties:

dim(V) ≤ C(r)2 J for all V ∈ V r,J , log |V r,J | ≤ C ′ (r)2 J and for all α ∈ (1/p, r) and all F ∈ B α p,∞ (R) there exists G ∈ V∈V r,J V such that sup x∈[0,1] |F (x) -G(x)| ≤ C ′′ (r)R2 -Jα . For V ∈ V r,J , we define V = V (V) = {(G(x 1 ), . . . , G(x n )), G ∈ V} ⊂ R n and V r,J
as the collection of those linear subspaces V as V runs among V r,J . By applying Theorem 4 with V = r≥1,J≥0 V r,J and ∆(V ) = (C ′ (r) + 1)2 J + r for all V ∈ V r,J , r ≥ 1 and J ≥ 0 we deduce the following result.

Corollary 8. By using the family of linear spaces V defined above, the estimator s satisfies for all p ∈ [1, +∞], α > 1/p and R > 1/ √ n, 1+2α) ,

sup s∈S α p,∞ (R) n -1 H 2 (s, s) ≤ CR 2/(1+2α) n -2α/(
where C depends on c 0 , τ and r.

To our knowledge, Example 3 has received little attention in the literature, especially from a non-asymptotic point of view. The only exceptions we are aware of are Antoniadis, Besbeas and Sapatinas ( 2001) (see also Antoniadis and Sapatinas (2001)) and [START_REF] Kolaczyk | Multiscale likelihood analysis and complexity penalized estimation[END_REF]. In [START_REF] Antoniadis | Wavelet shrinkage for natural exponential families with cubic variance functions[END_REF], the authors estimate F 2 by a wavelet shrinkage procedure and show that the resulting estimator achieves the usual estimation rate of convergence over Sobolev classes with smoothness larger than 1/2. [START_REF] Kolaczyk | Multiscale likelihood analysis and complexity penalized estimation[END_REF] study the risk properties of some thresholding and partitioning estimators. Their approach assumes that s is bounded from above and below by positive numbers on X = {1, . . . , n}. Finally, [START_REF] Baraud | Estimating the intensity of a random measure by histogram type estimators[END_REF] tackled this problem but they restricted themselves to histogram-type estimators and smoothness α ≤ 1 only.

7.2. Lower bounds. Let V be a linear subspace of R n such that V ∩ R n + = {0}. By applying Theorem 4 with ∆(V ) = 1, we have

sup s∈S E H 2 (s, s) ≤ C(τ )D V where S = s, √ s ∈ V .
The aim of this section is bound the minimax risk over S from below. We assume the following.

Assumption 2. The distribution of the random vector X = (X 1 , . . . , X n ) belongs to an exponential family of the form

(38) dP θ (x 1 , . . . , x n ) = exp n i=1 (θ i T (x i ) -A(θ i )) n i=1 dν(x i ) with θ ∈ Θ n
where ν denotes some measure on R + , T is a map from R + to R, θ i are parameters belonging to an open interval Θ such that

Θ ⊂ a ∈ R, exp [aT (x)] dν(x) < +∞
and A denotes a smooth function from Θ into R satisfying A ′′ (a) = 0 for all a ∈ Θ.

Assumption 2 holds for the Poisson, Binomial and Gamma distributions (among others). For θ ∈ Θ n , P θ and E θ will denote the probability and the expectation over P θ . It is well-known that the function A is infinitely differentiable on Θ and that for all i = 1, . . . , n

s(i) = E θ [X i ] = A ′ (θ i ) and Var θ (X i ) = A ′′ (θ i ),
these quantities being all positive because we assume X i ≥ 0 for i ∈ {1, . . . , n}.

We set

S = s ∈ A ′ (Θ) n , √ s ∈ V .
Let us fix some compact interval I ⊂ Θ and set K = A ′ (I). Since A ′ and A ′′ are continuous and positive on I, there exists κ > 0 such that for all θ ∈ I n

(39) 0 < E θ (X i ) ≤ κVar θ (X i ) ∀i = 1, . . . , n.
The following result holds.

Theorem 5. Let R ∈ (0, (2 √ κ) -1 ) with κ given by ( 39). Assume that Assumption 2 holds and that the linear space V is such that for some

u 0 ∈ V (40) (u 2 1 , . . . , u 2 n ) u ∈ V , u -u 0 ≤ R ⊂ K n .
Then, whatever the estimator ŝ based on X 1 , . . . , X n ,

sup s∈S E H 2 (s, ŝ) ≥ R 2 30 D(V ).

Estimation and variable selection in non-Gaussian regression

In this section, we use the notations of Example 2 and consider the regression setting (41)

X i = f i + ε i , i = 1, . . . , n
where f = (f 1 , . . . , f n ) is an unknown vector belonging to the cube C = [-R, R] n (for some R > 0) and the ε i are i.i.d. random variables with density q on R. Both q and R are assumed to be known. Our aim is to estimate f from the observation of X = (X 1 , . . . , X n ) and to do so, we introduce a collection { fλ , λ ∈ Λ} of estimators of f based on X and a family V of linear subspaces V ⊂ R n with respective dimensions D(V ). As usual we consider a mapping ∆ from V into [1, +∞) satisfying (4).

8.1. The main assumption. For all d ≥ 1 and t ∈ R d , we set

q t (x) = (q t 1 (x 1 ), . . . , q t d (x d )) = (q(x 1 -t 1 ), . . . , q(x d -t d )) ∀x ∈ R d
and omit to specify the dependency of q t with respect to the dimension of t. We assume that the density q satisfies the following.

Assumption 3. For all real numbers t, t ′ ∈ [-R, R], (42) R t -t ′ ≤ h (q t , q t ′ ) ≤ R t -t ′
where h is the Hellinger distance between the densities q t and q t ′ .

Assumption 3 holds whenever √ q is regular enough (see Theorem 3A page 183 in [START_REF] Borovkov | Mathematical statistics. Gordon and Breach Science[END_REF]) and is therefore satisfied for the Cauchy distribution. Note that such a distribution admits no finite moments.

8.2. The procedure and the results. For each λ ∈ Λ, let V λ be a subset of V (possibly random depending on X). Associate to each linear space V ∈ V the discrete subset V of C obtained by applying Proposition 7 with η = η(V ) = R -1 10D(V ) 1/2 . Then, set S(V ) = {q t , t ∈ V } and define S (respectively S λ ) as the collection of those S(V ) as V runs among V (respectively V λ ). Take ∆(S(V )) = ∆(V ) for all V ∈ V and select the estimator s = ŝλ among the family ŝλ = q fλ , λ ∈ Λ with fλ = Π C fλ by applying the selection procedure described in Section 4.1 with τ given by ( 14), a = 1/12, c = √ 2/6 (hence τ only depends on c 0 ) and for all λ ∈ Λ,

pen λ (s ′ ) = c 0 τ inf 40D(V ) + ∆(V ) V ∈ V λ .
Our final estimator f = fλ satisfies the following.

Theorem 6. Consider the regression setting given by ( 41) where the mean f is known to belong to the cube C = [-R, R] n for some R > 0 and assume that the density q of the ε i is known and satisfies Assumption 3. Let V be a family of linear subspaces V of R n with dimension D(V ), ∆ a mapping from V into [1, +∞) satisfying ( 4) and { fλ , λ ∈ Λ} a collection of estimators of f based on X. By applying the selection procedure described above, the estimator f satisfies

CE f -Π C f 2 ≤ inf λ∈Λ E f -fλ 2 + E inf V ∈V λ fλ -Π V fλ 2 + D(V ) ∨ ∆(V )
where C depends on c 0 , R, R and Σ only. In particular if for all λ ∈ Λ fλ belongs to some V λ ∈ S λ with probability 1,

CE f -Π C f 2 ≤ inf λ∈Λ E f -fλ 2 + E D(V λ ) ∨ ∆(V λ ) .
8.3. Model Selection. In this section, we consider the problem of model selection among a collection V of linear spaces V . Among the examples we have in mind is that of variable selection.

Problem 1 (Variable selection). We assume that f is of the form

f = p j=1 β j v (j)
where β = (β 1 , . . . , β p ) is an unknown vector of R p and v (1) , . . . , v (p) are p ≥ 2 known vectors in R n that we call predictors. Since, the number p of those may be large and possibly larger than the number n of data, we assume that the vector β is sparse which means that |{j, β j = 0}| ≤ p max for some integer p max ≤ n. Our aim is to estimate f and the set {j, β j = 0}. To do so, we consider the index set M consisting of all the subsets of {1, . . . , p} with cardinality not larger than p max and V the family gathering the linear spaces V m spanned by the v (j) for j ∈ m when m varies among M. By convention, V ∅ = {0}.

One way to address the problem of model selection is to associate to each V ∈ V a family of points Λ(V ) which is countable and dense in V and then to apply the procedure described in Section 8.2 to the family of estimators (in fact points) given by fλ = λ for λ ∈ V ∈V Λ(V ) = Λ. By applying Theorem 6 with V λ = V for all λ ∈ Λ, we deduce the following result without assuming any finite moments on the distribution of the ε i .

Corollary 9. Under the assumptions of Theorem 6, one can build an estimator f based on X such that

(43) CE f -f 2 ≤ inf V ∈V f -Π V f 2 + D(V ) ∨ ∆(V ) .
To our knowledge, such a result without any assumption on the integrability of the ε i is new. In the context of variable selection with non-Gaussian errors, Theorems 5 and 6 by [START_REF] Dalalyan | Aggregation by exponential weighting, sharp pac-bayesian bounds and sparsity[END_REF] are probably the closest even though the risk bounds they get is slightly different and depend on the ℓ 1 -norm of the β j . These bound are derived from sharp PAC-Bayesian ones and are difficult to compare to ours. Let us just say that Dalalyan and Tsybakov achieve better constants and do not assume that the distribution of the errors is known but require stronger assumptions both on the integrability of the errors and on the predictors v (j) to control the ℓ 1 -norm of the β j .

If the ε i are centered and admit a finite variance σ 2 , the family of candidate estimators used in Corollary 9 can be reduced to that of the least-squares { fλ , λ ∈ Λ} defined with the choice Λ = V by fV = Π V X for all V ∈ V. Since the risk of fV satisfies

(44) E f -fV 2 = f -Π V f 2 + D(V )σ 2 ,
by applying Theorem 6 with V V = V for all V ∈ V, we deduce the following.

Corollary 10. Assume that the assumptions of Theorem 6 hold and that the ε i are centered and admit a finite variance σ 2 . By applying the selection procedure of Section 8.2 to the family of least-squares estimators Π V X, V ∈ V , one can select from the data X some linear space V among V such that

(45) CE f -Π C Π V X 2 ≤ inf V ∈V f -Π V f 2 + D(V ) ∨ ∆(V )
where C depends on c 0 , σ, R, R and Σ only.

Under the assumption that the ε i are Gaussian, results of the same flavor (without any boundedness assumption on the vector f and therefore for C = R) were previously obtained by [START_REF] Birgé | Gaussian model selection[END_REF] when the variance is known and in [START_REF] Baraud | Gaussian model selection with an unknown variance[END_REF] when it is not. Nevertheless, these approaches as well as ours suffer from the same drawback: in the context of variable selection with p max large enough, the collection { fV , V ∈ V}, though finite, is very large and the selection procedure becomes NP-hard and hence practically useless. In the recent years, many efforts have been done to design (practical) selection rules for the purpose of performing variable selection. Among the most popular ones, we mention the Lasso and the Dantzig selectors described respectively in [START_REF] Tibshirani | Regression shrinkage and selection via the lasso[END_REF] and [START_REF] Candès | The Dantzig selector: statistical estimation when p is much larger than n[END_REF]. A theoretical analysis of these two procedures (separately and comparatively) has been done in [START_REF] Bickel | Simultaneous analysis of lasso and Dantzig selector[END_REF]. Others based on random forest (see Genuer, Poggi and Tuleau-Malot ( 2010)) or PLS regression (see Agnar [START_REF] Höskuldsson | Variable and subset selection in PLS regression[END_REF]) are also used in practice even though less is known on their theoretical performances. In any case, it seems that none of such procedures outperforms the others and it may therefore be reasonable to consider them all and let the data decide which is the most appropriate to estimate the truth.

In what follows we consider an arbitrary collection Λ of model selection procedures among V, denote V λ the model selected by the procedure λ ∈ Λ and fλ the leastsquares estimator on V λ . By applying the selection procedure of Section 8.2 with V λ = V λ for all λ ∈ Λ, we deduce the following result.

Corollary 11. Under the assumptions of Theorem 6, the estimator

Π C f satisfies CE f -Π C f 2 ≤ inf λ∈Λ E f -fλ 2 + E D V λ ∨ ∆ V λ
for some C depending on c 0 , R, R and Σ only.

The selection procedure we propose is unfortunately not practical in the general context of model selection. Not because the computations are NP-hard, at least as long as the family Λ of candidate procedures keeps to a reasonable size, but rather because our selection rule relies on a discretization device that is not practical yet. However, we mention that in the specific context of variable selection the procedure can be made feasible indeed by using the alternative family of models

V = V m ∩ C, m ∈ M .
By assumption, the unknown parameter s belongs to one of these models and as subsets of linear spaces, their discretization can be easily done. Furthermore, only the models V λ ∩ C with λ ∈ Λ actually need to be discretized.

8.4.

Selecting among linear estimators. In this section, we assume that the collection { fλ , λ ∈ Λ} consists of linear estimators of f . More precisely, we shall assume that Λ is an arbitrary collection of (deterministic) symmetric matrices and that fλ = λX for all λ ∈ Λ. Among the examples we have in mind is the following one.

Example 5 (Kernel estimation). Assume that f is of the form

f = (F (1/n), . . . , F (n/n))
for some real-valued function F on [0, 1]. Given a symmetric kernel K, that is, a function from R 2 into R, such that the matrix (K(i/n, j/n)) 1≤i,j≤n is symmetric, we can associate the kernel estimator of f defined by fλ(K) = λ(K)X with λ i,j (K) = K(i/n, j/n) n for all i, j = 1, . . . , n.

This estimator corresponds to the [START_REF] Priestley | Non-parametric function fitting[END_REF] estimator evaluated at points k/n for k = 1, . . . , n.

Other examples of linear estimators with symmetric matrices λ can be found in [START_REF] Arlot | Data-driven calibration of linear estimators with minimal penalties[END_REF] (kernel ridge regression, spline smoothing, multiple kernel learning...). As also mentioned there, the classical Nadaraya-Watson kernel estimator is beyond the scope of this study because it corresponds to a non-symmetric matrix λ.

In view of selecting among the family { fλ , λ ∈ Λ}, we consider the family of models V = V λ , λ ∈ Λ defined as follows. For λ ∈ Λ, let λ (1) ≥ . . . ≥ λ (n) be the the eigenvalues of λ sorted by non-increasing order, v (1) , . . . , v (n) the corresponding eigenvectors and

D λ = max k, λ (k) ≥ 1/2 with the convention D λ = 0 if k, λ (k) ≥ 1/2 = ∅.
The linear space V λ corresponds to the linear space generated by the v (k) for k ≤ D λ with the convention V λ = {0} if D λ = 0. The threshold 1/2 involved in the definition of D λ is not magical and has been chosen for convenience. Any other choice of a constant in (0, 1) would lead to a result which is similar to the one below (with a possibly different constant C).

Corollary 12. Assume that the assumptions of Theorem 6 hold and that the ε i are centered and admit a finite variance σ 2 . Consider a collection { fλ , λ ∈ Λ} of linear estimators associated to symmetric matrices λ. By using the selection procedure described in Section 8.2 with the family of linear spaces V defined above and any mapping ∆ from V into [1, +∞) satisfying (4) the selected estimator satisfies

CE f -Π C f 2 ≤ inf λ∈Λ E f -fλ 2 ∨ ∆(V λ )
for some constant C depending on c 0 , R, R, σ and Σ only.

The selection procedure allows to minimize the risk among the family of linear estimators {λX, λ ∈ Λ}. In particular, it can be used to select a window or a kernel among a collection of those. A result of the same flavour as that of Corollary 12 can be found in [START_REF] Arlot | Data-driven calibration of linear estimators with minimal penalties[END_REF] but under more restrictive assumptions on the matrices λ, the cardinality of Λ and the distribution of the ε i . Nevertheless, their point of view is more practical than ours and leads to a concrete algorithm. In the Gaussian white noise model, [START_REF] Goldenshluger | Structural adaptation via L p -norm oracle inequalities[END_REF] addressed the problem of structural adaptation by means of a suitable selection procedure among linear estimators based on kernels. Their assumptions on the family of kernels are slightly different from ours: they consider some convolution-type assumption between the kernels of the collection while we assume that the kernels are symmetric.

The proof of Corollary 12 is postponed to Section 9.11. 9. Proofs 9.1. Proof of Proposition 2. For S, S ′ ∈ S and ξ > 0,

y 2 ≥ τ 4 η 2 (S) ∨ η 2 (S ′ ) + ξ ≥ 4τ η 2 (S) ∨ η 2 (S ′ ) .
We set C 0 = (S ∩ B(s, y)) × (S ′ ∩ B(s, y)) and for j ≥ 1,

C j = (t, t ′ ) ∈ S × S ′ , 2 j-1 y 2 < H 2 (s, t) + H 2 s, t ′ ≤ 2 j y 2 .
Note that for all j ≥ 0, C j ⊂ S ∩ B(s, 2 j/2 y) × S ′ ∩ B(s, 2 j/2 y) and that for (t, t ′ ) ∈ C j , w 2 (t, t ′ , y) = H 2 (s, t) + H 2 (s, t ′ ) ∨ y 2 ≥ (2 j-1 ∨ 1)y 2 . Using ( 13) and ( 14), we have

P sup (t,t ′ )∈S×S ′ Z(N, t, t ′ ) w 2 (t, t ′ , y) > c 0 ≤ (t,t ′ )∈C 0 P Z(N, t, t ′ ) ≥ c 0 y 2 + j≥1 (t,t ′ )∈C j P Z(N, t, t ′ ) ≥ c 0 2 j-1 y 2 ≤ b |S ∩ B(s, y)| S ′ ∩ B(s, y) exp - ac 2 0 y 4 y 2 + cc 0 y 2 +b j≥1 |S ∩ B(s, 2 j/2 y)||S ′ ∩ B(s, 2 j/2 y)| exp - ac 2 0 2 2(j-1) y 4 2 j y 2 + cc 0 2 j-1 y 2 ≤ bM 2 exp 1 τ - ac 2 0 1 + cc 0 y 2 + bM 2 j≥1 exp 1 τ - ac 2 0 2(2 + cc 0 ) 2 j y 2 ≤ bM 2 j≥0 exp - 2 j y 2 τ ,
recalling that τ = 4(2+cc 0 )/(ac 2 0 ). By using that τ -1 y 2 ≥ 4(η 2 (S)∨η 2 (S ′ ))+ξ ≥ 1+ξ and the inequality 2 j ≥ j + 1 which holds for all j ≥ 0, we finally obtain

P sup (t,t ′ )∈S×S ′ Z(N, t, t ′ ) w 2 (t, t ′ , y) > c 0 ≤ bM 2 j≥0 exp [-(j + 1)(1 + ξ)] ≤ bM 2 e -ξ .
9.2. Proof of Proposition 3.

Cases of Examples 1 and 2. It suffices to prove the result in the case of Example 2, the result for Example 1 being obtained similarly by changing Z(N, t, t ′ ) into Z(N, t, t ′ )/n. Note that for all t, t ′ ∈ L 0 ,

Z(N, t, t ′ ) = n i=1 ψ(t i , t ′ i , X i ) -E ψ(t i , t ′ i , X i )
is a sum of independent and centered random variables bounded by √ 2. Besides, by setting r i = (t i + t ′ i )/2 for i = 1, . . . , n and using that for all x i ∈ X i , (t(x i ) ∨

t ′ i (x i ))/r i (x i ) ≤ 2 we have 4E Z 2 (N, t, t ′ ) ≤ n i=1 X i √ t i -t ′ i 2 s i r i dµ i = n i=1 X i √ t i -t ′ i 2 s i r i -1 + 1 2 dµ i ≤ 2 n i=1 X i √ t i -t ′ i 2 s i r i -1 2 dµ i +2 n i=1 X i √ t i -t ′ i 2 dµ i = 2 n i=1 X i √ t i -t ′ i 2 r i ( √ s i - √ r i ) 2 dµ i + 4H 2 t, t ′ ≤ 8 H 2 (s, r) + H 2 (s, t) + H 2 (s, t ′ ) .
Since the concavity of u → √ u implies that 2H 2 (s, r) ≤ H 2 (s, t) + H 2 (s, t ′ ), we obtain that for t, t ′ ∈ B(s, y)

Var Z(N, t, t ′ ) = E Z 2 (N, t, t ′ ) ≤ 3 H 2 (s, t) + H 2 s, t ′ ≤ 6y 2 .
Applying Bernstein's inequality, we obtain that ( 13) is fulfilled with b = 1, a = 1/12 and c = √ 2/6.

Case of Example 3. Under (15), for all u = (u 1 , . . . , u n ) ∈ R n such that n i=1 u 2 i s(i) ≤ v 2 and max n i=1 |u i | ≤ γ, and all λ ∈ (0, 1/(βγ)), we have

E e λ n i=1 u i (X i -s(i)) = n i=1 E e λu i (X i -s(i)) ≤ n i=1 exp λ 2 σu 2 i s(i) 2(1 -λγβ) ≤ exp λ 2 σv 2 2(1 -λγβ) . (46)
Under ( 46), we derive from Bernstein's inequality (see [START_REF] Massart | Concentration inequalities and model selection[END_REF], Corollary 2.10), ( 47)

P n i=1 u i (X i -s(i)) ≥ ξ ≤ exp - ξ 2 2(σv 2 + γβξ) .
For t, t ′ ∈ B(s, y), let us now take u = (ψ(t, t ′ , 1), . . . , ψ(t, t ′ , n)) (where ψ is defined by ( 12) on X = {1, . . . , n}) and note that

n i=1 ψ(t, t ′ , i) (X i -s(i)) = Z(N, t, t ′ ) max i=1,...,n ψ(t, t ′ , i) ≤ 1 √ 2 = γ.
Besides, arguing as for the case of Example 2, we get

n i=1 ψ 2 (t, t ′ , i)s(i) = 1 4 n i=1 t(i) -t ′ (i) 2 s(i) (t(i) + t ′ (i))/2 ≤ 3H 2 (s, t) + 3H 2 (s, t ′ ) ≤ 6y 2 = v 2 .
Consequently, we deduce from ( 47) that ( 13) is satisfied with a = 1/(12σ), b = 1 and c = β √ 2/(12σ) (then τ ≤ 96c -2 0 (σ + β)). Case of Example 4. In this case,

Z(N, t, t ′ ) = X ψ t, t ′ , x (dN (x) -s(x)dµ)
where ψ is bounded with values in [-1/ √ 2, 1/ √ 2] and, arguing as for Example 2, we see that it satisfies X ψ 2 t, t ′ , x s(x)dµ ≤ 3 H 2 (s, t) + H 2 (s, t ′ ) ≤ 6y 2 for all t, t ′ ∈ B(s, y). By applying Proposition 7 in [START_REF] Reynaud-Bouret | Adaptive estimation of the intensity of inhomogeneous Poisson processes via concentration inequalities[END_REF] we obtain that Z(N, t, t ′ ) satisfies ( 13 All t ∈ S m and t ′ ∈ S m ′ are constant on the cells I ∈ m ∨ m ′ with value t I , t ′ I respectively and therefore so is ψ(t, t ′ , .). Namely, for all x ∈ I ψ(t, t ′ , x) = ψ(t I , t

′ I ) = 1 √ 2 1 1 + t I /t ′ I - 1 1 + t ′ I /t I .
Using that |ψ(t I , t ′ I )| ≤ 1/ √ 2 and Cauchy-Schwarz inequality, we get

Z(N, t, t ′ ) = I∈m∨m ′ ψ(t I , t ′ I ) (N (I) -E [N (I)]) = I∈m∨m ′ ψ(t I , t ′ I ) N (I) -E [N (I)] N (I) + E [N (I)] = I∈m∨m ′ ψ(t I , t ′ I ) N (I) -E [N (I)] 2 +2 I∈m∨m ′ ψ(t I , t ′ I ) E(N (I)) N (I) -E [N (I)] ≤ X 2 (m ∨ m ′ ) √ 2 + 2 I∈m∨m ′ ψ 2 (t I , t ′ I )E(N (I)) 1/2 X (m ∨ m ′ ) = X 2 (m ∨ m ′ ) √ 2 + 2 ψ 2 (t, t ′ , x)sdµ 1/2 X (m ∨ m ′ ).
Besides, arguing as in Section 9.2 (Example 2), we have

X ψ 2 (t, t ′ , x)sdµ ≤ 3 H 2 (s, t) + H 2 (s, t ′ )
and thus, using that w 2 (t, t ′ , y) ≥ y 2 and w 2 (t, t ′ , y) ≥ H 2 (s, t) + H 2 (s, t ′ ) 1/2 y, we derive sup

(t,t ′ )∈Sm×S m ′ Z(N, t, t ′ ) w 2 (t, t ′ , y) ≤ X 2 (m ∨ m ′ ) √ 2y 2 + 2 √ 3 X (m ∨ m ′ ) y ≤ 2 √ 6 + 1 √ 2 X 2 (m ∨ m ′ ) y 2 ∨ X (m ∨ m ′ ) y . Since c 0 ∈ (0, 1), sup (t,t ′ )∈Sm×S m ′ Z(N, t, t ′ ) w 2 (t, t ′ , y) ≥ c 0 ⊂ X 2 (m ∨ m ′ ) y 2 ∨ X (m ∨ m ′ ) y ≥ c 0 √ 2 2 √ 6 + 1 ⊂ X 2 (m ∨ m ′ ) y 2 ≥ 2c 2 0 2 √ 6 + 1 2 and therefore P sup (t,t ′ )∈Sm×S m ′ Z(N, t, t ′ ) w 2 (t, t ′ , y) ≥ c 0 ≤ P X 2 (m ∨ m ′ ) ≥ 2c 2 0 y 2 2 √ 6 + 1 2 .
We conclude by using ( 16) together with the fact that under (17),

y 2 ≥ τ (D(S m ) ∨ D(S m ′ ) + ξ) ≥ 2 √ 6 + 1 2 2c 2 0 × a |m ∨ m ′ | + ξ . 9.4. Proof of Theorem 1. Throughout we set κ = c 0 + 1/ √ 2 ∈ (1/ √ 2, 1
) and fix some estimator ŝλ . We start with a preliminary result.

Preliminary result: For ξ > 0 and S, S ′ ∈ S, let us set

y 2 (S, S ′ , ξ) = τ D(S) ∨ D(S ′ ) + ∆(S) + ∆(S ′ ) + ξ ,
and

Ω ξ = (S,S ′ )∈S 2 sup (t,t ′ )∈S×S ′ Z(N, t, t ′ ) w 2 (t, t ′ , y(S, S ′ , ξ)) ≤ c 0 .
Note that under Assumption 1, P (Ω ξ ) ≥ 1 -γΣ 2 e -ξ . Let us prove that on the set Ω ξ , (48)

D(s λ ) ≤ 12 1 -κ H 2 (s, ŝλ ) + A(ŝ λ , S λ ) + 1 6 c 0 τ ξ .
Proof. Since D(s λ ) = 0 whenever E(s λ ) = ∅, we shall assume from now on that E(s λ ) = ∅. Hence, there exists sλ ′ ∈ E(s λ ) with sλ ′ = sλ . Using Proposition 1 with r = (s λ + sλ ′ )/2 and the fact that T(N, sλ , sλ ′ ) ≥ 0, we get

H 2 (s, sλ ′ ) -H 2 (s, sλ ) = ρ (s, sλ ) - 1 2 X sλ dµ -ρ (s, sλ ′ ) - 1 2 X sλ ′ dµ = -T(N, sλ , sλ ′ ) + pen(s λ ) -pen(s λ ′ ) + [ρ (s, sλ ) -ρ r (s • µ, sλ )] -[ρ (s, sλ ′ ) -ρ r (s • µ, sλ ′ )] + [ρ r (s • µ, sλ ) -ρ r (N, sλ )] -[ρ r (s • µ, sλ ′ ) -ρ r (N, sλ ′ )] ≤ 1 √ 2 H 2 (s, sλ ) + H 2 (s, sλ ′ ) +Z(N, sλ , sλ ′ ) + pen(s λ ) -pen(s λ ′ )
and therefore,

1 - 1 √ 2 H 2 (s, sλ ′ ) ≤ 1 + 1 √ 2 H 2 (s, sλ ) + Z(N, sλ , sλ ′ ) + pen(s λ ) -pen(s λ ′ ). On Ω ξ , Z(N, sλ , sλ ′ ) ≤ c 0 H 2 (s, sλ ) + c 0 H 2 (s, sλ ′ ) +c 0 inf y 2 (S, S ′ , ξ), (S, S ′ ) ∈ S λ (s λ ) × S λ ′ (s λ ′ ) ≤ c 0 H 2 (s, sλ ) + c 0 H 2 (s, sλ ′ ) +c 0 τ inf (S,S ′ )∈S λ (s λ )×S λ ′ (s λ ′ ) D(S) + D(S ′ ) + ∆(S) + ∆(S ′ ) + ξ
and since for all λ ∈ Λ, pen λ (s λ ) ≥ c 0 τ inf {D(S) + ∆(S), S ∈ S λ (s λ )} , we have

(1 -κ) H 2 (s, sλ ′ ) ≤ (1 + κ) H 2 (s, sλ ) + 2 pen λ (s λ ) + c 0 τ ξ.
Since sλ ′ is arbitrary in E(s λ ), we deduce that on Ω ξ ,

D(s λ ) = sup sλ ′ ∈E(s λ ) H 2 (s λ , sλ ′ ) ≤ 2H 2 (s, sλ ) + 2 sup s λ ′ ∈E(s λ ) H 2 (s, sλ ′ ) ≤ 2 1 + 1 + κ 1 -κ H 2 (s, sλ ) + 4 1 -κ pen λ (s λ ) + 2 1 -κ c 0 τ ξ ≤ 4 1 -κ 3H 2 (s, ŝλ ) + 3 2 H 2 (ŝ λ , sλ ) + pen λ (s λ ) + 1 2 c 0 τ ξ
and we conclude by using that H

2 (ŝ λ , sλ ) + pen λ (s λ ) ≤ A(ŝ λ , S λ ) + c 0 τ ≤ 2A(ŝ λ , S λ ) because ∆(.) ≥ 1 on S.
End of the proof of Theorem 1. Using the triangular inequality and the fact that

H sλ , s ≤ H sλ , ŝλ + √ c 0 τ ≤ H sλ , sλ + H (s λ , ŝλ ) + √ c 0 τ ,
we have

H (s, s) ≤ H(s, ŝλ ) + 2H(ŝ λ , sλ ) + 2H(s λ , sλ ) + √ c 0 τ , which with c 0 τ ≤ A(ŝ λ , S λ ) and H 2 (ŝ λ , sλ ) ≤ 2A(ŝ λ , S λ ) gives 3 -1 H 2 (s, s) ≤ H 2 (s, ŝλ ) + (2 √ 2 + 1) 2 A(ŝ λ , S λ ) + 4H 2 (s λ , sλ ).
On Ω ξ , we deduce from ( 48)

H 2 sλ , sλ ≤ D(s λ ) ∨ D(s λ) ≤ D(s λ ) + c 0 τ ≤ 12 1 -κ H 2 (s, ŝλ ) + A(ŝ λ , S λ ) + 1 6 c 0 τ ξ + A(ŝ λ , S λ ),
and get

3 -1 H 2 (s, s) ≤ 1 + 4 × 12 1 -κ H 2 (s, ŝλ ) + 8 1 -κ c 0 τ ξ + (2 √ 2 + 1) 2 + 4 13 -κ 1 -κ A(ŝ λ , S λ ).
Finally, we conclude the first part by using that P (Ω ξ ) ≥ 1 -γΣ 2 e -ξ and the fact that ŝλ is arbitrary. For the second part, it suffices to integrate with respect to ξ and to note that under the assumption that ∆ ≥ 1 on S,

A(ŝ λ , S λ ) ≥ inf S∈S inf t∈S pen λ (t) ≥ c 0 τ, ∀λ ∈ Λ.
9.5. Proof of Corollary 4. For Examples 1 and 4, we know from Proposition 5 that inequality (16) hold with a = 200/n and a = 6 respectively. Besides, inequality (17) holds with δ = 2 and since ( 25) is satisfied for all λ ≥ 1, we may apply Theorem 2.

To get the result, it remains to bound E [pen λ (ŝ λ )] from above. Let us first consider the case of density estimation. Note that if n ≥ 2

M = min M, min i =j |X i -X j | ≥ 1 M ≤ max i =j 1 |X i -X j | + 1
hence, by Hölder inequality with q > 1, q = q/(q -1) > 1 and for p = 2q > 1

E M 1/p ≤ 1 + E max i =j 1 |X i -X j | 1/p ≤ 1 + n(n -1) 2 E 1 |X 1 -X 2 | 1/p ≤ 1 + n(n -1) 2 [0,1) [0,1) 1 |x -y| 1/p s(y)dy s(x)dx ≤ 1 + 2n(n -1) s Lq . ( 49
)
Since for n = 1, M = 1 a.s., this inequality remains true in this case. By using the concavity of the logarithm and of the map t → t 1/p , for all λ ≥ 1,

E [pen λ (ŝ λ )] = 2c 0 τ (nλ -1 + 1)E log(e + M ) ≤ 2c 0 τ (nλ -1 + 1)pE log(e + M 1/p ) ≤ 2c 0 τ (nλ -1 + 1)p log e + E M 1/p ) ≤ 2c 0 τ (nλ -1 + 1)p log e + 1 + 2n(n -1) s Lq ≤ 4τ nλ -1 + 1 p log e + n(n -1) s Lq , (50)
and we conclude by using that in the density case τ equals 1/n up to a universal constant.

Let us now turn to the Poisson case. We decompose E [pen λ (ŝ λ )] as follows

E [pen λ (ŝ λ )] = E [pen λ (ŝ λ )1l n=0 ] + E [pen λ (ŝ λ )1l n≥1 ] = E [pen λ (ŝ λ )1l n=0 ] + E [E [pen λ (ŝ λ )1l n≥1 |n]] .
On the event {n = 0}, pen λ (ŝ λ ) = 2c 0 τ for all λ ≥ 1 and therefore,

E [pen λ (ŝ λ )1l n=0 ] ≤ 2c 0 τ. (51)
Since n > 0, P(n = k) > 0 for all k ≥ 1 and conditionally on the event {n = k}, X 1 , . . . , X k are i.i.d. with density s/n. Hence, using (50), we deduce that

E [pen λ (ŝ λ )1l n≥1 |n] ≤ 4τ nλ -1 + 1 p log e + n(n -1) s Lq n 1l n≥1 .
We now use the following inequality E(U log V ) ≤ E 1/2 (U 2 ) log(E(V )) which holds for all random variables U, V such that U ≥ 0 and V ≥ e. This inequality derives from Cauchy-Schwarz inequality together with the fact that the map v → log 2 v is concave on [e, +∞). We obtain

E [pen λ (ŝ λ )1l n≥1 ] ≤ 4pτ E 1/2 nλ -1 + 1 2 log e + E (n(n -1)) s Lq n
and using the fact that n is distributed as a Poisson random variable, we get

(52) E [pen λ (ŝ λ )1l n≥1 ] ≤ 4pτ nλ -1 + 1 log e + n s Lq .
We conclude by putting ( 51) and ( 52) together and by using that τ is a universal constant in the Poisson case. 9.6. Proof of Proposition 6. Clearly, the result is true for |m| = 1. Let us now assume D = |m| ≥ 2. For t = D j=1 q i j t i j ∈ S m , define

π m t = D-1 j=1 q i j t j +   1 - D-1 j=1 qi j   t i D with qi j = q i j ε -1 ε
Note that for all j ∈ {1, . . . , D}, qi j ≥ 0, q i j ε -1 ∈ 0, . . . , ε -1 , D-1 j=1 qi j ≤ D-1 j=1 q i j ≤ 1 and therefore π m t ∈ S m [η]. Besides, for all j ∈ {1, . . . , D} |q j -qj | ≤ ε and hence, by using that for all u, v ∈ L 0 , 2H 2 (u, v) ≤ X |u -v| dµ we get Let now p ∈ [1, +∞], α > 1/p and R > 0. There exists some r ∈ N * such that α ∈ (1/p, r) and it follows from Proposition 8 that for all J ≥ 0 there exits V ∈ V r,J such that D(V ) ≤ C(r)2 J and for all s F ∈ S α p,∞ (R)

2H 2 (t, S m [η]) ≤ 2H 2 (t, π m t) ≤ X D j=1 (q j -qj )t j dµ ≤ ε D-1 j=1 X t j dµ + D-1 j=1 q i j -qi j X t i D dµ ≤ 2ε(D -1) t 1 ≤ 2η 2 τ.
inf v∈V √ s F -v ≤ n inf V∈V r,J inf G∈V F -G ∞ ≤ C ′′ (r)nR2 -Jα .
Hence, we deduce from (35) that for some constant C (depending on c 0 , τ and r),

CE n -1 H 2 (s, s) ≤ inf J≥0 R 2 2 -2Jα + 2 J n
and the result follows by choosing 2 J of order (nR 2 ) 1/(1+α) ≥ 1. 9.9. Proof of Theorem 5. Hereafter, ρ(P, Q) and h(P, Q) denote the Hellinger affinity and the Hellinger distance between the probabilities P, Q. For θ ∈ Θ n , A ′ (θ) corresponds to the vector t = (A ′ (θ 1 ), . . . , A ′ (θ n )). Since the mapping from Θ n into R n + defined by θ → A ′ (θ) is one to one, for s ∈ S we denote (abusively) P s and E s the probability and expectation with respect to the probability P θ where θ is the unique element of Θ n satisfying s = A ′ (θ). We start with the following lemma.

Lemma 1. Under Assumption 2, for all θ, θ ′ ∈ I n , t = A ′ (θ) and t ′ = A ′ (θ ′ ), h 2 (P θ , P θ ′ ) ≤ - . Summing up over i, it is enough to show the inequality for n = 1, what we shall do. Let θ, θ ′ in I such that t = A ′ (θ) and t ′ = A ′ (θ ′ ). With no loss of generality, we may assume that θ ′ < θ and set δ = (θ -θ ′ )/2. The Hellinger affinity between P θ and P θ ′ is given by ρ(P θ , P θ ′ ) = exp -A(θ) + A(θ ′ ) 2 -A θ + θ ′ 2 and therefore

-log ρ (P θ , P θ ′ ) = A(θ) + A(θ ′ ) 2 -A θ + θ ′ 2 = 1 2 [A(θ) + A(θ -2δ) -2A (θ -δ)] = 1 2 θ θ-δ A ′ (u) -A ′ (u -δ) du = 1 2 θ θ-δ u u-δ
A ′′ (v)dv du.

Since t, t ′ ∈ R + \ {0} and since A ′′ do not vanish on [θ ′ , θ] and A ′ is nondecreasing, for all u ∈ [θ -δ, θ] and v ∈ [u -δ, u]

A ′′ (v) = A ′′ (v) 2 A ′ (v) A ′′ (u) 2 A ′ (u) 4 A ′ (v)A ′ (u) A ′′ (u) ≤ A ′′ (v) 2 A ′ (v) A ′′ (u) 2 A ′ (u) 4A ′ (u) A ′′ (u) ≤ 4κ A ′′ (v) 2 A ′ (v)
A ′′ (u) 2 A ′ (u) .

giving thus,

-log ρ (P θ , P θ ′ ) ≤ 2κ

S K = s ∈ K n , √ s ∈ V ⊂ S.
Let t 0 ∈ S be such that u 0 = √ t 0 . Under (40), there exists β 1 , . . . , β D such that √ t 0 = D j=1 β j u j and that for all ε ∈ {0, 1} D one can find t ε ∈ S K such that √ t ε = D j=1 (β j + Rε j ) u j . Note that the for all ε, ε ′ ∈ {0, 1} D ,

2H 2 (t ε , t ε ′ ) = √ t ε - √ t ε ′ 2 = R 2 d(ε, ε ′ ).
Besides, whatever the estimator ŝ and s ∈ S

sup s∈S E s H 2 (s, ŝ) ≥ sup s∈S K E s H 2 (s, ŝ) ≥ sup ε∈{0,1} D E t ε H 2 (t ε , ŝ) ≥ inf ε sup ε∈{0,1} D E t ε H 2 t ε , t ε = R 2 2 inf ε sup ε∈{0,1} D E t ε [d (ε, ε)] ,
where the two last infima run among all estimators ε based on the observations (X 1 , . . . , X n ) with values in {0, 1} D . Theorem 2.10 in [START_REF] Tsybakov | Introduction à l'estimation non-paramétrique[END_REF] asserts that inf

ε sup ε∈{0,1} D E t ε [d (ε, ε)] ≥ D 2 1 -α(2 -α)
provided that for all ε, ε ′ such that d(ε, ε ′ ) = 1, h 2 P θ ε , P θ ε ′ ≤ α < 1 where the parameters θ ε and θ ε ′ are such that t ε = A ′ (θ ε ) and t ε ′ = A ′ (θ ε ′ ). Taking α = 1/2 and using Lemma 1, we get for all ε, ε ′ such that d(ε, ε ′ ) = 1,

h 2 P θ ε , P θ ε ′ ≤ 4κH 2 (t ε , t ε ′ ) ≤ 2κR 2 ≤ 1 2 = α,
and hence,

inf ŝ sup s∈S E s H 2 (s, ŝ) ≥ 1 - √ 3/2 4 DR 2 ,
which concludes the proof. 9.10. Proof of Theorem 6. Let L 0 = q t , t ∈ C . Under Assumption 3, the mapping t → q t is a quasi isometry between (C, ) and (L 0 , H). In particular, by using Proposition 7, for all V ∈ V, q t ∈ L 0 and number r ≥ 2(Rη(V )),

q t ′ ∈ S(V ), H(q t , q t ′ ) ≤ r √ τ ≤ t ′ ∈ V, t -t ′ ≤ R -1 r √ τ ≤ exp 5D(V ) R -1 r η(V ) 2 ≤ exp r 2 2 .
Consequently, we deduce from Propositions 2 and 3 that S satisfies Assumption 1 with γ = 1 and D(S(V )) = 40D(V ) for all V ∈ V. We may therefore apply Theorem 1 and get (recalling that fλ = Π C fλ and ŝλ = q fλ ) CE H 2 (q f , ŝλ ) ≤ inf λ∈Λ E H 2 (q f , q fλ ) + E A(q fλ , S λ ) , and deduce that for some constant C ′ depending on c 0 , R, R and Σ only,

C ′ E f -Π C f 2 ≤ inf λ∈Λ E f -Π C fλ 2 + E A(q fλ , S λ ) .
It remains to bound A(q fλ , S λ ) from above for all λ ∈ Λ. By using Proposition 7, for all λ ∈ Λ and V ∈ V λ ,

A(ŝ λ , S λ ) ≤ inf t ′ ∈V
H 2 (q fλ , q t ′ ) + pen(q t ′ )

≤ R 2 inf t ′ ∈V Π C fλ -t ′ 2 + c 0 τ 40D(V ) + ∆(V ) ≤ 8R 2 inf t ′ ∈V Π C fλ -t ′ 2 + η 2 (V )τ + c 0 τ 40D(V ) + ∆(V ) ≤ 8R 2 inf t ′ ∈V Π C fλ -f + f -fλ + fλ -t ′ 2 + η 2 (V )τ +c 0 τ 40D(V ) + ∆(V ) ≤ 24R 2 f -Π C fλ 2 + f -fλ 2 + inf t ′ ∈V fλ -t ′ 2 + η 2 (V )τ +c 0 τ 40D(V ) + ∆(V ) ≤ C ′′ f -fλ 2 + fλ -Π V fλ 2 + D(V ) + ∆(V )
for some C ′′ depending on R, R and c 0 only which concludes the proof. 9.11. Proof of Corollary 12. For each λ ∈ Λ, let λ be the linear map which coincides with λ on V λ and takes the value 0 on its orthogonal. On the one hand,

E f -fλ 2 = f -λf 2 + tr λ 2 σ 2 = n k=1 (1 -λ (k) ) 2 f, v (k) 2 + σ 2 n k=1 λ 2 (k) .
On the other hand, the definition of D = D λ entails that for all k > D, λ (k) ≤ 1/2 (with the convention λ (n+1) = 0) and therefore

E inf t∈V λ fλ -t 2 ≤ E λX -λX 2 = E (λ -λ)f + (λ -λ)ε 2 = (λ -λ)f 2 + tr (λ -λ) 2 σ 2 = n k=1+D λ 2 (k) f, v (k) 2 + σ 2 n k=1+D λ 2 (k) ≤ n k=1 (1 -λ (k) ) 2 f, v (k) 2 + σ 2 n k=1 λ 2 (k) = E f -fλ 2 and, since D/4 ≤ D k=1 λ 2 (k) ≤ σ -2 E f -fλ 2
, we conclude the proof by using Theorem 6.

  ) and build an η √ τ -net S m [η] of S m for all η > 0 and m ∈ M. Concerning ∆, we choose ∆ S m = |m|(1 + log(eM/|m|)) for all m ∈ M. It satisfies (4) since for all D ∈ {1, . . . , M }, M D ≤ log(eM/D) and hence m∈M fix some η > 0 and m ∈ M and discretize S m . If for some i ∈ {1, . . . , M } m = {i}, we can merely take S {i} [η] = S i . If |m| ≥ 2, write m as i 1 , . . . , i |m| with 1 ≤ i 1 < . . . < i |m| ≤ M and for (33) ε = min η 2 τ (|m| -1) t 1 ; 1 with t 1 = max i=1,...,M X t i dµ define S m [η] ⊂ S m as the set gathering the elements of the form |m|-1 j=1

  ) with a = 1/12, b = 1 and c = √ 2/36. 9.3. Proof of Proposition 4. Let us fix m, m ′ ∈ M, ξ > 0 and y such that y 2 ≥ τ (D(S m ) ∨ D(S m ′ ) + ξ) .

9. 7 . 2 .

 72 Proof of Proposition 7. We set D = D(V ) and consider an orthonormal basis u j , j = 1, . . . , D of V . It follows from Proposition 9 of[START_REF] Birgé | Model selection via testing: an alternative to (penalized) maximum likelihood estimators[END_REF] that the set j , (k j ) j=1,...,D ∈ Z D    is a η √ τ -net for V and for all R ≥ 2η and h ∈ R n t ∈ T , h -t ≤ R √ τ ≤ exp 0.458D R ηThe result follows by applying Proposition 12 of[START_REF] Birgé | Model selection via testing: an alternative to (penalized) maximum likelihood estimators[END_REF] (with π = Π C , (M ′ , d) = (R n , ), M 0 = C, T = T and λ = 1 = ε).9.8. Proof of Corollary 8. Let us denote by ∞ the supremum norm on [0, 1]. First note that (4) holds sinceV ∈V e -∆(V ) ≤ r≥1 J≥0|V r,J | e -(C ′ (r)+1)2 J -r ≤ r≥1 e -r J≥0 e -2 J < +∞.

  j =ε ′ j . Besides, we set

  In view of estimating s at best, an ideal choice of m is given by the index m * , usually called the oracle, for which ŝm * achieves the best possible risk bound among the collection of estimators {ŝ m , m ∈ M}. In practice, this oracle is inaccessible since it depends on the unknown parameter s and the art of model selection is to design a rule solely based on the data in order to mimic ŝm

. More precisely, given a model S with a finite metric dimension, the construction of Birgé's estimators (called T -estimators) is based on a good discretization of S and on the use of a robust test in view of selecting among the discretization points. T -estimators are naturally robust under misspecification and, from this point of view, may outperform the well-known maximum likelihood estimators which are not. If one considers discretization points as candidate estimators, T -estimators result from an estimator selection procedure which crucially relies on the ability of finding a robust test with respect to a given distance d and, to our knowledge, no general recipe for this is available. Our approach provides a general machinery to build such robust tests for Hellinger-type distances and allows us to build T -estimators in various statistical settings, recovering Birgé's results in the contexts of Examples 1 and 4 and establishing new ones in the cases of Examples 2 and 3. Unlike Birgé's approach which allows to select among deterministic points only, ours can deal with arbitrary collection of estimators. 1.4. Connections with model selection. Consider a collection of models S = {S m , m ∈ M} (say linear spaces) together with a mapping ∆ from S into [1, +∞) satisfying (4) and associate to each m ∈ M an estimator ŝm with values in S m such that, for some distance d on L 0 and some positive constant C, CE d 2 (s, ŝm ) ≤ inf t∈Sm d 2 (s, t) + τ D(S m ). * . From this point of view, the model selection problem is a particular case of that of estimator selection. The following oracle-type inequality is typical of what is usually proved in the literature: for all s ∈ L 0 (8)

  (t i , t ′ i ) = 4κH 2 (t, t ′ ) Proof. Since h 2 (P θ , P θ ′ ) = 1 -ρ (P θ , P θ ′ ) = 1 -exp

	n			n
	i=1	log ρ P θ i , P θ ′ i	≤ 4κ	i=1	H 2 n
					log ρ P θ i , P θ ′ i
					i=1
	≤ -

n i=1 log ρ P θ i , P θ ′ i , it suffices to show that -n i=1 log ρ P θ i , P θ ′ i ≤ 4κ n i=1 H 2 (t i , t ′ i )

Acknowledgement. The author is thankful to Lucien Birgé for his careful reading of the paper and his thoughtful comments. We also thank the two referees for their comments and questions that have led to an improved version of the present paper.