William Wheeler

Ali Sari

Pascal Venet

Yann Bultel

Elie Rivière

Frédéric Meniere

on Materials for Energy Storage and Conversion Akyaka, Mugla

UMR5005, 69622 Villeurbanne, France 2 Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, Grenoble INP, LEPMI, F-38000 Grenoble, France 3 EVE System, 69440 Taluyers, France

At the end of the first use of batteries in industrial electric vehicles (EV), the remaining capacity is expected to be around 80-70% of its initial capacity. Therefore, autonomy becomes insufficient for the original application, although these batteries could be reused for less demanding applications. The management of recycling is the responsibility of the manufacturer, and its cost is substantial for lithium-ion and in particular for lithium iron phosphate (LFP) batteries. Indeed, second life applications for lithiumion batteries are an opportunity for the industry to postpone recycling costs, to improve lithium-ion batteries' economic value, and to reduce environmental impact.

Capacity and power limitations need to be predictable to meet required specifications throughout the second life. The purpose of this work is to identify the necessary parameters to sort battery modules in view of a second life application.

To this aim, it is necessary to assess capacity decrease, resistance increase, and other events in the whole lifetime of the battery in order to allow second-life-battery manufacturers to certify that any battery pack it sells can fulfil second life requirements. The sorting process is considered at the module level because cells are assembled with the Battery Management System (BMS) and this helps modularity in reassembly for a second use unlike a battery pack. Moreover, the sorting process at the module level is expected to be faster, cheaper and easier than at the single cell level.

This work presents an aging study that intends to continue cell aging over the end of the first life. In this context, changing behaviour or unexpected events may appear: there is no certainty that cell aging will follow the same trend during the second life. The test conditions are determined according to different possible uses of industrial EV: medium or fast charge and discharge current, or high depth of discharge. Those different parameters of use are suspected to imply a sudden shift in cell aging: acceleration of capacity fade or resistance increase. By the way, some conditions can improve battery aging. Loss of lithium inventory is commonly reported in the literature as the main aging cause in first life, but loss of active material may accelerate cell aging. The predictability of those aging process is required to set a sorting process for lithium-ion batteries in order to guarantee their use in second life. This work aims at shedding light on harmful phenomena in some normal battery use, and collect data related to their appearance. In a long-term research, this experiment data will establish a diagnosis model to estimate remaining cell capacity and a specific aging process that may lead to sudden capacity fade. Then, the prognosis of the remaining useful life will decide potential reuse in a second life application. Presentating author: William Wheeler, e-mail: william.wheeler@univ-lyon1.fr tel: +33761365228

  Tests are being carried out in two laboratories with four test benches on new industrial cells. Three LFP industrialelectric-vehicle 72Ah high capacity prismatic cells and twenty 18650 1.1Ah cells are cyclied on different test benches. All experiments are conducted under accelerated aging at a temperature of 50°C. An electric-vehicle-type discharged profile is calculated from the Worldwide harmonized Light vehicles Test Procedures (WLTP) standard. The cycle is used to approximate a typical operation of an electric vehicle during the aging study, with a C-rate peak of 2.7C in discharge and regenerative mode. The WLTP average discharge C-rate is C/3. Table1 summarizesaging conditions for all 23 cells. The variations depend on Depth of Discharge (DoD), charge type, Charge cut-off voltage, Charge C-rate, or discharge scenario. The change in aging rate is reported as being due to a change in aging mode. For each cell, a characterization procedure is carried out on fresh cells and every 100 cycles at 50°C. Then, the characterization procedure is repeated every 100 aging cycles to follow parameters evolution. The characterization process consists of two tests: a C/3 CCCV charge and discharge from 0 to 100% State of Charge (SoC), and a C/25 charge and discharge for pseudo-open circuit voltage measurement. The first test allow calculating the battery's remaining capacity, and the second one electrodes capacity. Born in 1995 in France, William Wheeler gets his engineering degree in mechatronics from Université de Technologie de Compiègne in 2018. Its experimental project on the topic of electric vehicles is a stepping stone to begin a Ph.D. thesis in the field of electric vehicles batteries. He currently works at the University Claude Bernard Lyon 1 and EVE System. He focuses his research on electric vehicles batteries, aging study and reuse in a second life.

Table 1 .

 1 Aging conditions for each cell.

	Cell reference	DoD (%)	Charge type	Charge cut-off voltage	Charge C-rate	Discharge condition
	prismatic cell 1	70	CC	3.65V	C/3	WLTP
	prismatic cell 2	70 CC-CV 3.65V	C/3	WLTP
	prismatic cell 3	70 CC-CV 3.55V	C/3	WLTP
	18650 cells 4, 5	70	CC	3.65V	C/3	WLTP
	18650 cells 6, 7, 8	100	CC	3.65V	C/3	WLTP
	18650 cells 9, 10, 11	70 CC-CV 3.55V	C/3	WLTP
	18650 cells 12, 13, 14 70 CC-CV 3.65V	2C	WLTP
	18650 cells 15, 16, 17 70 CC-CV 3.55V	2C	WLTP
	18650 cells 18, 19, 20 70 CC-CV 3.65V	2C	CC (2C)


Michel Broniatowski

Amor Keziou

PARAMETRIC ESTIMATION AND TESTS THROUGH DIVERGENCES AND DUALITY TECHNIQUE

Keywords: Parametric estimation, Parametric test, Maximum likelihood, Mixture, Boundary valued parameter, Power function, Duality, φ-divergence

We introduce estimation and test procedures through divergence optimization for discrete or continuous parametric models. This approach is based on a new dual representation for divergences. We treat point estimation and tests for simple and composite hypotheses, extending maximum likelihood technique. An other view at the maximum likelihood approach, for estimation and test, is given. We prove existence and consistency of the proposed estimates. The limit laws of the estimates and test statistics (including the generalized likelihood ratio one) are given both under the null and the alternative hypotheses, and approximation of the power functions is deduced. A new procedure of construction of confidence regions, when the parameter may be a boundary value of the parameter space, is proposed. Also, a solution to the irregularity problem of the generalized likelihood ratio test pertaining to the number of components in a mixture is given, and a new test is proposed, based on χ 2 -divergence on signed finite measures and duality technique.

 MSC (2000)Classification: 62F03; 62F10; 62F30.

Introduction and notation

Let (X , B) be a measurable space and P be a given probability measure (p.m.) on (X , B).

Denote M the real vector space of all signed finite measures on (X , B) and M(P ) the vector subspace of all signed finite measures absolutely continuous (a.c.) with respect to (w.r.t.) P . Denote also M 1 the set of all p.m.'s on (X , B) and M 1 (P ) the subset of all p.m.'s a.c. w.r.t. P . Let φ be a proper 1 closed 2 convex function from ] -∞, +∞[ to [0, +∞] with φ(1) = 0 and such that its domain domφ := {x ∈ R such that φ(x) < ∞} is an interval with endpoints a φ < 1 < b φ (which may be finite or infinite). For any signed Date: December 22, 2007. 1 We say a function is proper if its domain is non void. 2 The closedness of φ means that if a φ or b φ are finite numbers then φ(x) tends to φ(a φ ) or φ(b φ ) when

x ↓ a φ or x ↑ b φ , respectively. 1 finite measure Q in M(P ), the φ-divergence between Q and P is defined by D φ (Q, P ) := X φ dQ dP (x) dP (x).

(1.1)

When Q is not a.c. w.r.t. P , we set D φ (Q, P ) = +∞. The φ-divergences were introduced by [START_REF] Csiszár | Eine informationstheoretische Ungleichung und ihre Anwendung auf den Beweis der Ergodizität von Markoffschen Ketten[END_REF] as "f -divergences". For all p.m. P , the mappings Q ∈ M → D φ (Q, P ) are convex and take nonnegative values. When Q = P then D φ (Q, P ) = 0. Furthermore, if the function x → φ(x) is strictly convex on a neighborhood of x = 1, then the following fundamental property holds D φ (Q, P ) = 0 if and only if Q = P.

(1.2)

All these properties are presented in [START_REF] Csiszár | Eine informationstheoretische Ungleichung und ihre Anwendung auf den Beweis der Ergodizität von Markoffschen Ketten[END_REF]Csiszár ( , 1967a,b) ,b) and [START_REF] Liese | Convex statistical distances[END_REF] chapter 1, for φ-divergences defined on the set of all p.m.'s M 1 . When the φ-divergences are defined on M, then the same properties hold. Let us conclude these few remarks quoting that in general D φ (Q, P ) and D φ (P, Q) are not equal. Hence, φ-divergences usually are not distances, but they merely measure some difference between two measures.

Of course a main feature of divergences between distributions of random variables X and Y is the invariance property with respect to common smooth change of variables.

1.1. Examples of φ-divergences. When defined on M 1 , the Kullback-Leibler (KL), modified Kullback-Leibler (KL m ), χ 2 , modified χ 2 (χ 2 m ), Hellinger (H), and L 1 divergences are respectively associated to the convex functions φ(x) = x log xx + 1, φ(x) = log x + x -1, φ(x) = 1 2 (x -1) 2 , φ(x) = 1 2 (x -1) 2 /x, φ(x) = 2( √ x -1) 2 and φ(x) = |x -1|. All these divergences except the L 1 one, belong to the class of the so called "power divergences" introduced in Cressie and Read (1984) (see also [START_REF] Liese | Convex statistical distances[END_REF] chapter 2). They are defined through the class of convex functions

x ∈]0, +∞[ → φ γ (x) := x γ -γx + γ -1 γ(γ -1) (1.3) if γ ∈ R \ {0
, 1}, φ 0 (x) :=log x + x -1 and φ 1 (x) := x log xx + 1. (For all γ ∈ R, we define φ γ (0) := lim x↓0 φ γ (x)). So, the KL-divergence is associated to φ 1 , the KL m to φ 0 , the χ 2 to φ 2 , the χ 2 m to φ -1 and the Hellinger distance to φ 1/2 .

We extend the definition of the power divergences functions Q ∈ M 1 → D φγ (Q, P ) onto the whole vector space of all signed finite measures M via the extension of the definition of the convex functions φ γ : For all γ ∈ R such that the function x → φ γ (x) is not defined on ] -∞, 0[ or defined but not convex on whole R, set .4) Note that for the χ 2 -divergence, the corresponding φ function φ 2 (x) := 1 2 (x-1) 2 is defined and convex on whole R.

x ∈] -∞, +∞[ → φ γ (x) if x ∈ [0, +∞[, +∞ if x ∈] -∞, 0[. ( 1
In this paper, we are interested in estimation and test using φ-divergences. An i.i.d. sample X 1 , . . . , X n with common unknown distribution P is observed and some p.m. Q is given.

We aim to estimate D φ (Q, P ) and, more generally, inf Q∈Ω D φ (Q, P ) where Ω is some set of measures, as well as the measure Q * achieving the infimum on Ω. In the parametric context, these problems can be well defined and lead to new results in estimation and tests, extending classical notions.

Statistical examples and motivations.

1.2.1. Tests of fit. Let Q 0 and P be two p.m.'s with same support S. Introduce a finite partition A 1 , . . . , A k of S (when S is finite this partition is the support of Q 0 ). The quantization method consists in approximating D φ (Q 0 , P ) by k j=1 φ Q 0 (A j ) P (A j ) P (A j ) which is estimated by

D φ (Q 0 , P ) = k j=1 φ Q 0 (A j ) P n (A j ) P n (A j ),
where P n is the empirical measure associated to the data. In this vein, goodness of fit tests have been proposed by Zografos et al. (1990) for fixed number of classes, and by [START_REF] Menéndez | Asymptotic distributions of φ-divergences of hypothetical and observed frequencies on refined partitions[END_REF] and [START_REF] Györfi | Asymptotic distributions for goodness-of-fit statistics in a sequence of multinomial models[END_REF] when the number of classes depends on the sample size. We refer to [START_REF] Pardo | Statistical inference based on divergence measures, volume 185 of Statistics: Textbooks and Monographs[END_REF] which treats these problems extensively and contains many more references.

1.2.2. Parametric estimation and tests. Let {P θ ; θ ∈ Θ} be some parametric model with Θ a set in R d . On the basis of an i.i.d. sample X 1 , . . . , X n with distribution P θ T , we want to estimate θ T , the unknown true value of the parameter and perform statistical tests on the parameter using φ-divergences. When all p.m.'s P θ share the same finite support S, [START_REF] Liese | Convex statistical distances[END_REF], [START_REF] Lindsay | Efficiency versus robustness: the case for minimum Hellinger distance and related methods[END_REF] and [START_REF] Morales | Asymptotic divergence of estimates of discrete distributions[END_REF] introduced the socalled "Minimum φ-divergences estimates" (MφDE's) (Minimum Disparity Estimators in [START_REF] Lindsay | Efficiency versus robustness: the case for minimum Hellinger distance and related methods[END_REF]) of the parameter θ T , defined by θ φ := arg inf θ∈Θ D φ (P θ , P n ).

(1.5)

Various parametric tests can be performed based on the previous estimates of φ-divergences;

see [START_REF] Lindsay | Efficiency versus robustness: the case for minimum Hellinger distance and related methods[END_REF] and [START_REF] Morales | Asymptotic divergence of estimates of discrete distributions[END_REF]. The class of estimates (1.5) contains the maximum likelihood estimate (MLE). Indeed, when φ(x) = φ 0 (x) =log x + x -1, we obtain θ KLm := arg inf θ∈Θ KL m (P θ , P n ) = arg inf θ∈Θ j∈S

log(P θ (j))P n (j) = M LE.

(1.6)

The MφDE's (1.5) are motivated by the fact that a suitable choice of the divergence may lead to an estimate more robust than the ML one (see e.g. [START_REF] Lindsay | Efficiency versus robustness: the case for minimum Hellinger distance and related methods[END_REF], [START_REF] Basu | Minimum disparity estimation for continuous models: efficiency, distributions and robustness[END_REF] and [START_REF] Jiménez | On robustness and efficiency of minimum divergence estimators[END_REF]).

When interested in testing hypotheses

H 0 : θ T = θ 0 against alternatives H 1 : θ T = θ 0 ,
where θ 0 is a given value, we can use the statistic D φ (P θ 0 , P n ), the plug-in estimate of the φ-divergence between P θ 0 and P θ T , rejecting H 0 for large values of the statistic; see e.g. [START_REF] Cressie | Multinomial goodness-of-fit tests[END_REF]. In the case when φ(x) =log x + x -1, the corresponding test based on KL m (P θ 0 , P n ) does not coincide with the generalized likelihood ratio one, which defined through the generalized likelihood ratio (GLR) λ n := 2 log

sup θ∈Θ n i=1 p θ (X i ) n i=1 p θ 0 (X i )
. The new estimate KL m (P θ 0 , P θ T ) of KL m (P θ 0 , P θ T ), which is proposed in this paper, leads to the generalized likelihood ratio test; see remark 3.7 below.

When the support S is continuous, the plug-in estimates (1.5) are not well defined; [START_REF] Basu | Minimum disparity estimation for continuous models: efficiency, distributions and robustness[END_REF] investigate the so-called "minimum disparity estimators" (MDE's)

for continuous models, through some common kernel smoothing method of P n and P θ .

When φ(x) =log x + x -1, this estimate clearly, due to smoothing, does not coincide generally with the ML one. Also, the test based on the associated estimator of the KL m (P θ 0 , P θ T ) is different from the generalized likelihood ratio one. Further, their estimates poses the problem of the choice of the kernel and the window. For Hellinger distance, see [START_REF] Beran | Minimum Hellinger distance estimates for parametric models[END_REF]. For nonparametric goodness-of-fit test, [START_REF] Berlinet | About the asymptotic accuracy of Barron density estimates[END_REF][START_REF] Berlinet | How to get central limit theorems for global errors of estimates[END_REF] proposed a test based on the estimation of the KL m -divergence using the smoothed kernel estimate of the density. The extension of their results to other divergences remains an open problem; see [START_REF] Berlinet | How to get central limit theorems for global errors of estimates[END_REF], [START_REF] Györfi | Distribution estimates consistent in χ 2 -divergence[END_REF][START_REF] Berlinet | About the asymptotic accuracy of Barron density estimates[END_REF]. All those tests are stated for simple null hypotheses; the case of composite null hypotheses seems difficult to handle by the above technique. In the present paper, we treat this problem in the parametric setting.

When the support S is discrete infinite or continuous, then the plug-in estimate D φ (P θ , P n ) usually takes infinite value when no use is done of some partition-based approximation.

In [START_REF] Broniatowski | Estimation of the Kullback-Leibler divergence[END_REF], a new estimation procedure is proposed in order to estimate the KL-divergence between some set of p.m.'s Ω and some p.m. P , without making use of any partitioning nor smoothing, but merely making use of the well known "dual" representation of the KL-divergence as the Fenchel-Legendre transform of the moment generating function. Extending the paper by [START_REF] Broniatowski | Estimation of the Kullback-Leibler divergence[END_REF], we will use the new dual representations of φ-divergences (see [START_REF] Broniatowski | Estimation of the Kullback-Leibler divergence[END_REF]Keziou (2006) theorem 4.4 and[START_REF] Keziou | Dual representation of φ-divergences and applications[END_REF] theorem 2.1) to define the minimum φ-divergence estimates in both discrete and continuous parametric models. These representations are the starting point for the definition of estimates of the parameter θ T , which we will call "minimum dual φ-divergence estimates" (MDφDE's). They are defined in parametric models {P θ ; θ ∈ Θ}, where the p.m.'s P θ do not necessarily have finite support; it can be discrete or continuous, bounded or not. Also the same representations will be applied in order to estimate D φ (P θ 0 , P θ T )

and inf θ∈Θ 0 D φ (P θ , P θ T ) where θ 0 is a given value in Θ and Θ 0 is a given subset of Θ, which leads to various simple and composite tests pertaining to θ T , the true unknown value of the parameter. When φ(x) =log x+x-1, the MDφD estimate coincides with the maximum likelihood one (see remark 3.2 below); since our approach includes also test procedures, it will be seen that with this peculiar choice for the function φ, we recover the classical likelihood ratio test for simple hypotheses and for composite hypotheses (see remark 3.7 and remark 3.10 below). A similar approach has been proposed by [START_REF] Liese | On divergences and informations in statistics and information theory[END_REF];

see their formula (118).

In any case, an exhaustive study of MφDE's seems necessary, in a way that would include both the discrete and the continuous support cases. This is precisely the main scope of this paper.

The remainder of this paper is organized as follows. In section 2, we recall the dual representations of φ-divergences obtained by [START_REF] Broniatowski | Minimization of φ-divergences on sets of signed measures[END_REF]) theorem 4.4, Broniatowski and Keziou (2004) theorem 2.4 and Keziou (2003) theorem 2.1. Section 3 presents, through the dual representation of φ-divergences, various estimates and tests in the parametric framework and deals with their asymptotic properties both under the null and the alternative hypotheses. The existence and consistency of the proposed estimates are proved using similar arguments as developed in [START_REF] Qin | Empirical likelihood and general estimating equations[END_REF] lemma 1. We use the limit laws of the proposed test statistics, in a similar way to [START_REF] Morales | Some approximations to power functions of φdivergences tests in paramtric models[END_REF], to give an approximation to the power functions of the tests (including the GLR one). Observe that the power functions of the likelihood ratio type tests are not generally known; one of our contributions is to provide explicit power functions in the general case for simple or composite hypotheses. As a by-product, we obtain the minimal sample size which ensures a given power, for quite general simple or composite hypotheses. In section 4, we give a solution to the irregularity problem of the GLR test of the number of components in a mixture; we propose a new test based on the χ 2 -divergence on signed finite measures, and a new procedure of construction of confidence regions for the parameter in the case where θ T may be a boundary value of the parameter space Θ. All proofs are in the Appendix. We sometimes write P f for f dP for any measure P and any function f , when defined.

Fenchel Duality for φ-divergences

In this section, we recall a version of the dual representations of φ-divergences obtained in [START_REF] Broniatowski | Minimization of φ-divergences on sets of signed measures[END_REF], using Fenchel duality technique. First, we give some notations and some results about the conjugate (or Fenchel-Legendre transform) of real convex functions; see e.g. [START_REF] Rockafellar | Asymptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditions[END_REF] The strict convexity of φ is equivalent to the condition that its conjugate φ * is essentially smooth, i.e., differentiable with

lim t↓a φ * φ * ′ (t) = -∞ if a φ * > -∞, lim t↑b φ * φ * ′ (t) = +∞ if b φ * < +∞. ( 2
Imφ ′ to [a φ , b φ ]. If φ is differentiable, then for all x ∈]a φ , b φ [, φ * φ ′ (x) = xφ ′ (x) -φ (x) . (2.5)
If additionally φ is strictly convex, then for all t ∈ Imφ ′ we have φ * (t) = tφ ′ -1 (t)φ φ ′ -1 (t) and φ * ′ (t) = φ ′ -1 (t).

(2.6)

On the other hand, if φ is essentially smooth, then the interior of the domain of φ * coincides with that of Imφ ′ , i.e., (a φ * , b φ * ) = (φ ′ (a φ ), φ ′ (b φ )).

Let F be some class of B-measurable real valued functions f defined on X , and denote M F , the real vector subspace of M, defined by

M F := Q ∈ M such that |f | d|Q| < ∞, for all f ∈ F .
In the following theorem, we recall a version of the dual representations of φ-divergences obtained by [START_REF] Broniatowski | Minimization of φ-divergences on sets of signed measures[END_REF] (for the proof, see [START_REF] Broniatowski | Minimization of φ-divergences on sets of signed measures[END_REF] theorem 4.4).

Theorem 2.1. Assume that φ is differentiable. Then, for all Q ∈ M F such that D φ (Q, P ) is finite and φ ′ dQ dP belongs to F, the φ-divergence D φ (Q, P ) admits the dual representation

D φ (Q, P ) = sup f ∈F f dQ -φ * (f ) dP , (2.7)
and the function f := φ ′ dQ dP is a dual optimal solution4 . Furthermore, if φ is essentially smooth5 , then f := φ ′ (dQ/dP ) is the unique dual optimal solution (P -a.e.).

Parametric estimation and tests through minimum φ-divergence approach and duality technique

We consider an identifiable parametric model {P θ ; θ ∈ Θ} defined on some measurable space (X , B) and Θ is some set in R d , not necessarily an open set. For simplicity, we write D φ (θ, α) instead of D φ (P θ , P α ). We assume that for any θ in Θ, P θ has density p θ with respect to some dominating σ-finite measure λ, which can be either with countable support or not. Assume further that the support S of the p.m. P θ does not depend upon θ. On the basis of an i.i.d. sample X 1 , ..., X n with distribution P θ T , we intend to estimate θ T , the true unknown value of the parameter, which is assumed to be an interior point of the parameter space Θ. We will consider only strictly convex functions φ which are essentially smooth. We will use the following assumption

φ ′ p θ (x) p α (x) dP θ (x) < ∞. (3.1)
Note that if the function φ satisfies there exists 0 < δ < 1 such that for all c in [1δ, 1 + δ],

we can find numbers c 1 , c 2 , c 3 such that

φ(cx) ≤ c 1 φ(x) + c 2 |x| + c 3 , for all real x, (3.2)
then the assumption (3.1) is satisfied whenever D φ (θ, α) < ∞; see e.g. [START_REF] Broniatowski | Minimization of φ-divergences on sets of signed measures[END_REF] lemma 3.2. Also the real convex functions φ γ (1.4), associated to the class of power divergences, all satisfy the condition (3.2), including all standard divergences.

For a given θ ∈ Θ, consider the class of functions

F = F θ := x → φ ′ p θ (x) p α (x) ; α ∈ Θ . (3.3)
By application of Theorem 2.1 above, when assumption (3.1) holds for any α ∈ Θ, we obtain

D φ (θ, θ T ) = sup f ∈F θ f dP θ -φ * (f ) dP θ T ,
which, by (2.5), can be written as

D φ (θ, θ T ) = sup α∈Θ φ ′ p θ p α dP θ - p θ p α φ ′ p θ p α -φ p θ p α dP θ T . (3.4)
Furthermore, the supremum in this display is unique and it is achieved at α = θ T independently upon the value of θ. Hence, it is reasonable to estimate D φ (θ, θ T ) := φ(p θ /p θ T ) dP θ T , the φ-divergence between P θ and P θ T , by

D φ (θ, θ T ) := sup α∈Θ φ ′ p θ p α dP θ - p θ p α φ ′ p θ p α -φ p θ p α dP n , (3.5)
in which we have replaced P θ T by its estimate P n , the empirical measure associated to the data.

For a given θ ∈ Θ, since the supremum in (3.4) is unique and it is achieved at

α = θ T , define the following class of M-estimates of θ T α φ (θ) := arg sup α∈Θ φ ′ p θ p α dP θ - p θ p α φ ′ p θ p α -φ p θ p α dP n (3.6)
which we call "dual φ-divergence estimates" (DφDE's); (in the sequel, we sometimes write

α instead of α φ (θ)). Further, we have inf θ∈Θ D φ (θ, θ T ) = D φ (θ T , θ T ) = 0.
The infimum in this display is unique and it is achieved at θ = θ T . It follows that a natural definition of minimum φ-divergence estimates of θ T , which we will call "minimum dual φ-divergence estimates" (MDφDE's), is

θ φ := arg inf θ∈Θ sup α∈Θ φ ′ p θ p α dP θ - p θ p α φ ′ p θ p α -φ p θ p α dP n . (3.7)
In order to simplify formulas (3.5), (3.6) and (3.7), define the functions (3.13) Formula (3.4) can be written then as

g(θ, α) : x → g(θ, α, x) := p θ (x) p α (x) φ ′ p θ (x) p α (x) -φ p θ (x) p α (x) , (3.8) f (θ, α) : x → f (θ, α, x) := φ ′ p θ (x) p α (x) (3.9) and h(θ, α) : x → h(θ, α, x) := P θ f (θ, α) -g(θ, α, x). ( 3
D φ (θ, θ T ) = sup α∈Θ P θ T h(θ, α). (3.14)
If the supremum in (3.12) is not unique, we define the estimate α φ (θ) as any value of α ∈ Θ that maximizes the function α ∈ θ → P n h(θ, α). Also, if the infimum in (3.13) is not unique, the estimate θ φ is defined as any value of θ ∈ Θ that minimizes the function θ → sup α∈Θ P n h(θ, α). Conditions assuring the existences of the above estimates are given in section 3.1 and 3.2 below.

Remark 3.1. For the L 1 distance, i.e. when φ(x) = |x -1|, formula (3.4) does not apply since the corresponding φ function is not differentiable. However, using the general dual representation of divergences given in [START_REF] Broniatowski | Minimization of φ-divergences on sets of signed measures[END_REF] theorem 4.1, we can obtain an explicit formula for L 1 distance avoiding the differentiability assumption. A methodology on estimation and testing in L 1 distance has been proposed by [START_REF] Devroye | Combinatorial methods in density estimation[END_REF], and its consequences for composite hypothesis testing and for model selection based density estimates for nested classes of densities are presented in [START_REF] Devroye | A note on robust hypothesis testing[END_REF] and [START_REF] Biau | Density estimation by the penalized combinatorial method[END_REF].

Remark 3.2. (An other view at the ML estimate). The maximum likelihood estimate belongs to both classes of estimates (3.12) and (3.13). Indeed, it is obtained when φ So, the MLE can be seen as the estimate of θ T that minimizes the estimate of the KL mdivergence between the parametric model {P θ ; θ ∈ Θ} and the p.m. P θ T .

(x) = -log x + x -1, that
3.1. The asymptotic properties of the DφDE's α φ (θ) and D φ (θ, θ T ) for a given θ in Θ. This section deals with the asymptotic properties of the estimates (3.11) and (3.12). We will use similar arguments as developed in van der Vaart (1998) section 5. (c.3) for any positive ǫ, there exists some positive η such that for all α ∈ Θ satisfying αθ T > ǫ we have

P θ T h(θ, α) < P θ T h(θ, θ T ) -η. Remark 3.3. Condition (c.1) is fulfilled for example if the function α ∈ Θ → P n h(θ, α) is continuous and Θ is compact. Condition (c.2) is satisfied if {x → h(θ, α, x); α ∈ Θ} is a Glivenko-Cantelli class of functions. Condition (c.3) means that the maximizer α = θ T of the function α → P θ T h(θ, α
) is well-separated. This condition holds, for example, when the function α ∈ Θ → P θ T h(θ, α) is strictly concave and Θ is convex, which is the case for the following two examples:

Example 3.1. Consider the case φ(x) =log x + x -1 and the normal model

{N (α, 1); α ∈ Θ = R} .
Hence, we obtain

P θ T h(θ, α) = 1 2 (θ -θ T ) 2 - 1 2 (α -θ T ) 2 . (3.15)
We see that condition (c.3) is satisfied; we can choose η = ǫ 2 2 .

Example 3.2. Consider the case φ(x) =log x + x -1 and the exponential model

p α (x) = α exp(-αx); α ∈ Θ = R * + .
Hence, we obtain

P θ T h(θ, α) = -log θ + θ θ T + log α - α θ T , (3.16)
which is strictly concave (in α). Hence, condition (c.3) is satisfied.

Proposition 3.1.

(1) Under assumption (c.1-2), the estimate D φ (θ, θ T ) converges a.s. (resp. in probability) to D φ (θ, θ T ).

(2) Assume that the assumptions (c.1-2-3) hold. Then the estimate α φ (θ) converges in probability to θ T .

3.1.2. Asymptotic Normality. Assume that θ T is an interior point of Θ, the convex function φ has continuous derivatives up to 4th order, and the density p α (x) has continuous partial derivatives up to 3th order (for all x λa.e). Denote I θ T the Fisher information matrix

I θ T := p ′ θ T p ′ θ T T p θ T dλ.
In the following theorem, we give the limit laws of the estimates α φ (θ) and D φ (θ, θ T ). We will use the following assumptions.

(A.0) The estimate α φ (θ) exists and is consistent;

(A.1) There exists a neighborhood N (θ T ) of θ T such that the first and second order partial derivatives (w.r.t α) of f (θ, α, x)p θ (x) are dominated on N (θ T ) by some λ-integrable functions. The third order partial derivatives (w.r.t α) of h(θ, α, x)

are dominated on N (θ T ) by some P θ T -integrable functions;

(A.

2) The integrals P θ T (∂/∂α)h(θ, θ T ) 2 and P θ T (∂ 2 /∂α 2 )h(θ, θ T ) are finite, and

the matrix P θ T (∂ 2 /∂α 2 )h(θ, θ T ) is non singular; (A.3) The integral P θ T h(θ, θ T ) 2 is finite.
Theorem 3.2. Assume that assumptions (A.0-1-2) hold. Then, we have

(a)
√ n ( α φ (θ)θ T ) converges in distribution to a centered multivariate normal random variable with covariance matrix

V φ (θ, θ T ) = S -1 M S -1 (3.17) with S := -P θ T (∂ 2 /∂α 2 )h(θ, θ T ) and M := P θ T (∂/∂α)h(θ, θ T )(∂/∂α) T h(θ, θ T ). If θ T = θ, then V φ (θ, θ T ) = V (θ T ) = I -1 θ T . (b) If θ T = θ, then the statistic 2n φ ′′ (1) D φ (θ, θ T ) converges in distribution to a χ 2 random variable with d degrees of freedom. (c) If additionally assumption (A.3) holds, then when θ = θ T , we have √ n D φ (θ, θ T ) -D φ (θ, θ T ) converges in distribution to a centered normal random variable with variance σ 2 φ (θ, θ T ) = P θ T h(θ, θ T ) 2 -(P θ T h(θ, θ T )) 2 . (3.18)
Remark 3.4. Our first result (proposition 3.1 above) provides a general solution for the consistency of the global maximum (3.12) under strong but usual conditions, also difficult to be checked; see van der Vaart (1998) chapter 5. Moreover, in practice, the optimization in (3.12) is handled through gradient descent algorithms, depending on some initial guess α 0 ∈ Θ, which may provide a local maximum (not necessarily global) of

P n h(θ, .
). Hence, it is desirable to prove that in a "neighborhood" of θ T there exits a maximum of P n h(θ, .) which indeed converges to θ T ; this is the scope of theorem 3.3, in the following subsection, which states that for some "good" α 0 (near θ T ) the algorithm provides a consistent estimate. It is well known that, in various classical models, the global maximizer of the likelihood function may not exist or be inconsistent. Typical examples are provided in mixture models. Consider the Beta-mixture model given in [START_REF] Ferguson | An inconsistent maximum likelihood estimate[END_REF] section 3

p θ (x) = θg(x|1, 1) + (1 -θ)g(x|γ(θ), β(θ))
,

where Θ = [1/2, 1], g(x|γ(θ), β(θ))
is the Be(γ, β)-density and γ(θ) = θδ(θ) and β(θ) =

(1θ)δ(θ) with δ(θ) → +∞ sufficiently fast as θ → 1. The ML estimate converges to 1 (a.s.) whatever the value of θ T in Θ; see [START_REF] Ferguson | An inconsistent maximum likelihood estimate[END_REF] section 3 for the proof. However, if we take for example θ T = 3/4, theorem 3.3 hereafter proves the existence and consistency of a sequence of local maximizers under weak assumptions which hold for this example.

Other motivations for the results of theorem 3.3 are given in remark 3.5 below.

3.1.3. Existence, consistency and limit laws of a sequence of local maxima. We use similar arguments as developed in [START_REF] Qin | Empirical likelihood and general estimating equations[END_REF] lemma 1. Assume that θ T is an interior point of Θ, the convex function φ has continuous derivatives up to 4th order, and the density p α (x) has continuous partial derivatives up to 3th order (for all x λa.e). In the following theorem, we state the existence and the consistency of a sequence of local maxima α φ (θ) and D φ (θ, θ T ). We give also their limit laws.

Theorem 3.3. Assume that assumptions (A.1) and (A.2) hold. Then, we have

(a) Let B(θ T , n -1/3 ) := α ∈ Θ; α -θ T ≤ n -1/3
. Then, as n → ∞, with probability one, the function α → P n h(θ, α) attains its maximum value at some point α φ (θ) in the interior of the ball B, and satisfies

P n (∂/∂α)h(θ, α φ (θ)) = 0. (b) √ n ( α φ (θ) -θ T )
converges in distribution to a centered multivariate normal random variable with covariance matrix

V φ (θ, θ T ) = S -1 M S -1 . (3.19) (c) If θ T = θ, then the statistic 2n φ ′′ (1) D φ (θ, θ T ) converges in distribution to a χ 2 random variable with d degrees of freedom. (d) If additionally assumption (A.3) holds, then when θ = θ T , we have √ n D φ (θ, θ T ) -D φ (θ, θ T ) converges in distribution to a centered normal random
variable with variance σ 2 φ (θ, θ T ).

Remark 3.5. The results of this theorem are motivated by the following statements -The estimates α φ (θ) can be calculated if the statistician disposes of some preknowledge of the true unknown parameter θ T .

-The hypotheses are satisfied for a large class of parametric models for which the support does not depend upon θ, such normal, log normal, exponential, Gamma, Beta, Weibull, ... etc; see for example van der Vaart (1998) paragraph 5.43.

-The maps h(θ, α) : x → h(θ, α, x) and (θ, α) → P θ T h(θ, α) are allowed to take the value -∞; for example, take φ(x) =log x + x -1, and consider the model

{P α = αCauchy(0) + (1 -α)N (0, 1); α ∈ Θ} , with Θ = [0, 1] and θ T = 1/2. Then, P θ T h(θ, 1) = -∞ for all θ ∈]0, 1[.
-The theorem states both existence, consistency and asymptotic normality of the estimates.

-The estimate α φ (θ) may exist and be consistent whereas α φ (θ) does not in many cases.

-One interesting situation also is if the map α ∈ Θ → P n h(θ, α) = 0 is strictly concave and Θ is convex; the estimates α φ (θ) and α φ (θ) are the same.

Remark 3.6. Using theorem 3.2 part (c), the estimate D φ (θ 0 , θ T ) can be used to perform statistical tests (asymptotically of level ǫ) of the null hypothesis H 0 : θ T = θ 0 against the alternative H 1 : θ T = θ 0 for a given value θ 0 . Since D φ (θ 0 , θ T ) is nonnegative and takes value zero only when θ T = θ 0 , the tests are defined through the critical region

C φ (θ 0 , θ T ) := 2n φ ′′ (1) D φ (θ 0 , θ T ) > q d,ǫ (3.20)
where q d,ǫ is the (1ǫ)-quantile of the χ 2 distribution with d degrees of freedom. Note that these tests are all consistent, since D φ (θ 0 , θ T ) are n-consistent estimates of D φ (θ 0 , θ T ) = 0 under H 0 , and √ n-consistent estimate of D φ (θ 0 , θ T ) > 0 under H 1 ; see part (c) and (d) in theorem 3.2 above. Further, the asymptotic result (d) in theorem 3.2 above can be used to give approximation of the power function θ T → β(θ T ) := P θ T (C φ (θ 0 , θ T )). We obtain then the following approximation

β(θ T ) ≈ 1 -F N √ n σ φ (θ 0 , θ T ) φ ′′ (1) 2n q d,ǫ -D φ (θ 0 , θ T ) (3.21)
where F N is the cumulative distribution function of a normal random variable with mean zero and variance one. An important application of this approximation is the approximate sample size (3.22) below that ensures a power β for a given alternative θ T = θ 0 . Let n 0 be the positive root of the equation

β = 1 -F N √ n σ φ (θ 0 , θ T ) φ ′′ (1) 2n q d,ǫ -D φ (θ 0 , θ T ) i.e., n 0 = (a+b)- √ a(a+2b) 2D φ (θ 0 ,θ T ) 2 where a = σ 2 φ (θ 0 , θ T ) F -1 N (1 -β) 2 and b = φ ′′ (1)q d,ǫ D φ (θ 0 , θ T ).
The required sample size is then

n * = [n 0 ] + 1 (3.22)
where [.] is used here to denote "integer part of ".

Remark 3.7. (An other view at the generalized likelihood ratio test and approximation of the power function through KL m -divergence). In the particular case of the KL m -divergence, i.e., when φ(x) = φ 0 (x) :=log x + x -1, we obtain from (3.20)

the critical area

C KLm (θ 0 , θ T ) := 2n sup α∈Θ P n log p α p θ 0 > q d,ǫ = 2 log sup α∈Θ n i=1 p α (X i ) n i=1 p θ 0 (X i ) > q d,ǫ ,
which is to say that the test obtained in this case is precisely the generalized likelihood ratio one. The power approximation and the approximate sample size guaranteeing a power β for a given alternative (for the GLRT) are given by (3.21) and (3.22), respectively, where φ is replaced by φ 0 and D φ by KL m .

3.2. The asymptotic behavior of the MDφDE's. We now explore the asymptotic properties of the estimates θ φ and α φ ( θ φ ) defined in (3.13) and (3.12). We assume that condition (3.1) holds for any α, θ ∈ Θ.

3.2.1. Consistency. We state consistency under the following assumptions (c.4) The estimates θ φ and α φ ( θ φ ) exist.

(c.5) sup {α,θ∈Θ} |P n h(θ, α) -P θ T h(θ, α)| tends to 0 in probability;

(a) for any positive ǫ, there exists some positive η, such that for any α in Θ with αθ T > ǫ and for all θ ∈ Θ, it holds P θ T h(θ, α) < P θ T h(θ, θ T )η;

(b) there exists a neighborhood of θ T , say N (θ T ), such that for any positive ǫ, there exists some positive η such that for all α ∈ N (θ T ) and all θ ∈ Θ satisfying

θ -θ T > ǫ, it holds P θ T h(θ T , α) < P θ T h(θ, α) -η;
(c.6) there exists some neighborhood N (θ T ) of θ T and a positive function

H such that for all α in N (θ T ), h(θ T , α, x) ≤ H(x) (P θ T -a.s.) with P θ T H < ∞. Remark 3.8. Condition (c.5) is fulfilled if x → h(θ, α); (θ, α) ∈ Θ 2 is a
(θ T , θ T ), of (θ, α) ∈ Θ × Θ → P n h(θ, α)
, is well-separated. Note that theses two conditions are not very restrictive, they are satisfied for example when Θ is convex and the function

(θ, α) ∈ Θ × Θ → P n h(θ, α
) is concave in α (for all θ) and convex in θ (for all α), which is the case for example 3.1 and 3.2 above, both conditions (c.5.a) and (c.5.b) are satisfied;

we can take η = ǫ 2 2 .

Proposition 3.4. Assume that conditions (c.4-5-6) hold. Then,

(1) sup θ∈Θ α φ (θ)θ T tends to 0 in probability.

(2) The MDφ estimate θ φ converges to θ T in probability.

Asymptotic normality.

Assume that θ T is an interior point of Θ, the convex function φ has continuous derivatives up to 4th order, and the density p θ (x) has continuous partial derivatives up to 3th order (for all x λ-a.e.). In the following theorem we sate the asymptotic normality of the estimates θ φ and α φ ( θ φ ). We will use the following assump- (A.6) The integrals

P θ T (∂/∂α)h(θ T , θ T ) 2 , P θ T (∂/∂θ)h(θ T , θ T ) 2 , P θ T (∂ 2 /∂α 2 )h(θ T , θ T ) , P θ T (∂ 2 /∂θ 2 )h(θ T , θ T ) and P θ T (∂ 2 /∂θ∂α)h(θ T , θ T )
are finite, and the matrix I θ T is non singular.

Theorem 3.5. Assume that conditions (A.4-5-6) hold. Then, both √ n θ φθ T and √ n α φ ( θ φ )θ T converge in distribution to a centered multivariate normal random variable with covariance matrix V = I -1 θ T .

3.3.1. Existence, consistency and limit laws of a sequence of local minima-maxima. Assume that θ T is an interior point of Θ, the convex function φ has continuous derivatives up to 4th order, and the density p θ (x) has continuous partial derivatives up to 3th order (for all x λ-a.e.). In the following theorem we sate the existence and consistency of a sequence of local minima-maxima θ φ and α φ ( θ φ ). We give also their limit laws.

Theorem 3.6. Assume that conditions (A.5) and (A.6) hold.

(a) Let B := θ ∈ Θ; θθ T ≤ n -1/3 . Then, as n → ∞, with probability one, the function (θ, α) → P n h(θ, α) attains its min-max value at some point θ φ , α φ ( θ φ )

in the interior of B × B, and satisfies P n (∂/∂α)h θ φ , α φ ( θ φ ) = 0 and

P n (∂/∂θ)h θ φ , α φ ( θ φ ) = 0. (b) Both √ n θ φ -θ T and √ n α φ ( θ φ ) -θ T converge
H 0 : θ T ∈ Θ 0 versus H 1 : θ T ∈ Θ\Θ 0 . (3.23)
This is equivalent to

H 0 : θ T ∈ s(B 0 ) versus H 1 : θ T ∈ Θ\s(B 0 ).
Using (3.14), the φ-divergence D φ (Θ 0 , θ T ), between the set of distributions {P θ such that θ ∈ Θ 0 } and the p.m. P θ T , can be written as D φ (Θ 0 , θ T ) = inf θ∈Θ 0 sup α∈Θ P θ T h(θ, α). Hence, it can be estimated by

D φ (Θ 0 , θ T ) := inf θ∈Θ 0 D φ (θ, θ T ) := inf θ∈Θ 0 sup α∈Θ P n h(θ, α).
We use D φ (Θ 0 , θ T ) to perform statistical test pertaining to (3.23). Since D φ (Θ 0 , θ T ) := inf θ∈Θ 0 D φ (θ, θ T ) is positive under H 1 and takes value 0 only under H 0 (provided that the infimum is attained on Θ 0 ), we reject H 0 whenever D φ (Θ 0 , θ T ) takes large values. The following theorem provides the limit distribution of D φ (Θ 0 , θ T ) under the null hypothesis H 0 .

Theorem 3.7. Let us assume that the conditions in theorem 3.5 are satisfied. Under H 0 , the statistics 2n φ ′′ (1) D φ (Θ 0 , θ T ) converge in distribution to a χ 2 random variable with l degrees of freedom.

The following theorem gives the limit laws of the test statistics 2n φ ′′ (1) D φ (Θ 0 , θ T ) under the alternative hypothesis H 1 : θ T ∈ Θ\Θ 0 . We will use the following assumptions. (C.3) The integrals P θ T (∂/∂α)h(s(β * ), θ T ) 2 , P θ T (∂/∂β)h(s(β * ), θ T ) 2 , P θ T (∂ 2 /∂α 2 )h(s(β * ), θ T ) , P θ T (∂ 2 /∂β 2 )h(s(β * ), θ T ) and P θ T (∂ 2 /∂β∂α)h(s(β * ), θ T ) are finite, and the matrix

A := A 11 A 12 A 21 A 22
is non singular, where

A 11 := P θ T (∂ 2 /∂β 2 )h(s(β * ), θ T ), A 22 := P θ T (∂ 2 /∂α 2 )h(s(β * ), θ T )
and

A 12 = A T 21 := P θ T (∂ 2 /∂β∂α)h(s(β * ), θ T ). (C.4) The integral P θ T h(s(β * ), θ T ) 2 is finite.
Denote β φ and α φ ( β φ ) the min-max optimal solution of

D φ (Θ 0 , θ T ) := inf β∈B 0 sup α∈Θ P n h(s(β), α),
and let B(β * , n -1/3 ) := β ∈ B 0 ; β -β * ≤ n -1/3 , c n := ( β T φ , α φ ( β φ ) T ) T , c * := (β * T , θ T
V = A -1 F A -1 . (b) If additionally the condition (C.6) holds, then √ n D φ (Θ 0 , θ T ) -D φ (Θ 0 , θ T ) con-
verges in distribution to a centered normal random variable with variance

σ 2 φ (β * , θ T ) = P θ T h(s(β * ), θ T ) 2 -(P θ T h(s(β * ), θ T )) 2 . (3.24)
Remark 3.9. Using theorem 3.7, the estimate D φ (Θ 0 , θ T ) can be used to perform statistical tests (asymptotically of level ǫ) of the null hypothesis H 0 : θ T ∈ Θ 0 against the alternative H 1 : θ T ∈ Θ\Θ 0 . Since D φ (Θ 0 , θ T ) is nonnegative and takes value zero only when θ T ∈ Θ 0 , the tests are defined through the critical region

C φ (Θ 0 , θ T ) := 2n φ ′′ (1) D φ (Θ 0 , θ T ) > q l,ǫ , (3.25)
where q l,ǫ is the (1ǫ)-quantile of the χ 2 distribution with l degrees of freedom. Note that these tests are all consistent, since D φ (Θ 0 , θ T ) are n-consistent estimates of D φ (Θ 0 , θ T ) = 0 under H 0 , and √ n-consistent estimate of D φ (Θ 0 , θ T ) > 0 under H 1 ; see theorem 3.7 and theorem 3.8 part (c). Further, the asymptotic result (c) in theorem 3.8 above can be used to give an approximation to the power function θ T → β(θ T ) := P θ T (C φ (Θ 0 , θ T )). We obtain then the following approximation

β(θ T ) ≈ 1 -F N √ n σ φ (β * , θ T ) φ ′′ (1) 2n q l,ǫ -D φ (Θ 0 , θ T ) (3.26)
where F N is the cumulative distribution function of a normal variable with mean zero and variance one. An important application of this approximation is the approximate sample size (3.27) below that ensures a power β for a given alternative θ T ∈ Θ\Θ 0 . Let n 0 be the positive root of the equation

β = 1 -F N √ n σ φ (β * , θ T ) φ ′′ (1) 2n q l,ǫ -D φ (Θ 0 , θ T ) i.e., n 0 = (a+b)- √ a(a+2b) 2D φ (Θ 0 ,θ T ) 2 where a = σ 2 φ (β * , θ T ) F -1 N (1 -β) 2 and b = φ ′′ (1)q l,ǫ D φ (Θ 0 , θ T ).
The required sample size is then

n * = [n 0 ] + 1 (3.27)
where [.] is used here to denote "integer part of ".

Remark 3.10. (An other view at the generalized likelihood ratio test for composite hypotheses, and approximation of the power function through KL mdivergence). In the particular case of the KL m -divergence, i.e., when φ(x) = φ 0 (x) := log x + x -1, we obtain from (3.25) the critical area

C KLm (Θ 0 , θ T ) = 2 log sup α∈Θ n i=1 p α (X i ) sup θ∈Θ 0 n i=1 p θ (X i ) > q l,ǫ ,
which is to say that the test obtained in this case is precisely the generalized likelihood ratio test associated to (3.23). The power approximation and the approximate sample size guaranteeing a power β for a given alternative (for the GLRT) are given by (3.26) and

(3.27), respectively, where φ is replaced by φ 0 and D φ by KL m .

Non regular models. A simple solution for the case of mixture models

The test problem for the number of components of a finite mixture has been extensively treated when the total number of components k is equal to 2, leading to a satisfactory solution; the limit distribution of the generalized likelihood ratio statistic is non standard, since it is 0.5δ 0 + 0.5χ 2 (1), a mixture of a Dirac mass at 0 and a χ 2 (1) with weights equal to 1/2; see e.g. [START_REF] Titterington | Statistical analysis of finite mixture distributions[END_REF] and Self and Liang (1987). When k > 2, the problem is much more involved. Self and Liang (1987) obtained the limit distribution of the generalized likelihood ratio statistic, which is non standard and complex. This result yields formidable numerical difficulties for the calculation of the critical value of the test. In section 5.1 below, we propose a unified treatment for all these cases, with simple and standard limit distribution both when the parameter θ T is an interior or a boundary point of the parameter space Θ. On the other hand, confidence regions for the mixture parameter θ T even when k = 2 are intractable through the generalized likelihood ratio statistic. Indeed, the limit law of the generalized likelihood ratio statistic depends heavily on the fact that θ is a boundary or an interior point of the parameter space. For example, when k = 2, the limit distribution of the generalized likelihood ratio statistic is 0.5δ 0 + 0.5χ 2 (1) when θ = 0 and χ 2 (1) when 0 < θ < 1. Therefore, the confidence level is not defined uniquely. At the opposite, we will prove in section 5.3 that the proposed dual χ 2 -statistic yields quite standard confidence regions even when k > 2.

Notations. Let P

(1)

a 1 ; a 1 ∈ A 1 , . . ., P (k) a k ; a k ∈ A k be k-parametric models where A 1 , . . . , A k are k (k ≥ 2) sets in R d 1 , . . . , R d k and d 1 , . . . , d k ∈ N * . Denote P θ the mixture model P θ := k i=1 w i P (i) a i (4.1)
where 0 ≤ w i ≤ 1, w i = 1 and

θ ∈ Θ := (w 1 , . . . , w k , a 1 , . . . , a k ) T ∈ [0, 1] k × A 1 × • • • × A k such that k i=1 w i = 1 , (4.2)
and assume that the model is identifiable. Let k 0 ∈ {1, . . . , k -1}. We test if (kk 0 ) components in (4.1) have null coefficients. We assume that their labels are k 0 + 1, ..., k.

Denote Θ 0 the subset of Θ defined by

Θ 0 := {θ ∈ Θ such that w k 0 +1 = • • • = w k = 0} .
On the basis of an i.i.d sample X 1 , . . . , X n with distribution P θ T , θ T ∈ Θ, we intend to perform tests of the hypothesis

H 0 : θ T ∈ Θ 0 against the alternative H 1 : θ T ∈ Θ \ Θ 0 . (4.3)
It is known that the generalized likelihood ratio test, based on the statistic

2 log λ := 2 log sup θ∈Θ n i=1 p θ (X i ) sup θ∈Θ 0 n i=1 p θ (X i ) , (4.4)
is not valid for this problem, since the asymptotic approximation by χ 2 distribution does not hold in this case; the problem is due to the fact that the null value of θ T is not in the interior of the parameter space Θ. We clarify now this problem. For simplicity, consider a mixture of two known densities p 0 and p 1 with p 0 = p 1 :

p θ = (1 -θ)p 0 + θp 1 where θ ∈ Θ := [0, 1]. (4.5)
Given data X 1 , . . . , X n with distribution P θ T , θ T ∈ [0, 1], consider the test problem

H 0 : θ T = 0 against the alternative H 1 : θ T > 0. (4.6)
The generalized likelihood ratio statistic for this test problem is

W n (0) := 2log L( θ) L(0) , (4.7)
where

L(θ) := n i=1 [(1 -θ)p 0 (X i ) + θp 1 (X i )] for all θ ∈ [0, 1],
and θ is the MLE of θ. Using the strict concavity of the function θ ∈ [0, 1] → l(θ) := log L(θ), it is clear that θ = 0 whenever l ′ + (0), the derivative on the right at θ = 0 of θ → l(θ), is nonpositive. Hence, we can write

P 0 {W n = 0} ≥ P 0 θ = 0 = P 0 l ′ + (0) ≤ 0 = P 0 n i=1 p 0 (X i ) p 1 (X i ) -n ≤ 0 = P 0 √ n 1 n n i=1 p 0 (X i ) p 1 (X i ) -1 ≤ 0 (4.8)
which, by the CLT, tends to 1/2 (if 1 = E(Y 2 i ) < ∞ where Y i := p 0 (X i )/p 1 (X i )) since the random variables Y i are i.i.d with E(Y i ) = 1 under H 0 . This proves that the convergence in distribution of the generalized likelihood ratio statistic W n (0) to a χ 2 random variable (under H 0 ) does not hold. Under suitable regularity conditions we can prove that the limit distribution of the statistic W n in (4.7) is 0.5δ 0 + 0.5χ 2 1 , a mixture of the χ 2 -distribution and the Dirac measure at zero; see Self and Liang (1987).

Moreover, in the case of more than two components and kk 0 ≥ 2, the limit distribution of the GLR statistic (4.4) under H 0 is complicate and not standard (not a χ 2 distribution) which poses some difficulty in determining the critical value that will give correct asymptotic size; see Self and Liang (1987). On the other hand, the likelihood ratio statistic

W n (θ) := 2log L( θ) L(θ) (4.9)
can not be used to construct asymptotic confidence region for the parameter θ T since its limit law is not the same when θ T = 0 and θ T > 0. In figure 1, we illustrate the accuracy of the approximation of distribution of the GLR by its limit 0.5δ 0 + 0.5χ 2 1 ; we plot the cumulative distribution function (c.d.f) of both the limit law, and the observed GLR's obtained from 1000 independent runs of samples with sizes n = 200, n = 500 and n = 1000, with P 0 = N (0, 1) and P 1 = N (0.5, 1).

4.2.

A simple solution to the problem of testing the number of components in a mixture. We propose the following simple solution : Consider the following set of signed finite measures p θ := (1 -θ)p 0 + θp 1 where θ ∈ Ê. This set (of signed finite measures with mass one) obviously contains the mixture model (4.5). In particular, the null value of θ T (i.e., θ T = 0) is an interior point of the parameter space Ê. The likelihood ratio test (for a model of signed measures) cannot be used since the log-likelihood l(θ) may be infinite (when θ < 0 or θ > 1). In the context of divergences, this means that the estimate KL m (P 0 , P θ T ) may be infinite if we consider the model (4.10), which is due to the fact that the corresponding convex function φ(x) =log x + x -1 is infinite on R -. This suggests to use a divergence associated to a convex function φ which is finite on all R, for instance, the χ 2 -divergence (which is associated to the convex function φ(x) = 1 2 (x -1) 2 ). So, in order to perform a test asymptotically of level ǫ for (4.6), we propose to use the following estimate of the χ 2 -divergence between P 0 and P θ T

χ 2 (0, θ T ) = sup α∈Θe {P 0 f (0, α) -P n g(0, α)} , (4.11)
where f (0, α) = p 0 /p α -1 and g(0, α) = 1/2(p 0 /p α + 1)(p 0 /p α -1) as a consequence of definitions (3.9) and (3.8), and Θ e is the new parameter space which we define as follows

Θ e := α ∈ Ê such that |f (0, α)| dP 0 is finite .
The value of the parameter θ T under the null hypothesis H 0 , i.e., θ T = 0, is in the interior of the new parameter space Θ e which is generally non void. Hence, under conditions of theorem 3.2 where Θ is replaced by Θ e and θ by zero, under H 0 the statistic 2n χ 2 (0, θ T ) converges in distribution to a χ 2 random variable with one degree of freedom; the critical region takes then the form

CR := 2n χ 2 (0, θ T ) > q 1,ǫ , (4.12)
where q 1,ǫ is the (1ǫ)-quantile of the χ 2 distribution with one degree of freedom. Obviously other divergences which are associated to convex functions finite on all R can be used. The use of the χ 2 -divergence is recommended. Indeed, for regular cases (for example for multinomial goodness-of-fit tests) χ 2 -test is equivalent (in Pitman sense) to the generalized likelihood ratio one; see also [START_REF] Cressie | Multinomial goodness-of-fit tests[END_REF] sections 3.1 and 3.2 for other motivations in favor of the χ 2 approach.

In figure 2, we illustrate the accuracy of the approximation of the distribution of the proposed dual χ 2 -statistic by the χ 2 (1); we plot the cumulative distribution function (c.d.f) of both the limit law, and the dual χ 2 -statistic obtained from 1000 independent runs of samples with sizes n = 200, n = 500 and n = 1000, with P 0 = N (0, 1) and P 1 = N (0.5, 1). We observe that the approximation is as satisfactory as it is in figure 1 for the GLR case, so that the extension of the model to signed finite measures does not affect the quality of the approximation of the limit distribution.

4.3.

Confidence regions for the mixture parameters. We propose the following solution to the confidence region problem when the parameter may be a boundary value of the parameter space: The estimate

χ 2 (θ, θ T ) = sup α∈Θe(θ) {P θ f (θ, α) -P n g(θ, α)} , (4.13)
where Θ e (θ) := α ∈ Ê such that |f (θ, α)| dP θ is finite , (1ǫ) defined by

C := θ ∈ Θ such that 2n χ 2 (θ, θ T ) ≤ q 1,ǫ .
In fact, lim n→∞ P θ T (θ T ∈ C) = 1ǫ both when θ T = 0 or θ T > 0 since the statistic 2n χ 2 (θ T , θ T ) converges in distribution to χ 2 random variable with one degree of freedom both when θ T = 0 or θ T > 0. We give now the form of the critical region and the confidence region in the multivariate case, i.e., in the case of the general model (4.1). For all θ ∈ Θ, define the set

Θ e (θ) := α ∈ Ê k × A 1 × • • • × A k such that k i=1 α i = 1 and |f (θ, α)| dP θ is finite ,
and the statistic

χ 2 (Θ 0 , θ T ) := inf θ∈Θ 0 χ 2 (θ, θ T ) := inf θ∈Θ 0 sup α∈Θe(θ) {P θ f (θ, α) -P n g(θ, α)} .
Under some conditions similar to that in theorems 3.1, 3.2 and 3.3, we can prove, under the null hypothesis H 0 in (4.3), that the statistic 2n χ 2 (Θ 0 , θ T ) converges in distribution to χ 2 random variable with (kk 0 ) degrees of freedom. Also, the statistic 2n χ 2 (θ, θ T ) when θ = θ T converges in distribution to χ 2 random variable with d :

= k -1 + d 1 + • • • + d k
degrees of freedom in both case when θ T is a boundary value or not. Hence, the critical region is given by

CR := 2n χ 2 (Θ 0 , θ T ) > q k-k 0 ,ǫ , and
C := θ ∈ Θ such that 2n χ 2 (θ, θ T ) ≤ q d,ǫ
is an asymptotic confidence region for θ T of level ǫ both when θ T is a boundary value or not.

4.4. Approximation of the power function of the likelihood ratio statistic: simulation results. In the context of the exponential model p θ (x) = θ exp {θx}, we consider the problem of testing

H 0 : θ T = 1 versus H 1 : θ T = 1
using the GLR. We recall that the power function of the GLR test is

θ T → β(θ T ) := P θ T 2n KL m (1, θ T ) ≥ q 1,0.5 (4.14)
and its approximation is

β(θ T ) = 1 -F N √ n σ φ (1, θ T ) 1 2n q 1,0.05 -KL m (1, θ T ) (4.15)
where F N is the cumulative distribution function of a normal random variable with mean zero and variance one, and φ(x) =log x + x -1; see remarks 3.3 and 3.4 above. The power function (4.14) is plotted (with continuous line) for sample sizes n = 50, n = 100, n = 300 and n = 500, and for different values of θ T . Each power entry was obtained from 1000 independent runs. The approximation (4.15) is plotted as a function of θ T by a dashed line. We observe (see figure 3) that the approximation is accurate for alternatives which are not "close to" the null hypothesis even for moderate sample sizes.

Concluding remarks and possible developments

We have addressed the parametric estimation and test problems. We have introduced new estimation and test procedure using divergence minimization and duality technique for discrete or continuous parametric models, avoiding the smoothing method. The procedure leads to optimal estimates for the parameter model and for the divergences. It includes both the discrete (finite or infinite) and the continuous support cases. It extends the maximum likelihood method for both estimation and test problems. Moreover, the procedure and the divergences framework permit to obtain the limit laws of the proposed Many problems remain to be studied in the future, such as the choice of the divergence which leads to an "optimal" (in some sense) estimate or test in terms of efficiency and robustness, construction of convergent estimates and test statistics by divergence when the maximum likelihood is not consistent (for example for location family for which the expectation does not exists), the Bartlett correctability and the large deviation properties of the proposed statistics D φ .

Appendix

Proof of proposition 3.1. (1) We will prove the consistency of the estimate D φ (θ, θ T ). We have

D φ (θ, θ T ) -D φ (θ, θ T ) = |P n h(θ, α φ (θ)) -P θ T h(θ, θ T )| := |A|, which implies P n h(θ, θ T ) -P θ T h(θ, θ T ) ≤ A ≤ P n h(θ, α φ (θ)) -P θ T h(θ, α φ (θ)).
Both the RHS and the LHS terms in the above display go to 0, under condition (c.2). This implies that A tends to 0.

(2) For the consistency of α φ (θ), we refer to van der Vaart (1998) theorem 5.7.

Proof of theorem 3.2. (a) Using (A.1), simple calculus give

P θ T (∂/∂α)h(θ, α) = 0 (6.1)
and

P θ T (∂ 2 /∂α 2 )h(θ, θ T ) = -φ ′′ (p θ /p θ T )(p 2 θ /p 3 θ T )p ′ θ T p ′ θ T
T dλ =: -S. (6.2)

Observe that the matrix S is symmetric and positive since the second derivative φ ′′ is nonnegative by the convexity of φ. Let U n (θ T ) := P n (∂/∂α)h(θ, θ T ), and use (6.1) and (A.2) in connection with the Central Limit Theorem (CLT) to see that √ nU n (θ T ) → N (0, M ). ( 6.3) Also, let V n (θ T ) := P n (∂ 2 /∂α 2 )h(θ, θ T ), and use (6.2) and (A.2) in connection with the Law of Large Numbers (LLN) to conclude that

V n (θ T ) → -S (a.s). (6.4)

Using the fact that P n (∂/∂α)h(θ, α) = 0 and a Taylor expansion of P n (∂/∂α)h(θ, α) in α around θ T , we obtain

0 = P n (∂/∂α)h(θ, α) = P n (∂/∂α)h(θ, θ T ) + ( α -θ T ) T P n (∂ 2 /∂α 2 )h(θ, θ T ) + o p (n -1/2 ). Hence, √ n ( α -θ T ) = -V n (θ T ) -1 √ nU n (θ T ) + o p (1). (6.5)
Using (6.3) and (6.4) and Slutsky theorem, we conclude then

√ n ( α -θ T ) → N (0, V φ (θ, θ T )) (6.6) where V φ (θ, θ T ) is given in part (a) of theorem 3.2. When θ T = θ, direct calculus shows that V φ (θ, θ T ) = I -1 θ T .
(b) Assume that θ T = θ. From (6.5), using the convergence (6.4), we get

√ n ( α -θ T ) = S -1 √ nU n (θ T ) + o p (1). (6.7)
On the other hand, a Taylor expansion of [2n/φ ′′ (1)] D φ (θ, θ T ) = [2n/φ ′′ (1)] P n (∂/∂α)h(θ, α) in α around θ T , using the fact that P n h(θ, θ T ) = 0 when θ T = θ, gives

2n φ ′′ (1) D φ (θ, θ T ) = 2n φ ′′ (1) U T n ( α -θ T ) + 2n φ ′′ (1) ( α -θ T ) T V n ( α -θ T ) + o p (1).
Use (6.4), (6.7) and the fact that S = -φ ′′ (1)I θ T when θ T = θ to conclude that

2n φ ′′ (1) D φ (θ, θ T ) = φ ′′ (1) -2 √ nU T n I -1 θ T √ nU n + o p (1).
Finally, use the convergence (6.3) and the fact that M = φ ′′ (1) 2 I θ T when θ = θ T , to conclude that [2n/φ ′′ (1)] D φ (θ, θ T ) converges in distribution to a χ 2 variable with d degrees of freedom when θ = θ T .

(c) Assume that θ T = θ. A Taylor expansion of D φ (θ, θ T ) = P n h(θ, α), in α around θ T , using the fact that

P θ T (∂/∂α)h(θ, θ T ) = 0, gives D φ (θ, θ T ) = P n h(θ, θ T ) + o p (n -1/2 ). Hence, √ n D φ (θ, θ T ) -D φ (θ, θ T ) = √ n [P n h(θ, θ T ) -P θ T h(θ, θ T )] + o p (1),
which under assumption (A.3), by the CLT, converges in distribution to a centred normal variable with variance σ

2 φ (θ, θ T ) = P θ T h(θ, θ T ) 2 -(P θ T h(θ, θ T )) 2 .
Proof of theorem 3.3. (a) For any α = θ T + un -1/3 with |u| ≤ 1, consider a Taylor expansion of P n h(θ, α) in α around θ T , and use (A.1) to see that

nP n h(θ, α) -nP n h(θ, θ T ) = n 2/3 u T U n + 2 -1 n 1/3 u T V n u + O(1) (a.s.)
uniformly on u with |u| ≤ 1. Now, use (6.4) and the fact that

U n = O n -1/2 (log log n) 1/2 (a.s) to conclude that nP n h(θ, α) -nP n h(θ, θ T ) = O n 1/6 (log log n) 1/2 -2 -1 u T Sun 1/3 + O(1) (a.s.)
uniformly on u with |u| ≤ 1. Hence, uniformly on the surface of the ball B (i.e., uniformly on u with |u| = 1), we have

nP n h(θ, α) -nP n h(θ, θ T ) ≤ O n 1/6 (log log n) 1/2 -2 -1 cn 1/3 + O(1) (a.s.) (6.8)
where c is the smallest eigenvalue of the matrix S. Note that c is positive since S is positive definite (it is symmetric, positive and non singular by assumption A.2). In view of (6.8), by the continuity of α → P n h(θ, α) -nP n h(θ, θ T ) and since it takes value zero on α = θ T and is asymptotically negative on the surface of B, it holds that as n → ∞, with probability one, α → P n h(θ, α) attains its maximum value at some point α φ (θ) in the interior of the ball B, and therefore the estimate α φ (θ) satisfies P n (∂/∂α)h(θ, α) = 0 and αθ T = O(n -1/3 ).

The proofs of parts (b), (c) and (d) are similar to those of parts (a), (b) and (d) in theorem 3.2. Hence, they are omitted.

Proof of proposition 3.4. We prove (1). For all θ ∈ Θ, under condition (c.4-5-6), we prove that sup θ∈Θ α φ (θ)θ T tends to 0. By the very definition of α φ (θ) and the condition (c.5), we have

P n h(θ, α φ (θ)) ≥ P n h(θ, θ T ) ≥ P θ T h(θ, θ T ) -o p (1),
where o p (1) does not depend upon θ (due to condition (c.5)). Hence, we have for all θ ∈ Θ

P θ T h(θ, θ T ) -P θ T h(θ, α φ (θ)) ≤ P n h(θ, α φ (θ)) -P θ T h(θ, α φ (θ)) + o p (1). (6.9)
The RHS term is less than sup {θ,α∈Θ} |P n h(θ, α) -P θ T h(θ, α)| +o p (1) which, by (c.5), tends to 0. Let ǫ > 0 be such that sup θ∈Θ α φ (θ)θ T > ǫ. There exists some a n ∈ Θ such that α φ (a n )θ T > ǫ. Together with (c.5.a), there exists some η > 0 such that

P θ T h(a n , θ T ) -P θ T h(a n , α φ (a n )) > η.
We then conclude that

P sup θ∈Θ α φ (θ) -θ T > ǫ ≤ P {P θ T h(a n , θ T ) -P θ T h(a n , α φ (θ)) > η} ,
and the RHS term tends to 0 by (6.9). This concludes the proof of part (1).

We prove (2). By the very definition of θ φ , conditions (c.5) and (c.6) and part (1), we have

P n h( θ φ , α φ ( θ φ )) ≤ P n h(θ T , α φ (θ T )) ≤ P θ T h(θ T , α φ ( θ φ )) -o p (1), from which P θ T h( θ φ , α φ ( θ φ )) -P θ T h(θ T , α φ ( θ φ )) ≤ P θ T h( θ φ , α φ ( θ φ )) -P n h( θ φ , α φ ( θ φ )) + o p (1) ≤ sup {θ,α∈Θ} |P n h(θ, α) -P θ T h(θ, α)| + o p (1).(6.10)
Further, by part (1) and condition (c.5.b), for any positive ǫ, there exists η > 0 such that

P θ φ -θ T > ǫ ≤ P P θ T h( θ φ , α φ ( θ φ )) -P θ T h(θ T , α φ ( θ φ )) > η ,
and the RHS term, under condition (c.5), tends to 0 by (6.10). This concludes the proof.

Proof of theorem 3.5. Under condition (A.5), simple calculus give (6.12) and

P θ T ∂ ∂α h(θ T , θ T ) = P θ T ∂ ∂θ h(θ T , θ T ) = P θ T ∂ 2 ∂α∂θ h(θ T , θ T ) = P θ T ∂ 2 ∂θ∂α h(θ T , θ T ) = 0, (6.11) -P θ T ∂ 2 ∂α 2 h(θ T , θ T ) = P θ T ∂ 2 ∂θ 2 h(θ T , θ T ) = φ ′′ (1)I θ T ,
P θ T ∂ ∂θ h(θ T , θ T ) ∂ ∂θ h(θ T , θ T ) T = P θ T ∂ ∂α h(θ T , θ T ) ∂ ∂α h(θ T , θ T ) T = -P θ T ∂ ∂α h(θ T , θ T ) ∂ ∂θ h(θ T , θ T ) T = φ ′′ (1) 2 I θ T . (6.13) Denote U n (θ, θ T ) := P n (∂/∂α)h(θ, θ T ), V n (θ, θ T ) := P n (∂ 2 /∂α 2 )h(θ, θ T ), S(θ, θ T ) := -P θ T (∂ 2 /∂α 2 )h(θ, θ T ) and a T n := ( θ φ -θ T ) T , ( α φ ( θ φ ) -θ T ) T T
. Under conditions (A.4-5), by a Taylor expansion, we obtain

√ na n = √ n 1 φ ′′ (1) I -1 θ T 0 0 -1 φ ′′ (1) I -1 θ T -P n ∂ ∂θ h(θ T , θ T ) -P n ∂ ∂α h(θ T , θ T ) + o p (1).
We therefore deduce, by the CLT, that, under condition (A.6), √ na n converges in distribution to a centred normal variable with covariance matrix

V = I -1 θ T I -1 θ T I -1 θ T I -1 θ T ,
which completes the proof of theorem 3.5.

Proof of theorem 3.6. (a) Using condition (A.5) and (6.11), we can write (6.15) uniformly on θ ∈ B(θ T , n -1/3 ). On the other hand, for any α = θ T + un -1/3 with |u| ≤ 1, by a Taylor expansion using condition (A.5), we obtain

U n (θ, θ T ) := U n (θ T , θ T ) + o(n -1/3 ) (a.s.) (6.14) and V n (θ, θ T ) := V n (θ T , θ T ) + O(n -1/3 ) (a.s.),
nP n h(θ, α) -nP n h(θ, θ T ) = n 2/3 u T U n (θ, θ T ) + 2 -1 n 1/3 u T V n (θ, θ T )u + O(1) (a.s.)
uniformly on θ ∈ B(θ T , n -1/3 ) and u with |u| ≤ 1. Combining this with (6.14) and (6.15) to see that

nP n h(θ, α) -nP n h(θ, θ T ) = n 2/3 u T U n (θ T , θ T ) + 2 -1 n 1/3 u T V n (θ T , θ T )u + o(n 1/3 ) (a.s.)
uniformly on θ ∈ B(θ T , n -1/3 ) and u with |u| ≤ 1. Now, from this, using the fact that

U n (θ T , θ T ) = O n -1/2 (log log n) 1/2 (a.s.) and V n (θ T , θ T ) = -S(θ T , θ T ) + o(1) (a.s.), we obtain nP n h(θ, α) -nP n h(θ, θ T ) = O n 1/6 (log log n) 1/2 -2 -1 n 1/3 u T S(θ T , θ T )u + o(n 1/3 ) (a.s.) (6.16)
uniformly on θ ∈ B(θ T , n -1/3 ) and u with |u| ≤ 1. Hence, uniformly on α in the surface of the ball B(θ T , n -1/3 ) (i.e., uniformly on u with |u| = 1), we have nP n h(θ, α) -nP n h(θ, θ T ) ≤ O n 1/6 (log log n) 1/2 -2 -1 φ ′′ (1)cn 1/3 + o(n 1/3 ) (a.s.) (6.17) (uniformly on θ ∈ B(θ T , n -1/3 )) where c > 0 is the smallest eigenvalue of the matrix I θ T = φ ′′ (1) -1 S(θ T , θ T ). Hence, by the continuity of the function α → nP n h(θ, α) -nP n h(θ, θ T ) and since it takes value zero when α = θ T and is asymptotically negative with respect to α on the surface of B, it holds that, as n tends to ∞, with probability one, the function α → P n h(θ, α) attains it maximum value at some point α φ (θ) in the interior of B(θ T , n -1/3 ), and this holds for all θ ∈ B(θ T , n -1/3 ). Further, since (6.16) holds uniformly on θ ∈ B(θ T , n -1/3 ), we conclude that α φ (θ)θ T = O(n -1/3 ) (a.s.) uniformly on θ ∈ B(θ T , n -1/3 ). (6.18)

We now prove that, as n → ∞, with probability one, the function θ → P n (θ, α φ (θ)) attains its minimum value at some point θ φ in the interior of the ball B(θ T , n -1/3 ). Here, α φ (θ) is any value in the interior of B(θ T , n -1/3 ) which maximizes α → P n h(θ, α). It exists by the above arguments. For any θ = θ T + vn -1/3 with |v| ≤ 1, by a Taylor expansion of nP n h(θ, α φ (θ)) in θ and α φ (θ) around θ T , and a Taylor expansion of nP n h(θ T , α φ (θ T )) in α φ (θ T ) around θ T , using (6.18) and (6.11), we obtain nP n h(θ, α φ (θ)) -nP n h(θ T , α φ (θ T )) = n 2/3 v T P n (∂/∂θ)h(θ T , θ T ) + 2 -1 n 1/3 v T P n (∂ 2 /∂θ 2 )h(θ T , θ T ) v + o(n 1/3 ) (a.s.)

uniformly on v with |v| ≤ 1. Hence, from this, using the fact that P n (∂/∂θ)h(θ T , θ T ) = O n -1/2 (log log n) 1/2 (a.s.) and P n (∂ 2 /∂θ 2 )h(θ T , θ T ) = φ ′′ (1)I θ T + o(1) (a.s.), we conclude that nP n h(θ, α φ (θ))-nP n h(θ T , α φ (θ T )) = O n 1/6 (log log n) 1/2 +2 -1 φ ′′ (1)v T I θ T vn 1/3 +o(n 1/3 ) (a.s.)

uniformly on v with |v| ≤ 1. Hence, uniformly on θ in the surface of the ball B(θ T , n -1/3 ) (i.e., uniformly on v with |v| = 1), we obtain nP n h(θ, α φ (θ))-nP n h(θ T , α φ (θ T )) ≥ O n 1/6 (log log n) 1/2 +2 -1 φ ′′ (1)cn 1/3 +o(n 1/3 ) (a.s.)

where c > 0 is the smallest eigenvalue of I θ T . This implies that n 2/3 P n h(θ, α φ (θ))-n 2/3 P n h(θ T , α φ (θ T )) ≥ O n -1/6 (log log n) 1/2 +2 -1 φ ′′ (1)c+o(1) (a.s.)

uniformly on θ in the surface of the ball B(θ T , n -1/3 ). The left hand side of the above display equals zero when θ = θ T and is positive when θ is in the surface of the ball B(θ T , n -1/3 ) (for n sufficiently large). This implies that, as n → ∞, with probability one, the function θ → P n h(θ, α φ (θ)) attains its minimum value at some point θ φ in the interior of the ball B. This concludes the proof of part (a).

(b) See the proof of theorem 3.5.

Proof of theorem 3.7. We have Moreover, using the fact that φ ′ (1) = 0, we can see that ∂ ∂s(β) h (s(β T ), θ T ) = -∂ ∂α h (s(β T ), θ T ), which implies

P θ T ∂ ∂β h (s(β T ), θ T ) = [S(β T )] T -P θ T ∂ ∂α h (s(β T ), θ T ) .
In the same way, we obtain

P θ T ∂ 2 ∂β 2 h (

  .10) Hence, (3.5), (3.6) and (3.7) can be written as follows D φ (θ, θ T ) := sup α∈Θ P n h(θ, α),

  is as the dual modified KL-divergence estimate or as the minimum dual modified KL-divergence estimate, i.e., MLE=D(KL m )DE=MD(KL m )DE. Indeed, we then have P θ f (θ, α) = 0 and P n h(θ, α) =log p θ pα dP n . Hence by definitions (3θ ) dP n = M LE.

  2 and 5.6 under classical conditions, for the study of M-estimates. In the sequel, we assume that condition (3.1) holds for any α ∈ Θ, and use . to denote the Euclidean norm in Ê d . 3.1.1. Consistency. Consider the following conditions (c.1) The estimate α φ (θ) exists; (c.2) sup α∈Θ |P n h(θ, α) -P θ T h(θ, α)| converges to zero a.s. (resp. in probability);

  Glivenko-Cantelli class of functions. Conditions (c.5.a) and (c.5.b) mean that the saddle-point

  tions (A.4) The estimates θ φ and α φ ( θ φ ) exist and are consistent; (A.5) There exists a neighborhood N (θ T ) of θ T such that the first and second order partial derivatives (w.r.t. α and θ) of f (θ, α, x)p θ (x) are dominated on N (θ T ) × N (θ T ) by λ-integrable functions. The third partial derivatives (w.r.t. α and θ) of h(θ, α, x) are dominated on N (θ T ) × N (θ T ) by some P θ T -integrable functions;

  in distribution to a centered multivariate normal random variable with covariance matrix V = I -1 θ T . 3.4. Composite tests by minimum φ-divergence. Let Θ 0 be a subset of Θ. We assume that there exists an open set B 0 ⊂ R d-l and mappings r : Θ → R l and s : B 0 → R d such that the matrices R(θ) := ∂ ∂θ i r(θ) and S(β) := ∂ ∂β i s(β) exist, with elements continuous, and are of rank l and (dl), respectively, Θ 0 = {s(β); β ∈ B 0 } and r(θ) = 0 for all θ ∈ Θ 0 . Consider the composite null hypothesis

(C. 1 )

 1 The minimum of θ → D φ (θ, θ T ) on Θ 0 is attained at some point, say θ * := s(β * ) with β * ∈ B 0 ; uniqueness then follows by strict convexity of φ and model identifiability assumption; (C.2) There exists a neighborhood N (β * ) of β * and a neighborhood N (θ T ) of θ T such that the first and second order partial derivatives (w.r.t. α and β) of f (s(β), α, x)p s(β) (x) are dominated on N (β * ) × N (θ T ) by λ-integrable functions. The third partial derivatives (w.r.t. β and α) of h(s(β), α, x) are dominated on N (β * ) × N (θ T ) by some P θ T -integrable functions;

  T ) T and F the matrix defined by F := P θ T (∂/∂β)h(s(β * ), θ T ) (∂/∂α)h(s(β * ), θ T ) (∂/∂β)h(s(β * ), θ T ) (∂/∂α)h(s(β * ), θ T ) T . (C.5) The estimates β φ and α φ ( β φ ) exist and are consistent estimators for β * and θ T respectively. Theorem 3.8. Assume that conditions (C.1-2-3-4-5) hold. Then, under the alternative hypothesis H 1 , we have (a) √ n (c nc * ) converges in distribution to a centered multivariate normal random variable with covariance matrix

Figure 1 .

 1 Figure 1. Empirical distribution of the GLR and its limit distribution

Figure 2 .

 2 Figure 2. Empirical distribution of the dual χ 2 -statistic and its limit law

Figure 3 .

 3 Figure 3. Approximation of the power function

D

  φ (Θ 0 , θ T ) := inf β∈B 0 sup α∈Θ P n h (s(β), α) . = P n h s( β), α , in which as in the proof of theorem 3.5, s( β) and α are solutions of the system of equations  equation the partial derivative is intended w.r.t. the first variable β in s(β) and in the second one w.r.t. the second variable α. A Taylor expansion of P n ∂ ∂β h s( β), α and P n ∂ ∂α h s( β), α in a neighborhood of (β T , θ T ) gives-P n ∂ ∂β h(s(β T ), θ T ) -P n ∂ ∂α h(s(β T ), θ T ) = P θ T ∂ 2 ∂β 2 h(s(β T ), θ T ) P θ T ∂ 2 ∂β∂α h(s(β T ), θ T ) P θ T ∂ 2 ∂α∂β h(s(β T ), θ T ) P θ T ∂ 2 ∂β 2 h(s(β T ), θ T ) b n + o p (1), (6.19) where b n := ( ββ T ) T , ( αθ T ) T T. This implies that b n = O p (n -1/2 ). So, by a Taylorexpansion of D φ (Θ 0 , θ T ) around (β T , θ T ), we obtain 2n φ ′′ (1) T φ n = U T n A -1 U n -V T n B -1 V n + o p (1β T ), θ T ) , V n := √ n φ ′′ (1) P n ∂ ∂β h(s(β T ), θ T ), A := -1 φ ′′ (1) P θ T ∂ 2 ∂α 2 h(s(β T ), θ T ), B := 1 φ ′′ (1) P θ T ∂ 2 ∂β 2 h(s(β T ), θ T ).By (6.12), it holds A = I θ T . On the other hand, β T ), θ T ) = [S(β T )] T ∂ ∂s(β) h (s(β T ), θ T ) .

  s(β T ), θ T ) = [S(β T )] T -P θ T ∂ 2 ∂α 2 h (s(β T ), θ T ) [S(β T )] . It follows that V n = [S(β T )] T U n and B = [S(β T )] T I θ T S(β T ). Combining this result with (6.20), we get 2n φ ′′ (1) D φ (Θ 0 , θ T ) = U T n I -1 θ T -S(β T )B -1 S(β T ) T U n + o p (1),which is precisely the asymptotic expression for the Wilks likelihood ratio statistic for composite hypotheses. The proof is completed following therefore the same arguments as for the Wilks likelihood ratio statistic; see e.g.[START_REF] Sen | Large sample methods in statistics[END_REF] chapter 5.Proof of theorem 3.8. The proofs of part (a) and (b) are similar to the proofs of part (a) and (b) of theorem 3.7, hence they are omitted.(c) Using (3.4) and (3.14 ), we can see that D φ (Θ 0 , θ T ) can be written asD φ (Θ 0 , θ T ) := inf β∈B 0 D φ (s(β), θ T ) = D φ (s(β * ), θ T ) = sup α∈Θ P θ T h(s(β * ), α) = P θ T h(s(β * ), θ T ). (6.21)On the other hand, by a Taylor expansion of D φ (Θ 0 , θ T ) = P n h(s( β), α φ ( β)) in β and α φ ( β) around β * and θ T , we obtainD φ (Θ 0 , θ T ) = P n h(s(β * ), θ T ) + o p (n -1/2 ).Combining this with (6.21) to conclude that√ n D φ (Θ 0 , θ T ) -D φ (Θ 0 , θ T ) = √ n [P n h(s(β * ), θ T ) -P θ T h(s(β * ), θ T )] + o p (1)which, by the CLT, converges to a centred normal variable with variance σ 2 φ (β * , θ T ) = P θ T h(s(β * ), θ T ) 2 -(P θ T h(s(β * ), θ T )) 2 .

  (the domain of φ * ) will be denoted a φ * and b φ * with a φ * ≤ b φ * .

				for proofs. The Fenchel-Legendre transform
	of φ will be denoted φ * , i.e.,
	t ∈ R → φ * (t) := sup x∈R	{tx -φ(x)} ,	(2.1)
	and the endpoints of domφ Note that φ * is a proper closed convex function. In particular, a φ * < 0 < b φ * , φ * (0) = 0
	and
	a φ * = lim y→-∞	φ(y) y	, b φ * = lim y→+∞	φ(y) y	.	(2.2)
	By the closedness of φ, applying the duality principle, the conjugate φ * * of φ * coincides
	with φ, i.e.,
	φ * * (t) := sup x∈R	{tx -φ * (x)} = φ(t), for all t ∈ R.	(2.3)
	For the proper convex functions defined on R (endowed with the usual topology), the lower
	semi-continuity 3 and the closedness properties are equivalent. The function φ (resp. φ * )
	is differentiable if it is differentiable on ]a φ , b φ [ (resp. ]a φ * , b φ * [), the interior of its domain.
	Also φ (resp. φ

* * ) is strictly convex if it is strictly convex on ]a φ , b φ [ (resp. ]a φ * , b φ * [).

3 We say a function φ is lower semi-continuous if the level sets {x ∈ R such that φ(x) ≤ α}, α ∈ R are all closed.

  .4) Conversely, φ is essentially smooth if and only if φ * is strictly convex; see e.g.[START_REF] Rockafellar | Asymptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditions[END_REF] section 26 for the proofs of these properties. If φ is differentiable, we denote φ ′ the derivative function of φ, and we define φ ′ (a φ ) and φ ′ (b φ ) to be the limits (which may be finite or infinite) lim x↓a φ φ ′ (x) and lim x↑b φ φ ′ (x), respectively. We denote Imφ ′ the set of all values of the function φ ′ , i.e., Imφ′ := {φ ′ (x) such that x ∈ [a φ , b φ ]}. Ifadditionally the function φ is strictly convex, then φ ′ is increasing on [a φ , b φ ]. Hence, it is a one-to-one function from [a φ , b φ ] to Imφ ′ . In this case, φ ′ -1 denotes the inverse function of φ ′ from

i.e., the supremum in (2.7) is achieved at f := φ ′ (dQ/dP ) .

Note that this is equivalent to the condition that its conjugate φ * is strictly convex.