Nabil Rachdi
email: nabil.rachdi@eads.net

Jean-Claude Fort
email: jean-claude.fort@parisdescartes.fr

Thierry Klein
email: thierry.klein@math.univ-toulouse.fr

Risk bounds for new M-estimation problems

In this paper, we develop new algorithms for parameter estimation in the case of models type Input/Output in order to represent and to characterize a phenomenon Y . From experimental data Y 1 , ..., Y n supposed to be i.i.d from Y , we prove a risk bound qualifying the proposed procedures in terms of the number of experimental data n, computing budget m and model complexity. The methods we present are general enough which should cover a wide range of applications.

1

Introduction

As in many statistical problems, we are interested to investigate the stochastic behavior of a random variable Y . We have at disposal an i.i.d sample Y 1 , ..., Y n . These data come from experiments that could be real or the result of a computer code. In an industrial context, it is not rare that the size of the available set of data is small. This is due either to the cost of each real experiment or to the very long time needed for each run of a simulation code. It is encountered in various eld of industry: meteorology, oil extraction, nuclear security, aeronautic, mechanical engineering etc... Besides these costly experiments or codes, various reduced models are available. Even if they still are complicated, one can use them to simulate in a reasonable computing time and obtain large samples from simulations. Of course, these reduced models depend on parameters that are not well known and need to be estimated. So that the reduced models take the following form: (x, θ) ∈ X × Θ → h(x, θ). It is important to note that when the model h varies, the set of input variables X and parameters Θ may change too. Moreover, theses variables are not directly related to the "conditions" leading to the "experimental" data Y 1 , ..., Y n . Indeed, in our study, we don't suppose having the data (x 1 , Y 1 ), ..., (x n , Y n ) which diers our framework from the classical regression one. That's why we assume that the available data reduced to Y 1 , ..., Y n : this includes the cases where the experimental conditions are not available or where the input of the complex code, modeling the phenomenon, are not clearly related to the input of the reduced models. Let us take an example of particular interest coming from EADS 4 Research department: the eect of an electromagnetic eld on the behavior of an aircraft. When lightning or an electromagnetic eld strike an aircraft, sensors measure data corresponding to the intensity of such eld in various part of the aircraft. The data recorded are dispersed due to the intrinsic variability of the phenomenon. In our framework, information of one sensor is represented by the sample Y 1 , ..., Y n . On another side, we dispose of several computer codes h modeling the electromagnetic eld in function of input variables z. These input variables take the following form, z = (x, θ), where x represents variables not well controlled and θ a vector of parameters to be estimated, corresponding to the eld properties (angles, atmospheric conditions etc...). The uncontrolled variables x will be modeled by a random variable X with distribution P x . This distribution may be known or not, we suppose at least having at disposal a sample X 1 , ..., X m where m >> n .

The computer code are complex systems, i.e the result of interconnected disciplines providing a granular modeling. Actually, one disposes of a set of models H covering all available models: from the simplest to the most complicated. Hence, another important issue would be to "select" a model among the set H for a specic use. We don't treat this aspect in this paper, we work with one model h only.

So, shortly speaking, our goal is to construct a Random Simulator, X → ĥ(X, θ) with X some random variable, predicting as well as possible the observed data Y 1 , ..., Y n . In this setting, it may be non-signicant to talk about function approximation. For instance, suppose that Y ∼ U([0, 1]) (uniform distribution on [0, 1]) and consider the model h(X, θ) = θ 1 + θ 2 X where θ = (θ 1 , θ 2 ) and X ∼ U([0, 1]). The cases θ 1 = (0, 1) and θ 2 = (1, -1), corresponding to models h(x, θ 1 ) = x and h(x, θ 2 ) = 1 -x respectively, produce the (same) Random Simulator h(X, θ) ∼ U([0, 1]) ( θ = θ 1 or θ 2 ). Hence, this Random Simulator predicts like the variable of interest Y but with two dierent models (the models or the parameters are not identiable). Thus, the function approximation approach can be meaningless without preliminary precautions. This paper is the theoretical part of a work on industrial applications in the eld of "Uncertainty Management" [START_REF] De Rocquigny | Uncertainty in industrial practice[END_REF]. We aim at constructing a data-dependent model which outputs are "close to" some observed data (experimental data). The results we present are theoretical in that the estimation and selection algorithms we propose don't include practical implementations. The same is true for the modeling aspect: we deal with (input/output) models without specifying what can be done in practice. For instance, we do not deal with the pertinence of the possible metamodels (see [START_REF] Kleijnen | Design and analysis of simulation experiments[END_REF][START_REF] Vazquez | Modélisation comportementale de systèmes non-linéaires multivariables par méthodes à noyaux et applications[END_REF][START_REF] Santner | The design and analysis of computer experiments[END_REF][START_REF] Soize | Physical systems with random uncertainties: chaos representations with arbitrary probability measure[END_REF]). Here, we don't talk about the impact of modeling technics, this is let for a forthcoming paper where we will apply some results obtained in this study in an industrial context. The main tool of our development is the empirical processes theory. This theory constitutes the mathematical toolbox of asymptotics statistics and was rst explored in the 1950's by the work on Functional Central Limit Theorem [START_REF] Donsker | Justication and extension of Doob's heuristic approach to the Kolmogorov-Smirnov theorems[END_REF]. Along the years, the development of empirical processes theory increased successfully thanks to work of many contributors, R.M. Dudley [START_REF] Dudley | Weak convergence of measures on nonseparable metric spaces and empirical measures on euclidian spaces[END_REF], D. Pollard [START_REF] Pollard | Empirical processes: theory and applications[END_REF], P. Gaenssler [START_REF] Gaenssler | Empirical Processes[END_REF], Galen R. Shorack and Jon A. Wellner [START_REF] Shorack | Empirical processes with applications to statistics[END_REF] and others. More recently, many references give a general overview of this theory with its applications to statistics, for example [START_REF] Van Der Vaart | Weak Convergence and Empirical Processes[END_REF][START_REF] Van De Geer | Empirical processes in M-estimation[END_REF][START_REF] Kosorok | Introduction to empirical processes and semiparametric inference[END_REF]. Empirical processes give power tools for evaluating statistical estimation and inference problems. In particular, we use concentration inequalities to derive risk bounds following the work of [START_REF] Talagrand | Sharper bounds for Gaussian and empirical processes[END_REF], [START_REF] Massart | Risk bounds for statistical learning[END_REF] and [START_REF] Klein | Concentration around the mean for maxima of empirical processes[END_REF] among others. Estimation based on minimizing a function was introduced by Huber in 1964 [START_REF] Huber | Robust estimation of a location parameter[END_REF] where he proposed generalizing maximum likelihood estimation. The estimators resulting are called Mestimator ("M" for minimizing or maximizing ) [START_REF] Huber | Robust statistics[END_REF]. The class of M-estimators is a broad class because many estimation procedure can be viewed as M-estimation, maximum likelihood and least-squares estimators are some of the most important examples. Asymptotic properties of these estimators were widely studied in a general context, and many authors like [START_REF] Van De Geer | Empirical processes in M-estimation[END_REF] or [START_REF] Van Der | Asymptotic statistics[END_REF] used empirical processes theory which turn out to be a very valuable tool. We present a general method where the criterion to minimize depends on both experimental and simulated data. This paper is divided into ve parts. In Section 2 we describe our general framework. In Section 3 we establish Theorem 4.1 providing a risk bound for inverse problems based on both experimental and simulated data. In Section 6 we discuss about constants in Theorem 4.1. Let (Ω, A, P) be a probability space. We assume that all random variables are dened on this probability space. Let a complex phenomenon modeled by a random real valued variable Y ∈ Y, with distribution unknown Q and f the associated (Lebesgue) density function. Let assume that

Y ⊂ [-M, M ], M > 0.
Suppose that a n-sample Y 1 , ..., Y n is available: we call experimental data.

Next, we suppose that this complex phenomenon can be represented by the outputs h(x, θ)

given by models h which belong to a set H

h : X × Θ -→ Y (x, θ) -→ h(x, θ)
where X ⊂ R d (input space), Θ ⊂ R k compact (parameters space). We equip the input space X with a probability measure P x which forms a probability space (X , B, P x ). The probability measure P x is not supposed to be known, we will only dispose of a sample drawn from this distribution. In the case where P x is known, without loss of generality, one can simply consider the uniform distribution on [0, 1] provided to apply a well known probabilistic transformation. The input vector is a random vector X dened on this space, and so, the output vector h(X, θ) is a random real valued variable, for each θ ∈ Θ.

The space Y is equipped with a σ-algebra E so as to ensure the measurability of the functions

h(•, θ) : (X , B, P x ) -→ (Y, E) X -→ h(X, θ)
Moreover, we suppose given m realizations of the input random vector X, X 1 , ..., X m which provides m output simulated data h(X 1 , θ), ..., h(X m , θ) for all θ ∈ Θ.

Remark 2.1. In practice, the data X 1 , ..., X m may either arise from a data base (from experiments etc...) or simply arise from simulations of the random variable X with known distribution P x . In this paper, we develop a general method for estimating the parameter θ based on the training data

Y 1 , ..., Y n ; X 1 , ..., X m .
The method we propose is general enough to include some specic problems met in practice.

Indeed, two kinds of statistical analysis involving inverse problems can be considered: Identication and Prediction.

-Identication.

This analysis consists in estimating the "true" parameter θ . It aims at estimating "physical" parameters having a real signication like dimensions or material properties for instance.

-Prediction.

In prediction, one wants to estimate a parameter θ (not necessarily unique) in order to predict the random phenomenon Y . Informally, one hopes that

h (X, θ ) ≈ Y .
Here, the parameter θ may have no real signication. It is the case in model calibration for example.

Model performance

Tools for evaluating the model performance

Let introduce some tools to evaluate the quality of a model h ∈ H parameterized by θ ∈ Θ.

-Feature of probability measure, model, contrast and Risk function.

A feature of the distribution µ is dened following the Denition ?? in Chapter ??, that is, as a quantity ρ F (µ) ∈ F where F is called the feature space.

Notice that the feature space F can be either a scalar space (mean, threshold probability, etc...) or a functional space (density distribution, cumulative distribution function). We equip the feature space F with the norm || • || F which can be either the absolute value norm | • | when F ⊂ R, or a L r -norm (r ≥ 1) when F is a functional space (with functions dene on Y).

In all what follows, we denote by ρ h (θ) a feature of the distribution of the random model output h(X, θ). In the previous chapter, we used the notation ρ F (θ), but especially in this chapter, we will use the writing ρ h (θ) in order to emphasize the fact that the quantities of interest we deal with are relative to the numerical model h .

We call model (feature space) a subset F ⊂ F . In particular, we will deal with a model induced by h given by

F h,θ = {ρ h (θ) , θ ∈ Θ} ⊂ F . (1)
Denition 2.1. Contrast and risk function. A contrast function (with value in L 1 (Q)) is any function

Ψ : F -→ L 1 (Q) (2) ρ -→ Ψ(ρ, •) : y ∈ Y -→ Ψ(ρ, y) , such that ρ * = Argmin ρ∈F E Y Ψ (ρ, Y )
is unique.

We call risk function the application

∀ ρ ∈ F , R Ψ (ρ) := E Y Ψ (ρ, Y ) .
On the model F h,θ ⊂ F, we denote the risk by

R Ψ (h, θ) := E Y Ψ (ρ h (θ) , Y ) . (3)
Next, for a random variable ξ, we use the notation E ξ for the expectation under the variable ξ.

Example 2.1. Some classical features and associated contrasts.

-F = R : we may consider ρ(µ) = u µ(du) = E µ (ξ) (mean), ρ(µ) = 1 [s,+∞[ (u)µ(du) =
µ(ξ > s) (exceeding probability), etc...

Mean-contrast

Ψ (ρ, y) = (y -ρ) 2 -F = {set of density functions} log-contrast Ψ (ρ, y) = -log ρ(y) L 2 -contrast Ψ (ρ, y) = ||ρ|| 2 2 -2 ρ(y) -etc... See Table 1, page 11.
Example 2.2. Some classical risk functions.

By elementary calculus, we see that -the mean-contrast gives a distance between means (up to a constant term)

R Ψ (h, θ) = (E(Y ) -ρ h (θ)) 2 + V ar(Y )
-the log-contrast gives the Kullbach-Leibler divergence (up to a constant term)

R Ψ (h, θ) = KL(f, ρ h (θ)) -E(log(Y )) ,
where

KL(g 1 , g 2 ) = log( g 1 g 2 )(y) g 1 (y) dy, -the L 2 -contrast gives a L 2 distance between density functions (up to a constant term) R Ψ (h, θ) = ρ h (θ) -f 2 2 -f 2 2 .
In view of that examples, it make sense to investigate models h or/and parameters θ providing small risk values.

Let precise what we mean by complex models in view of statistical using.

-Complex models.

For θ ∈ Θ, let consider a feature ρ h (θ) of the random model output h(X, θ).

We say that h is complex if the feature ρ h (θ) is analytically unreachable in θ. For instance, if ρ h (θ) = X h(x, θ) P x (dx) , this integral is not necessarily tractable, even if the probability measure P x is known. Complex models can arise from several ways. For example, the function h(•, θ) can have a complicated form due to the high complexity of the modeling, or the function can be a black box function input/output and so, not with an analytical form. This situation is very common in engineering, where complex models exist and are only known through simulations

(X 1 , h(X 1 , θ)) , ..., (X m , h(X m , θ)) for all θ ∈ Θ .
This aspect is the principal motivation of our work. [START_REF] Donoho | Ideal spatial adaptation by wavelet shrinkage[END_REF] Inverse Problem.

Our goal is to compute a parameter θ ∈ Θ making the risk function R Ψ (h, θ) as small as possible.

-Oracle.

We want to estimate a parameter θ minimizing the risk (3), i.e

θ ∈ Argmin θ∈Θ R Ψ (h, θ) . (4)
In the literature, the parameter θ is also called the oracle. This term was introduce by Donoho and Johnstone [START_REF] Donoho | Ideal spatial adaptation by wavelet shrinkage[END_REF].

Notice that it may exist more than one parameter minimizing the risk R Ψ (h, θ). The minimal risk we can reach is R Ψ (h, θ ), also called ideal risk.

However, the risk function R Ψ (h, θ) is uncomputable (hence θ ) for two reasons. First, the measure Q is unknown, and second, because we are dealing with complex models.

We aim at computing a parameter θ that performs as well as the oracle θ , that is

R Ψ (h, θ) ≈ R Ψ (h, θ ) .
In what follows, we establish a risk bound of the form

R Ψ (h, θ) ≤ C R Ψ (h, θ ) + ∆ .
We propose the following estimation procedure to built θ.

As Q is unknown, we replace it by its empirical version

Q n := 1 n n i=1 δ Y i based on Y 1 , ..., Y n .
The approximation of the risk becomes

1 n n i=1 Ψ (ρ h (θ) , Y i ) .
Then, it remains the feature ρ h (θ) which is supposed analytically intractable (for each θ). We propose to estimate the feature as follows.

-Plug-in estimator.

We denote by ρ m h (θ) a plug-in estimator of ρ h (θ) based on h(X 1 , θ), ..., h(X m , θ). We suppose that ρ m h (θ) takes the following form

ρ m h (θ) := 1 m m j=1 ρ(h(X j , θ)) (5)
where

1 m ρ : Y → F is a weight function depending on the contrast Ψ considered.
For simplicity, we may also call ρ weight function. mean-contrast

1 m ρ(y) = y m -log-contrast or L 2 -contrast 1 m ρ(y)(•) = 1 m K b ( • -y)
where For instance, in the case where 1 m ρ(y) = y m , the function ρ(y)(λ) is constant in λ.

K b ( • -y) = 1 b K( • -y b ) for a kernel K()
For notation convenience, we may use the notation ξ 1..l for a sample ξ 1 , ..., ξ l of random variables, and E ξ 1..l will be the expectation under the joint law of (ξ 1 , ..., ξ l ).

Denition 3.1. We denote by σ m h (θ), called simulation error, the error committed estimating the feature ρ h (θ) by the estimator ρ m h (θ),

σ m h (θ) := ||ρ m h (θ) -ρ h (θ)|| F .
By triangular inequality and the fact that

E X ρ(h(X, θ)) = E X 1..m ρ m h (θ) , it holds σ m h (θ) = ||ρ m h (θ) -ρ h (θ)|| F = ||ρ m h (θ) -E X 1..m ρ m h (θ) + E X 1..m ρ m h (θ) -ρ h (θ)|| F = ||ρ m h (θ) -E X ρ(h(X, θ)) + E X ρ(h(X, θ)) -ρ h (θ)|| F ≤ ||ρ m h (θ) -E X ρ(h(X, θ))|| F + ||E X ρ(h(X, θ)) -ρ h (θ)|| F ≤ 1 m m j=1 [ ρ(h(X j , θ)) -E X ρ(h(X, θ))] F + b m h (θ) (6) with b m h (θ) := ||E X ρ(h(X, θ)) -ρ h (θ)|| F (7)
the bias error. For example, in the case where ρ(y)(•) = K b ( • -y), the bandwidth will depend on m (b m ).

The rst term in the right hand side of inequality ( 6) is a variance (random) term, and the second is a bias (deterministic) term.

For our statistical analysis, the variability term

1 m m j=1 [ ρ(h(X j , θ)) -E X ρ(h(X, θ))]
F will play a crucial role. Assumption 3.1. We assume that the plug-in estimator ρ m h (θ) (5) is uniformly asymptotically unbiased, i.e it exists some constant b h (m) depending on h and m such that the bias error [START_REF] Goldenshluger | Uniform bounds for norms of sums of independent random functions[END_REF] satises

sup θ∈Θ b m h (θ) < b h (m) < ∞ ,
and b h (m) → 0 with m.

For instance, in the case where ρ h (θ) is the density (of h(X, θ)), the quantity

E X ρ(h(X, θ)) = E X K b ( • -h(X, θ)
) may be choose to be the expectation of a kernel K b with bandwidth b. Under right conditions on the set of densities {ρ h (θ), θ ∈ Θ}, one can hope that

sup θ∈Θ b m h (θ) -→ b→0 0
See the example given in Subsection 6.2.

Finally, the criterion we propose to minimize has the form

1 n n i=1 Ψ   1 m m j=1 ρ(h(X j , θ)) , Y i   ,
which provides the estimator

θ = Argmin θ∈Θ 1 n n i=1 Ψ   1 m m j=1 ρ(h(X j , θ)) , Y i   , (8)
or

θ = Argmin θ∈Θ n i=1 Ψ   1 m m j=1 ρ(h(X j , θ)) , Y i   .
We give some examples of estimators θ.

Example 3.2. Examples of estimators.

mean-contrast We recall that the issue is the statistical properties of this procedure taking into account the two kinds of data: experimental and simulated data, which is non classical in statistics. Indeed, once we dene the procedure for computing θ, we have to qualify the quality of this procedure.

θ M = Argmin θ∈Θ n i=1   m j=1 (Y i -h(X j , θ))   2 -log-contrast θ log = Argmin θ∈Θ - n i=1 log   m j=1 K b (Y i -h(X j , θ))   -L 2 -contrast θ L 2 = Argmin θ∈Θ    m j=1 K b ( • -h(X j , θ)) 2 2 - 2 m n n i=1 m j=1 K b ( Y i -h(X j , θ))    . Remark 3.2.
It's the topic of the following section.

Main Result

In this section, we aim at establishing a risk bound which provides a qualication of the estimation procedure previously dened. We recall that

R Ψ (h, θ) = E Y Ψ (ρ h (θ), Y ) , θ ∈ Argmin θ∈Θ R Ψ (h, θ) ,
and

θ = Argmin θ∈Θ 1 n n i=1 Ψ   1 m m j=1 ρ(h(X j , θ)) , Y i   .
Now, we give some denitions and notations useful for setting the Theorem 4.1.

Denote by

G n = √ n(Q n -Q) and K x m = √ m(P x
m -P x ) , the Q-empirical process (based on Y 1 , ..., Y n ) and P x -empirical process (based on X 1 , ..., X m ), respectively. Let the classes of functions

W ( ρ,Ψ) = {y ∈ Y → Ψ( ρ(λ) , y) , λ ∈ Y} , (9)
P ( ρ,h) = {x ∈ X → ρ(h(x, θ))(λ) , (θ, λ) ∈ Θ × Y} . (10) W ( ρ,Ψ) P ( ρ,h) A Ψ mean-contrast y → (y -λ) 2 , x → h(x, θ), 4 M λ ∈ Y θ ∈ Θ log-contrast y → -log (K b (y -λ)) , x → K b (λ -h(x, θ)), f 2 /η λ ∈ Y (λ, θ) ∈ Θ × Y L 2 -contrast y → ||K b ( • -λ)|| 2 -2 K b (y -λ), idem 2 ( f 2 + B) λ ∈ Y
G n g = Y g(u)G n (du) = √ n Y g(u)(Q n -Q)(du) = 1 √ n n i=1 (g(Y i ) -E(g(Y ))) .
Also, for a class of functions G X , :

X → R K x m g = 1 √ m m j=1
(g(X j ) -E(g(X))) .

Remark 4.1. The quantities G n G Y and K x m G X are nonnegative real valued random variables.

In our applications, the class of functions G Y is W ( ρ,Ψ) and G X is P ( ρ,h) , respectively dened in ( 9) and [START_REF] Kleijnen | Design and analysis of simulation experiments[END_REF]. Denition 4.1. Tightness.

Let (ξ l ) l≥1 be a sequence of real value random variables dened on the probability space (Ω, A, P). This sequence is tight if for all ε > 0, it exists some compact

K ε ⊂ R such that ∀ l ≥ 1 , P (ξ l ∈ K ε ) ≥ 1 -ε .
In particular, if the ξ l are nonnegative, the sequence is tight if for all ε > 0 it exists some constant Kε ≥ 0 such that

∀ l ≥ 1 , P ξ l ≤ Kε ≥ 1 -ε .
We make the following assumption.

Assumption 4.1. We assume that the contrast Ψ satises -for all y ∈ Y, the function ρ → Ψ(ρ, y) is convex , -for all y ∈ Y and

ρ 1 , ρ 2 ∈ F |Ψ(ρ 1 , y) -Ψ(ρ 2 , y)| ≤ L Ψ (y) ||ρ 1 -ρ 2 || F with L Ψ : Y → R satisfying A Ψ := E Y L Ψ (Y ) < ∞ .
The function L Ψ (hence the constant A Ψ ) doesn't depend on ρ 1 and ρ 2 .

The contrasts given in Example 2.1 satisfy this assumption under right conditions on the distribution of Y . Let the feature space F equipped with either the absolute value norm, or some L r norm. Then, for all ε > 0, with probability at least 1 -2ε it holds

R Ψ (h, θ) ≤ inf θ∈Θ (R Ψ (h, θ)) + K ε ( ρ,Ψ) √ n 1 + n m (K ε ( ρ,h) + B m )
where the constants

K ε ( ρ,Ψ) , K ε ( ρ,h) depend on Kε ( ρ,Ψ) , Kε
( ρ,h) , A Ψ , M and r. B m is a bias factor depending on b h (m).

Some comments

It is of interest to compare the methodology we develop with the classical framework where the feature ρ h (θ) of the random model output h(X, θ) is analytically tractable. In this case, the estimation procedure ( 8) is classically

θ n = Argmin θ∈Θ 1 n n i=1 Ψ (ρ h (θ) , Y i ) ,
and we can derive immediately a risk bound.

Proposition 5.1. Basic risk bound.

It holds that

R Ψ (h, θ n ) ≤ inf θ∈Θ (R Ψ (h, θ)) + 2 √ n G n W Ψ , (11)
where

W Ψ = {y ∈ Y → Ψ(ρ h (θ) , y) , θ ∈ Θ} .
Proof. The proof comes from a classical calculus in M-estimation, see for example [START_REF] Van Der | Asymptotic statistics[END_REF] (p. 46) . Most of statistical procedures, as likelihood, regression, classication etc... can be written like [START_REF] Klein | Concentration around the mean for maxima of empirical processes[END_REF]. Such procedures have been widely studied with a large literature available. Recently, authors use the Empirical Processes theory (see [START_REF] Van De Geer | Empirical processes in M-estimation[END_REF][START_REF] Van Der | Asymptotic statistics[END_REF][START_REF] Van Der Vaart | Weak Convergence and Empirical Processes[END_REF][START_REF] Kosorok | Introduction to empirical processes and semiparametric inference[END_REF] among others) to derive limit theorems. Indeed, the asymptotic (and non-asymptotic) properties of the estimator θ n can be given from the behavior of the residual term 2 √ n G n W Ψ . In particular, for identication problem (i.e θ is unique), consistency and rate of convergence are derived from the uctuations of the random variable G n W Ψ , see for example [START_REF] Van De Geer | Empirical processes in M-estimation[END_REF]. Suppose for a moment that it exists some constant (uniform in n) such that with high probability

G n W Ψ ≤ K 2 ,
then by inequality [START_REF] Klein | Concentration around the mean for maxima of empirical processes[END_REF], with high probability

R Ψ (h, θ n ) ≤ inf θ∈Θ (R Ψ (h, θ)) + K √ n . (12)
Thus, depending on whether the constant K is sharp or not, one can bound properly the estimation error. To compute such (sharp) constant K is dicult in general, we can refer to [START_REF] Ledoux | The concentration of measure phenomenon[END_REF][START_REF] Talagrand | Sharper bounds for Gaussian and empirical processes[END_REF][START_REF] Van Der Vaart | Weak Convergence and Empirical Processes[END_REF][START_REF] Massart | Concentration inequalities and model selection: Ecole d'Eté de Probabilités de Saint-Flour XXXIII-2003[END_REF].

Inequality [START_REF] Klein | Concentration around the mean for maxima of empirical processes[END_REF] can not be applied to our framework because the induced procedure θ n involves the quantity ρ h (θ) intractable for complex models.

The result of Theorem 4.1 is non-asymptotic, i.e valid for all n ≥ 1 and m ≥ 1 under mentioned assumptions. The fundamental point of this theorem is the "concentration of the measure phenomenon" (Ledoux [START_REF] Ledoux | The concentration of measure phenomenon[END_REF], Billingsley [START_REF] Billingsley | Convergence of probability measures[END_REF]) presents in the assumptions, more precisely, when we supposed the tightness of the sequences of the random variables G n W ( ρ,Ψ) (Y 1..n -dependent) and K x m P ( ρ,h) (X 1..m -dependent). Moreover, we insist on the fact that the constants Kε ( ρ,Ψ)

(that bounds G n W ( ρ,Ψ) ) and Kε ( ρ,h) (that bounds K x m P ( ρ,h) ) are uniform (or decreasing) in n and m, respectively. The advantage of this uniformity is the explicit expression of the residual term

K ε ( ρ,Ψ) √ n 1 + n m (K ε ( ρ,h) + B m ) (13)
depending on the data (n and m) on one hand, and on the constants Kε ( ρ,Ψ) , Kε ( ρ,h) and B m on the other hand. However, although the existence of such constants are proved or supposed, their computation is more tedious. Indeed, we need results about tail bounds for Gaussian and Empirical Processes. We will discuss in Section 6.3 how to compute properly such constants using concentration inequalities. Let assume for a moment the existence of these constants. We showed that the estimation procedure θ dened in (8) "mimic" the ideal risk R Ψ (h, θ ) = inf θ∈Θ (R Ψ (h, θ)) up to the residual term [START_REF] Ledoux | The concentration of measure phenomenon[END_REF]. Making m -→ +∞ , this residual becomes simply

K ε ( ρ,Ψ)
√ n which has the same form as those found in classical cases [START_REF] Kosorok | Introduction to empirical processes and semiparametric inference[END_REF]. We nd the usual rate of convergence √ n. In our purpose, the factor

1 + n m (K ε ( ρ,h) + B m ) > 1
we call simulation factor, is due to simulation used estimating the feature ρ h (θ) of the random output h(X, θ) by a plug-in estimator ρ m h (θ) we dened in (5).

Example 5.1. For unbiased plug-in estimator ρ m h (θ) we have B m = 0, so the simulation factor is simply

1 + n m K ε ( ρ,h)
.

It appears that for xed n, one should have a number of simulation data m greater than n. For instance, for some β > 1, if we have

m = n β or n (log(n)) β ,
we can make the simulation factor close to 1.

Remark 5.1. The term inf θ∈Θ (R Ψ (h, θ)) in Theorem 4.1 appears as the best (smaller) error one can make. This kind of error is commonly called approximation error or systematic error. It can be understood as the "distance" between the a priori knowledge one has with the observed phenomenon.

By Examples (2.2) and (3.2), we can write the risk bound in Theorem 4.1 in specic cases as follows.

Example 5.2. Risk bounds in specic cases.

-mean-contrast

E(Y ) -ρ h ( θ M ) 2 ≤ inf θ∈Θ (E(Y ) -ρ h (θ)) 2 + K ε ( ρ,M ) √ n 1 + n m (K ε ( ρ,h) + B m )
In practice, B m = 0.

log-contrast

KL(ρ h ( θ log ), f ) ≤ inf θ∈Θ (KL(ρ h (θ), f )) + K ε ( ρ,log) √ n 1 + n m (K ε ( ρ,h) + B m ) -L 2 -contrast ρ h ( θ L 2 ) -f 2 2 ≤ inf θ∈Θ ρ h (θ) -f 2 2 + K ε ( ρ,L 2 ) √ n 1 + n m (K ε ( ρ,h) + B m ) . The terms inf θ∈Θ (E(Y ) -ρ h (θ)) 2 , inf θ∈Θ (KL(ρ h (θ), f )) and inf θ∈Θ ρ h (θ) -f 2 2 are the ideal risks inf θ∈Θ (R Ψ (h, θ)) in dierent situations.
These examples show clearly that these terms represent a "distance" between the "target" and the "best" information available, see Remark 5.1. If these terms are supposed equal to zero, it means that we believe that for instance the density f belongs to the family of densities {ρ h (θ), θ ∈ Θ}. In this case we obtain for example (L 2 -contrast)

ρ h ( θ L 2 ) -f 2 2 ≤ K ε ( ρ,L 2 ) √ n 1 + n m (K ε ( ρ,h) + B m ) .
However, such a priori has to be made with precautions. We will show how we obtain the constants A Ψ in Table [START_REF] Billingsley | Convergence of probability measures[END_REF]. Let recall that Y ∈ [-M, M ].

-mean-contrast.

Let y ∈ Y, ρ 1 , ρ 2 ∈ F ⊂ Y. We have

(y -ρ 1 ) 2 -(y -ρ 2 ) 2 = |ρ 1 -ρ 2 | |2y -(ρ 1 -ρ 2 )| ≤ |ρ 1 -ρ 2 | 4 M .
log-contrast.

Let y ∈ Y, ρ 1 , ρ 2 ∈ F, with F some set of density functions.

Moreover, suppose that it exists some η > 0 such that

∀ ρ ∈ F ρ > η
By Taylor Lagrange formula, it exists some τ ∈ (ρ 1 (y), ρ 2 (y)) such that

|log (ρ 1 (y)) -log (ρ 2 (y))| = 1 τ |ρ 1 (y) -ρ 2 (y)| ≤ 1 η |ρ 1 (y) -ρ 2 (y)|
since ρ > η for all ρ ∈ F and τ > η.

Taking the expectation under the measure Q (with Lebesgue density f ) involves the quantity

E Y (|ρ 1 (Y ) -ρ 2 (Y )|)
in the right member. By Cauchy-Schwarz inequality

E Y (|ρ 1 (Y ) -ρ 2 (Y )|) ≤ ρ 1 -ρ 2 2 f 2 , so E Y |log (ρ 1 (Y )) -log (ρ 2 (Y ))| ≤ f 2 η ρ 1 -ρ 2 2 .
-L 2 -contrast.

Let y ∈ Y, ρ 1 , ρ 2 ∈ F, with F some set of density functions. Suppose that it exists some B > 0 such that

sup ρ∈F ρ 2 < B .
By triangular inequality

( ρ 1 2 2 -2 ρ 1 (y)) -( ρ 2 2 2 -2 ρ 2 (y)) ≤ ρ 1 2 2 -ρ 2 2 2 + 2 |ρ 2 (y) -ρ 2 (y)| ≤ ρ 1 -ρ 2 2 2 + 2 |ρ 2 (y) -ρ 2 (y)| .
Taking the expectation under Q and by Cauchy-Schwarz inequality (as before) yields

E Y ( ρ 1 2 2 -2 ρ 1 (Y )) -( ρ 2 2 2 -2 ρ 2 (Y )) ≤ ρ 1 -ρ 2 2 2 + 2 ρ 1 -ρ 2 2 f 2 ≤ ρ 1 -ρ 2 2 ( ρ 1 -ρ 2 2 + 2 f 2 ) ≤ 2 (B + f 2 ) ρ 1 -ρ 2 2

Constant b h (m)

When the plug-in estimator ρ m h (θ) is unbiased, the bias term b m h (θ) dened in ( 7) is zero for all θ ∈ Θ and m > 0, hence b h (m) = 0 too. We study the example of the kernel estimator (biased), i.e when the weight function ρ is a function of the form ρ(y

)(•) = K b ( • -y)
where

K b ( • -y) = 1 b K( • -y b )
for some kernel K() and some bandwidth b.

Consider that • F = • 2 , then for all θ ∈ Θ we have b m h (θ) = E X (K b ( • -h(X, θ))) -ρ h (θ) 2 = Y X (K b ( y -h(x, θ)) -ρ h (θ)) P x (dx)
g(y) = 1 m m j=1 1 b K y -ξ m b .
If the following assumptions are valid

• g 2 < +∞
• y K(y) dy = 0

• I = y 2 K(y) dy < +∞ ,
then it exists a constant C g such that for all b > 0

E ξ 1..m g -g 2 2 ≤ C g 1 m b + b 4 .
In particular, the bias term E ξ 1..m g -g 2 is bounded above by

I g 2 √ 3 b 2 .
In our context, take g = ρ h (θ) and suppose that the assumptions of this Theorem are satised, then

b m h (θ) ≤ I ρ h (θ) 2 √ 3 b 2 .
Moreover, if sup θ∈Θ ρ h (θ) 2 is nite, it justies the existence of b h (m) = sup θ∈Θ b m h (θ) .

Constants Kε

( ρ,Ψ) and Kε ( ρ,h)

We detail the arguments for computing the constants Kε ( ρ,Ψ) and Kε ( ρ,h) . Since these constants are tightness constants relative to some empirical processes (see the assumptions of Theorem 4.1), we will give arguments with a generic empirical process W p = √ p(W p -W ) indexed by a generic class of functions G . Now, the goal is to compute some constant K(ε) such that

P( W p G ≤ K(ε)) ≥ 1 -ε for small ε > 0 . ( 14
)
For this, we propose to use the work of T. Klein and E. Rio [START_REF] Klein | Concentration around the mean for maxima of empirical processes[END_REF], in particular Theorem 1.1, that deal with right hand side deviations of the empirical process. They show that for an empirical process W p indexed by a countable class of functions G with values in [-1, 1]

P sup g∈G W p (g) ≥ E(sup g∈G W p (g)) + t ≤ exp - t 2 2v + 3 x/ √ p , (15)
for all positive t and some constant v . They also give left hand side deviations. In our purpose, we don't really work with sup g∈G W p (g) but rather with sup g∈G |W p (g)| = W p G corresponding to a two-side control. Hence, according to the work of T. Klein and E. Rio [START_REF] Klein | Concentration around the mean for maxima of empirical processes[END_REF], it exists some function ϕ G : R + → [0, 1] decreasing to zero such that for all positive t

P ( W p G ≥ E( W p G ) + t) ≤ ϕ G (t) . ( 16
)
Another point is missing before we apply this result in our context, it is the fact that the result is valid for countable classes of functions, and so, we need to extend the Theorem 1.1 in [START_REF] Klein | Concentration around the mean for maxima of empirical processes[END_REF]. We prove the following proposition. Proposition 6.1. Let W p be an empirical process indexed by a class of functions G taking values in [-1, 1] and parameterized by a compact set C of R l , l ≥ 1 . Suppose that the application

λ ∈ C -→ g λ ∈ G ⊂ L 2 (17)
is continuous. Then, it exists a function ϕ G decreasing to zero (given by [START_REF] Klein | Concentration around the mean for maxima of empirical processes[END_REF]) such that for all t ≥ 0

P ( W p G ≥ E( W p G ) + t) ≤ ϕ G (t) . (18)
Proof. For simplicity, we prove the proposition with G = W ( ρ,Ψ) where We dene the sets Y s = {y s 1 , ..., y s is } for s ≥ 1 recursively, as follows:

W ( ρ,Ψ) = {y ∈ Y → Ψ( ρ(λ) , y) , λ ∈ Y}
• Y 1 = {-M, 0, M } .
• Assume that the set Y s = {y s 1 , ..., y s is } is construct with increasing elements, i.e y s 1 < ... < y s is . For j = 1, ..., i s -1, let

ỹs j = y s j + y s j+1 2 and Y s = {ỹ s j , i = 1, ..., i s-1 -1} . • Dene Y s+1 = Y s ∪ Y s
with increasing elements.

Remark 6.1. One can verify that

Card(Y s ) = 2 s + 1 .
Now, dene the classes of functions

W s ( ρ,Ψ) = {y ∈ Y → Ψ( ρ(λ) , y) , λ ∈ Y s }
and notice that for all s ≥ 1,

W s-1 ( ρ,Ψ) W s ( ρ,Ψ) W ( ρ,Ψ) . (19)
By this previous display and the fact that s≥1 Y s is dense in [-M, M ] and by the continuous assumption [START_REF] Santner | The design and analysis of computer experiments[END_REF] , we have

lim s→∞ W s ( ρ,Ψ) = s≥1 W s ( ρ,Ψ) = W ( ρ,Ψ) . (20)
The classes of functions W s ( ρ,Ψ) , s ≥ 1 are countable (2 s + 1 elements) with values in [-1, 1]. Finally, we apply the inequality [START_REF] Pollard | Empirical processes: theory and applications[END_REF] to the classes W s ( ρ,Ψ) , we get for all t ≥ 0 and s ≥ 1

P W p W s ( ρ,Ψ) ≥ E( W p W s ( ρ,Ψ) ) + t ≤ ϕ s (t) . (21)
We wish to prove that the left and right member of this last inequality converge when s → ∞. Write the left member as follows

P W p W s ( ρ,Ψ) ≥ E( W p W s ( ρ,Ψ) ) + t = E 1 Wp W s ( ρ,Ψ) ≥ E( Wp W s ( ρ,Ψ) )+t = E 1 Wp W s ( ρ,Ψ) -E( Wp W s ( ρ,Ψ) ) ≥ t . (22)
The inclusions [START_REF] Soize | Physical systems with random uncertainties: chaos representations with arbitrary probability measure[END_REF] yields

W p W s-1 ( ρ,Ψ) ≤ W p W s ( ρ,Ψ) ≤ W p W ( ρ,Ψ) ∀s ≥ 1 , so the sequence W p W s ( ρ,Ψ) s≥1
is increasing and bounded, thus it converges. By monotone convergence, we obtain that the sequence E W p W s ( ρ,Ψ) s≥1 converges too provided that

E( W p W ( ρ,Ψ) ) < ∞ . Thus, the sequence W p W s ( ρ,Ψ) -E W p W s ( ρ,Ψ) s≥1
converges too, and by dominated convergence the quantity (22) converges to the wanted limit

E 1 Wp W ( ρ,Ψ) -E( Wp W ( ρ,Ψ) ) ≥ t = P W p W ( ρ,Ψ) ≥ E( W p W ( ρ,Ψ) ) + t .
For the right member of ( 21), by similar arguments, it can be shown that ϕ s (t) → ϕ(t) = ϕ G (t) .

That concludes the proof.

Next, since the function

t → ϕ G (t) is decreasing from R + into [0, 1], then it exists a unique function κ G : [0, 1] → R + such that ∀ t ≥ 0 κ -1 G (t) = ϕ G (t) . (23)
Then, we can write [START_REF] Shorack | Empirical processes with applications to statistics[END_REF] as follows, for all ε ∈]0, 1[

P ( W p G ≥ E( W p G ) + κ G (ε)) ≤ ε or equivalently P ( W p G ≤ E( W p G ) + κ G (ε)) ≥ 1 -ε .
Thus, for a constant K(ε) that should satisfy [START_REF] Massart | Concentration inequalities and model selection: Ecole d'Eté de Probabilités de Saint-Flour XXXIII-2003[END_REF], i.e

P( W p G ≤ K(ε)) ≥ 1 -ε , one can take K(ε) equal to E( W p G ) + κ G (ε) .
But, the quantity E( W p G ) remains not tractable. We propose to bound it. Indeed, maximal inequalities allow to bound such quantities in terms of entropy integrals we will dene. Although these methods are known to be not sharp, the bounds we will obtain are of interest for our purpose. Before, let recall some useful denitions.

Let G be a class of functions and W some probability measure. The covering number in the right member is bounded by 16 M 2 / , so that we nally get

N [ ] ( , W ( ρ,Ψ) , L 2 (Q)) ≤ 16 M 2 .
Now, we compute the bracketing integral

J [ ] G 2,Q , W ( ρ,Ψ) , L 2 (Q) = G 2,Q 0 log N [ ] ( , W ( ρ,Ψ) , L 2 (Q)) d ≤ 4 M 2 0 log 16 M 2 d ,
and with the variable substitution u = 2 log(16 M 2 / ), this integral becomes

4 √ 2 M 2 +∞ log (16)
√ u e -u/2 du .

Moreover, since

+∞ 0

√ u e -u/2 du = √ 2 π, the bracketing integral is bounded by

J [ ] G 2,Q , W ( ρ,Ψ) , L 2 (Q) ≤ 8 √ π M 2 .
Finally, we obtain the following constant

Kε ( ρ,Ψ) = 8 a 1 √ π M 2 + κ 1 (ε) . (26) 7.2

Kε

( ρ,h) with the weight function ρ(y) = y

In this case, the class of functions P ( ρ,h) is

P ( ρ,h) = {x ∈ X → h(x, θ) , θ ∈ Θ} (X ⊂ R d ) .
We assumed in the introduction that the models x → h(x, θ), θ ∈ Θ are uniformly bounded by M , thus denote by P an envelop of P ( ρ,h) , take P = M . Moreover, let suppose that the models x → h(x, θ), θ ∈ Θ belong to the Hölder space H(X , α, L) (α, L > 0) dened as

H(X , α, L) = {g : X → R continuous, g α ≤ L}
where

g α = max |ν|≤ α sup x∈X |D ν g(x)| + max ν:|ν|= α sup x,x ∈X |D ν g(x) -D ν g(x )| x -x α-α
with α the largest integer smaller than α, and the dierential operator D ν is dened as, We aim at computing the entropy integral J [ ] P 2,Q , P ( ρ,h) , L 2 (Q) by integrating the entropy log N [ ] ( , P ( ρ,h) , L 2 (Q)). Corollary 2.7.2 in [START_REF] Van Der Vaart | Weak Convergence and Empirical Processes[END_REF] (p. 157) gives an entropy bound for the Hölder space H(X , α, 1):

for ν = (ν 1 , ..., ν d ) ∈ N d D ν = ∂ |ν| ∂ν ν 1 1 ...
log N [ ] ( , H(X , α, 1), L 2 (Q)) ≤ K 1 d/α ∀ > 0 , (27)
where K depends on α, diam(X ) and d. We supposed that P ( ρ,h) ⊂ H(X , α, L), and one can easily check that

H(X , α, L) = L • H(X , α, 1) . ( 28
)
where L • H(X , α, 1) = {L g : g ∈ H(X , α, 1)} .

Remark 7.1. If P ( ρ,h) ⊂ H(X , α, L), then necessarily L ≥ M . It comes from the fact that g α ≥ g ∞ for all α > 0.

Next, we will use the following lemma. Lemma 7.1.

N [ ] ( , H(X , α, L), L 2 (Q)) = N [ ] ( , L • H(X , α, 1), L 2 (Q)) = N [ ] ( /L, H(X , α, 1), L 2 (Q)) .
Proof. The rst equality is clear by (28). Let ([l i , u i ]) i=1...N be a set of -brackets covering H(X , α, 1). Then the brackets

([L l i , L u i ]) i=1...N cover L • H(X , α, 1) since for g ∈ H(X , α, 1) l ≤ g ≤ u =⇒ L l ≤ L g ≤ L u .
Finally, the brackets [L l i , L u i ] are of size L , and the result follows.

Using (27), Lemma 7.1 and the inequality

J [ ] P 2,Q , P ( ρ,h) , L 2 (Q) ≤ J [ ] ( P 2,Q , H(X , α, L), L 2 (Q)) , it holds for d < 2α J [ ] P 2,Q , P ( ρ,h) , L 2 (Q) ≤ √ K M 0 L d/2α d , hence J [ ] P 2,Q , P ( ρ,h) , L 2 (Q) ≤ M √ K L M d/2α 1 1 -d/2α
.

Finally, under the condition d < 2 α, we get the constant

Kε ( ρ,h) = a 2 M √ K L M d/2α 1 1 -d/2α + κ 2 ( ) .
Remark 7.2. The condition d < 2α above, means that the dimension of the random input X (equal to d) is limited by the "smoothness" of the models x → h(x, θ), θ ∈ Θ. The smoother the models are (i.e α large), the larger the dimension d can be.

Remark 7.3. The computation of the constants Kε ( ρ,Ψ) and Kε ( ρ,h) are dicult enough to obtain, as we saw. However, we adopt a nonasymptotic point of view and so such computations are crucial in order to give sense to the risk bounds. [START_REF] Huber | Robust estimation of a location parameter[END_REF] Proofs

In order to to prove the risk bound of Theorem (4.1), we need the following lemmas. We have (a.s.)

sup θ∈Θ |G n (Ψ ( ρ(h(X, θ))))| ≤ ||G n || W ( ρ,Ψ) ,
where W ( ρ,Ψ) is dened in [START_REF] Huber | Robust statistics[END_REF].

Proof. The key ingredient is re-parametrization.

Since for all x ∈ X and θ

∈ Θ, h(x, θ) ∈ Y, conditionally to X = x 0 sup θ∈Θ |G n (Ψ ( ρ(h(x 0 , θ))))| ≤ sup λ∈Y |G n (Ψ ( ρ(λ)))| = ||G n || W ( ρ,Ψ) .
The right member does not depend on x 0 , and the result follows.

Remark 8.1. The left member of the inequality in the lemma (8.1) depends on the model h, contrary to the right member. Indeed, this last term depends only on the weight function with the associated contrast, and on n.

Lemma 8.2. Consider the P x -empirical process K x m and let

• F = | • | or • r and dene c = 1 if ρ(y) is constant, ∀y ∈ Y , ( 2
M ) 1/r else .
We have

sup θ∈Θ ||K x m ρ(h(•, θ))|| F ≤ c ||K x m | P ( ρ,h) ,
where P ( ρ,h) is dened in [START_REF] Kleijnen | Design and analysis of simulation experiments[END_REF].

Proof. Let notice that the quantity

K x m ρ(h(•, θ)) = 1 √ m m j=1 [ ρ(h(X j , θ)) -E X ρ(h(X, θ))]
can be (up to a factor) either a sum of independent random real variables or a sum of independent random functions.

-If ρ(y) ∈ R for all y ∈ Y (we have a sum of random variables).

Taking

• F = | • | the absolute value norm, it comes directly that sup θ∈Θ ||K x m ρ(h(•, θ))|| F = sup θ∈Θ 1 √ m m j=1 [ ρ(h(X j , θ)) -E X ρ(h(X, θ))] = K x m P ( ρ,h)
Remark 8.2. In this case, ρ(y)(λ) = ρ(y) for all y and λ in Y.

-If, for all y ∈ Y, ρ(y) is a real valued function dened on Y.

Take • F = • r , r ≥ 1, the L r norm. By integration properties and the fact that

sup z≥0 z r = (sup z≥0 z) r ,
we have and the result follows.

sup θ∈Θ ||K x m ρ(h(•, θ))|| r = sup θ∈Θ 1 √ m m j=1 [ ρ(h(X j , θ)) -E X ρ(h(X, θ))] r = sup θ∈Θ   Y 1 √ m m j=1 [ ρ(h(X j , θ))(λ) -E X ρ(h(X, θ))(λ)] r dλ   1/r ≤ sup θ∈Θ   Y   sup λ∈Y 1 √ m m j=1 [ ρ(h(X j , θ))(λ) -E X ρ(h(X, θ))(λ)]   r dλ   1/r = sup θ∈Θ sup λ∈Y 1 √ m m j=1 [ ρ(h(X j , θ))(λ) -E X ρ(h(X, θ))(λ)] Y dλ 1/r = (2 M ) 1/r sup (θ,λ)∈Θ×Y 1 √ m m j=1 [ ρ(h(X j , θ))(λ) -E X ρ(h(X, θ))(λ)] .
Remark 8.3. In the case where the weight function is a kernel K b (• -•), the quantity

K x m ρ(h(•, θ)) = 1 √ m m j=1 [K b ( • -h(X j , θ)) -E X K b ( • -h(X, θ))]
is treated as a sum of independent random functions in the recent work of A. Goldenshluger and O. Lepski [START_REF] Goldenshluger | Uniform bounds for norms of sums of independent random functions[END_REF]. Here we have made the restrictive assumption that Y ⊂ [-M, M ]. A valuable challenge would be to extend our results to the unbounded case using [START_REF] Goldenshluger | Uniform bounds for norms of sums of independent random functions[END_REF]. +M n,m (h, θ ) Finally, the following bound holds for the procedure risk

≤ -M m (h, θ) -M (h, θ) - 1 √ n G n Ψ ρ m h ( θ) + M n,m (h, θ ) -M m (h, θ ) + M m (h, θ ) ≤ -M m (h, θ) -M (h, θ) - 1 √ n G n Ψ ρ m h ( θ) + 1 √ n G n Ψ (ρ m h (θ )) + M m (h, θ ) ≤ -M m (h, θ) -M (h, θ) + 1 √ n G n Ψ (ρ m h (θ )) -Ψ ρ m h ( θ) + M m (h, θ ) -M (h, θ ) + M (h, θ ) ≤ 1 √ n G n Ψ (ρ m h (
R Ψ (h, θ) ≤ inf θ∈Θ (R Ψ (h, θ)) + 2 √ n ||G n || W ( ρ,Ψ) + 2 A Ψ √ m c K x m P ( ρ,h) + √ m b h (m) .
Now, let notice that for any 3 events E 1 , E

1

  Institut de Mathématiques de Toulouse -EADS Innovation Works, 12 rue Pasteur, 92152 Suresnes 2 Université Paris Descartes, 45 rue des saints pères, 75006 Paris 3 Institut de Mathématiques de Toulouse, 118 route de Narbonne F-31062 Toulouse 4 EADS : European Aeronautic Defense and Space Company

Figure 1 :

 1 Figure 1: Example of model outputs with 2 dierent parameters.

Example 3 . 1 .

 31 Examples of weight functions.

  and a bandwidth b.

Figure 2 : 1 m

 21 Figure 2: Example of weight function in the case of the mean (top) and the case of the density (bottom).

Theorem 4 . 1 .

 41 Risk bound for Parameter Estimation. Under the Assumptions (4.1) and (3.1), suppose that the sequences of random variables ||G n || W ( ρ,Ψ) and ||K x m || P ( ρ,h) are tight. Denote by Kε ( ρ,Ψ) and Kε ( ρ,h) the associated constants, uniform (or decreasing) in n and m, respectively.

6 About the constants in Theorem 4. 1 6. 1

 611 Constant A Ψ

(

  in fact we consider W p = G p ) and take Y = [-M, M ]. Moreover, without loss of generality, suppose that the functions in W ( ρ,Ψ) take values in [-1, 1] .

/ 2 .

 2 An envelope function of the class G is a function G : y → G(y) such that |g(y)| ≤ G(y), for all y and g ∈ G. Denote byg 2,W = g 2 (y) W (dy)1The three following denitions are from[START_REF] Van Der Vaart | Weak Convergence and Empirical Processes[END_REF] (p. 83-85). Denition 6.1. L 2 (W ) Covering numbers and Entropy. The covering number N (ε, G, L 2 (W )) is the minimal number of balls {j , j -g 2,W < } of radius needed to cover the class G. The centers of the balls need not belong to G, but they should have nite norm. The entropy is the logarithm of the covering number. Denition 6.2. L 2 (W ) Bracketing numbers and Entropy with bracketing. Given two functions l, u, the bracket [l, u] is the set of all functions g with l ≤ g ≤ u. An -bracket is a bracket [l, u] with ||u -l|| 2,W < . The bracketing number N [ ] ( , G, L 2 (W )) is the minimum number of -brackets needed to cover the class of functions G.

∂ν ν d d ,

 d

8. 1

 1 Preliminary lemmas Lemma 8.1. Consider the random functions y → Ψ( ρ(h(X, θ)) , y) , θ ∈ Θ .

Finally

  h(X j , θ))(y) -E X ρ(h(X, θ))(y)] = ||K x m | P ( ρ,h)

8. 2

 2 Proof of Theorem (4.1)Proof. We denote by-M (h, θ) = R Ψ (h, θ) = E Y Ψ (ρ h (θ), Y ) -M n, (h, θ) = 1 n n i=1 Ψ (ρ h (θ), Y i ) -M ,m (h, θ) = E Y Ψ (ρ m h (θ), Y ) -M n,m (h, θ) m h (θ), Y i ) -G n Ψ (ρ m h (θ)) = √ n (M n,m (h, θ) -M m (h, θ)) (X j , θ)) and recall that θ = Argmin θ∈Θ M n,m (h, θ) and θ = Argmin θ∈Θ M (h, θ) . (29)We have,R Ψ (h, θ) = M (h, θ) -M m (h, θ) + M m (h, θ) -M n,m (h, θ) + M n,m (h, θ) = -M m (h, θ) -M (h, θ) -1 √ n G n Ψ ρ m h ( θ n,m ) + M n,m (h, θ) -M n,m (h, θ ) ≤0(29)

  θ )) -Ψ ρ m h ( θ) + M m (h, θ ) -M (h, θ ) -M m (h, θ) -M (h, θ) + M (h, θ ) ≤ inf θ∈Θ (R Ψ (h, θ)) + 2 √ n sup θ∈Θ |G n (Ψ (ρ m h (θ)))| + 2 sup θ∈Θ |M m (h, θ) -M (h, θ)| since M (h, θ ) = R Ψ (h, θ ) = inf θ∈Θ (R Ψ (h, θ)).Now, we want to bound the second and third terms in the right member of the last inequality.Second term. Since ρ m h ((X j , θ)) and ρ -→ Ψ(ρ, y) is convex by Assumption (4.1), we have the inequality for all y ∈ Y, h(X j , θ)) , y) .Then, by the linearity of the measure G n , it yieldsG n (Ψ (ρ m h (θ))) ≤ 1 m m j=1 G n Ψ ( ρ(h(X j , θ))) .(30)By Lemma 8.1 we have (a.s)sup θ∈Θ |G n (Ψ ( ρ(h(X j , θ))))| ≤ ||G n || W ( ρ,Ψ)whereW ( ρ,Ψ) = {Ψ( ρ(λ) , •) , λ ∈ Y} , then (a.s) sup θ∈Θ |G n (Ψ (ρ m h (θ)))| ≤ ||G n || W ( ρ,Ψ) .Third term. We have|M m (h, θ) -M (h, θ)| = |E Y (Ψ (ρ m h (θ), Y ) -Ψ (ρ h (θ), Y ))| ≤ E Y |Ψ (ρ m h (θ), Y ) -Ψ (ρ h (θ), Y )| .By Assumption (4.1)|Ψ (ρ m h (θ), Y ) -Ψ (ρ h (θ), Y )| ≤ L Ψ (Y ) ||ρ m h (θ) -ρ h (θ)|| F , then |M m (h, θ) -M (h, θ)| ≤ ||ρ m h (θ) -ρ h (θ)|| F E Y L Ψ (Y ) . (31) Let A Ψ = E Y L Ψ (Y ) .Moreover, the inequality (6) yields||ρ m h (θ) -ρ h (θ)|| F ≤ 1 m m j=1 [ ρ(h(X j , θ)) -E X ρ(h(X, θ))] F + b m h (θ) .(32)Equivalently, by considering the empirical process K x m = √ m(P x m -P x ), we obtain||ρ m h (θ) -ρ h (θ)|| F ≤ 1 √ m ||K x m ρ(h(•, θ))|| F + b m h (θ) (33) ≤ 1 √ m ||K x m ρ(h(•, θ))|| F + √ m b m h (θ) .(34)Taking the supremum over Θ and combining the Lemma (8.2) and the Assumption (3.1) givessup θ∈Θ ||ρ m h (θ) -ρ h (θ)|| F ≤ 1 √ m c K x m P ( ρ,h) + √ m b h (m) .Hence, in (31) we obtainsup θ∈Θ |M m (h, θ) -M (h, θ)| ≤ A Ψ √ m c K x m P ( ρ,h) + √ m b h (m) .

  [START_REF] Billingsley | Convergence of probability measures[END_REF]. The estimator θ depends on the model h, the number of experimental data n and the number of simulation data m . 2. The number of simulations m have to be thought greater than n (number of experimental data). It appears natural to think that experimental data are dicult to obtain whereas simulated data are more reachable.

Table 1 :

 1 Example of classes of functions and constant A Ψ (see section (6.1)).Next, we use the following notation: let P be some measure and G a class of real valued functions. We denote by

	P g := g(u)P (du)	g ∈ G
	and
	P G := sup

g∈G |P g| . With this notation, for a class of functions G Y , : Y → R we have

The entropy with bracketing is the logarithm of the bracketing number.

The bracketing numbers measure the "size", the complexity of a class of functions. We also dispose of a denition providing at which "speed" the classes grow. Denition 6.3. L 2 (W ) Bracketing integral.

The bracketing integral is dened as

Now we apply Corollary 19.35 of [START_REF] Van Der | Asymptotic statistics[END_REF] (p. 288), it holds that

where

Finally, setting

provides the claimed constant. In particular, we should take G = W ( ρ,Ψ) (W = Q) and G = P ( ρ,h) (W = P x ) in order to compute Kε ( ρ,Ψ) and Kε ( ρ,h) , respectively.

This class is uniformly bounded by 4 M 2 , we take the envelop function G = 4 M 2 . Then, we have

with F (y) = |2y + 2 M |, and by Theorem (2.7.11) in [START_REF] Van Der Vaart | Weak Convergence and Empirical Processes[END_REF] (p. 164) it holds that

Take the following events

and

where Kε ( ρ,Ψ) and Kε ( ρ,h) are such that

Using the inequality (35) with the fact that

, we obtain

But note that P(E 1 ) = 1, so

Equivalently, we have with probability at least 1 -2 ε

and

That concludes the proof.