Michel Broniatowski
email: michel.broniatowski@upmc.fr

Amor Keziou
email: amor.keziou@upmc.fr

ESTIMATION AND TESTS FOR MODELS SATISFYING LINEAR CONSTRAINTS WITH UNKNOWN PARAMETER

Keywords: Empirical likelihood, Generalized Empirical likelihood, Minimum divergence, Efficiency, Robustness, Duality, Divergence projection. MSC (2000) Classification: 62G05, 62G10, 62G15, 62G20, 62G35. JEL Classification: C12, C13, C14

We introduce estimation and test procedures through divergence minimization for models satisfying linear constraints with unknown parameter. Several statistical examples and motivations are given. These procedures extend the empirical likelihood (EL) method and share common features with generalized empirical likelihood (GEL). We treat the problems of existence and characterization of the divergence projections of probability measures on sets of signed finite measures. Our approach allows for a study of the estimates under misspecification. The asymptotic behavior of the proposed estimates are studied using the dual representation of the divergences and the explicit forms of the divergence projections. We discuss the problem of the choice of the divergence under various respects. Also we handle efficiency and robustness properties of minimum divergence estimates. A simulation study shows that the Hellinger divergence enjoys good efficiency and robustness properties.

Introduction and notation

A model satisfying partly specified linear parametric constraints is a family of distributions M 1 all defined on a same measurable space (X , B), such that, for all Q in M 1 , the following condition holds g(x, θ) dQ(x) = 0.

The unspecified parameter θ belongs to Θ, an open set in R d . The function g := (g 1 , ..., g l ) T is defined on X ×Θ with values in R l , each of the g i 's being real valued and the functions g 1 , . . . , g l , ½ X are assumed linearly independent. So M 1 is defined through l-linear constraints indexed by some d-dimensional parameter θ. Denote M 1 the collection of all probability measures on (X , B), and

M 1 θ := Q ∈ M 1 such that g(x, θ) dQ(x) = 0 so that (1.1) M 1 = θ∈Θ M 1 θ .
Assume now that we have at hand a sample X 1 , ..., X n of independent random variables (r.v.'s) with common unknown distribution P 0 . When P 0 belongs to the model (1.1), we denote θ 0 the value of the parameter θ such that M θ0 contains P 0 . Obviously, we assume that θ 0 is unique.

The scope of this paper is to propose new answers for the classical following problems Problem 1 : Does P 0 belong to the model M 1 ?

Problem 2 : When P 0 is in the model, which is the value θ 0 of the parameter for which g(x, θ 0 ) dP 0 (x) = 0 ? Also can we perform simple and composite tests for θ 0 ? Can we construct confidence areas for θ 0 ? Can we give more efficient estimates for the distribution function than the usual empirical cumulative distribution function (c.d.f.) ?

We present some examples and motivations for the model (1.1) and Problems 1 and 2. Example 1.1. Suppose that P 0 is the distribution of a pair of random variables (X, Y ) on a product space X × Y with known marginal distributions P 1 and P 2 . [START_REF] Bickel | Efficient estimation of linear functionals of a probability measure P with known marginal distributions[END_REF] study efficient estimation of θ = h(x, y) dP 0 (x, y) for specified function h. This problem can be handled in the present context when the spaces X and Y are discrete and finite. Denote X = {x 1 , . . . , x k } and Y = {y 1 , . . . , y r }. Consider an i.i.d. bivariate sample (X i , Y i ), 1 ≤ i ≤ n of the bivariate random variable (X, Y ). The space M θ in this case is the set of all p.m.'s Q on X ×Y satisfying g(x, y, θ) dQ(x, y) = 0 where g = (g

(1) 1 , . . . , g

k , g

(2) 1 , . . . , g

(2) r , g 1 ) T , g

i (x, y, θ) = ½ {xi}×Y (x, y) -P 1 (x i ), g (2) j (x, y, θ) = ½ X ×{yj } (x, y) -P 2 (y j ) for all (i, j)∈{1, . . . , k} × {1, . . . , r}, and g 1 (x, y, θ) = h(x, y)θ. Problem 1 turns to be the test for "P 0 belongs to θ∈Θ M θ ", while Problem 2 pertains to the estimation and tests for specific values of θ. Motivation and references for this problem are given in [START_REF] Bickel | Efficient estimation of linear functionals of a probability measure P with known marginal distributions[END_REF].

An important estimator of θ 0 is the generalized method of moments (GMM) estimator of [START_REF] Hansen | Large sample properties of generalized method of moments estimators[END_REF]. The empirical likelihood approach developed by [START_REF] Owen | Empirical likelihood ratio confidence intervals for a single functional[END_REF] and [START_REF] Owen | Empirical likelihood ratio confidence regions[END_REF] has been adapted in the present setting by [START_REF] Qin | Empirical likelihood and general estimating equations[END_REF] and [START_REF] Imbens | One-step estimators for over-identified generalized method of moments models[END_REF] introducing the empirical likelihood estimator (EL). The recent literature in econometrics focusses on such models, the paper by [START_REF] Newey | Higher order properties of GMM and generalized empirical likelihood estimators[END_REF] provides an exhaustive list of works dealing with the statistical properties of GMM and generalized empirical likelihood (GEL) estimators.

Our interest also lays in the behavior of the estimates under misspecification. In the context of tests of hypothesis, the statistics to be considered is some estimate of some divergence between the unknown distributions of the data and the model. We are also motivated by the behavior of those statistics under misspecification, i.e., when the model is not appropriated to the data. Such questions have not been addressed until now for those problems in the general context of divergences. [START_REF] Schennach | Point estimation with exponentially tilted empirical likelihood[END_REF] consider the asymptotic properties of the empirical likelihood estimate under misspecification. As a by product, we will prove that our proposal leads to consistent test procedures; furthermore, the asymptotic behavior of the statistics, under H 1 , provides the fundamental tool in order to achieve Bahadur efficiency calculations (see [START_REF] Nikitin | Asymptotic efficiency of nonparametric tests[END_REF]).

An important result due to [START_REF] Newey | Higher order properties of GMM and generalized empirical likelihood estimators[END_REF] states that EL estimate enjoys optimality properties in term of efficiency when bias corrected among all GEL and GMM estimators. Also [START_REF] Corcoran | Bertlett adjustement of empirical discrepancy statistics[END_REF] and [START_REF] Baggerly | Empirical likelihood as a goodness-of-fit measure[END_REF] proved that in a class of minimum discrepancy statistics, EL ratio is the only that is Bartlett correctable. However, these results do not consider the optimality properties of the tests for Problems 1 and 2. Also, in connection with estimation problem, they do not consider the properties of EL estimate with respect to robustness. So, the question regarding divergence-based methods remains open at least in these two instances.

The approach which we develop is based on minimum descrepancy estimates, which have common features with minimum distance techniques, using merely divergences. We present wide sets of estimates, simple and composite tests and confidence regions for the parameter θ 0 as well as various test statistics for Problem 1, all depending on the choice of the divergence. Simulations show that the approach based on Hellinger divergence enjoys good robustness and efficiency properties when handling Problem 2. As presented in Section 5, empirical likelihood methods appear to be a special case of the present approach.

1.2. Minimum divergence estimates. We first set some general definition and notation. Let P be some probability measure (p.m.). Denote M 1 (P ) the subset of all p.m.'s which are absolutely continuous (a.c.) with respect to P . Denote M the space of all signed finite measures on (X , B) and M (P ) the subset of all signed finite measures a.c. w.r.t. P . Let ϕ be a convex function from [-∞, +∞] onto [0, +∞] with ϕ(1) = 0. For any signed finite measure Q in M (P ), the φ-divergence between Q and the p.m. P is defined through (1.2) φ(Q, P ) := ϕ dQ dP dP.

When Q is not a.c. w.r.t. P , we set φ(Q, P ) = +∞. This definition extends [START_REF] Rüschendorf | On the minimum discrimination information theorem[END_REF]'s one which applies for φ-divergences between p.m.'s; it also differs from [START_REF] Csiszár | Eine informationstheoretische Ungleichung und ihre Anwendung auf den Beweis der Ergodizität von Markoffschen Ketten[END_REF]'s one, which requires a common dominating σ-finite measure, noted λ, for Q and P . Since we will consider subsets of M 1 (P ) and subsets of M (P ), it is more adequate for our sake to use the definition (1.2). Also note that all the just mentioned definitions of φ-divergences coincide on the set of all p.m.'s a.c. w.r.t. P and dominated by λ.

For all p.m. P , the mappings Q ∈ M → φ(Q, P ) are convex and take nonnegative values. When Q = P then φ(Q, P ) = 0. Further, if the function x → ϕ(x) is strictly convex on neighborhood of x = 1, then the following basic property holds

(1.3) φ(Q, P ) = 0 if and only if Q = P.
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], [START_REF] Csiszár | On topology properties of f -divergences[END_REF] 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 arguments as developed on M 1 hold.

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 x -x + 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 those divergences except the L 1 one, belong to the class of 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

(1.4) x ∈ R * + → ϕ γ (x) := x γ -γx + γ -1 γ(γ -1) if γ ∈ R \ {0, 1} and by ϕ 0 (x) := -log x + x -1 and ϕ 1 (x) := x log x -x + 1. For all γ ∈ R, ϕ γ (0) := lim x↓0 ϕ γ (x) and ϕ γ (+∞) := lim x↑+∞ 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 . For all γ ∈ R, we sometimes denote φ γ the divergence associated to the convex function ϕ γ . We define the derivative of ϕ γ at 0 by ϕ ′ γ (0) := lim x↓0 ϕ ′ γ (x). We extend the definition of the power divergences functions Q ∈ M 1 → φ γ (Q, P ) onto the whole set of signed finite measures M as follows. When the function x → ϕ γ (x) is not defined on (-∞, 0[ or when ϕ γ is defined on R but is not a convex function we extend the definition of ϕ γ through Let Ω be some subset in M . The φ-divergence between the set Ω and a p.m. P , noted φ(Ω, P ), is φ(Ω, P

(1.5) x ∈ [-∞, +∞] → ϕ γ (x)½ [0,+∞] (x) + (ϕ ′ γ (0)x + ϕ γ (0))½ [-∞,0[ (x). For any convex function ϕ, define the domain of ϕ through (1.6) D ϕ = {x ∈ [-∞, +∞] such that ϕ(x) < +∞} . Since ϕ is convex, D ϕ is
) := inf Q∈Ω φ(Q, P ). Definition 1.2. Assume that φ(Ω, P ) is finite. A measure Q * ∈ Ω such that φ(Q * , P ) ≤ φ(Q, P ) for all Q ∈ Ω
is called a φ-projection of P onto Ω. This projection may not exist, or may be not defined uniquely.

We will make use of the concept of φ-divergences in order to perform estimation and tests for the model (1.1). So, let X 1 , ..., X n denote an i.i.d. sample of r.v.'s with common distribution P 0 . Let P n be the empirical measure pertaining to this sample, namely

P n := 1 n n i=1 δ Xi
in which δ x is the Dirac measure at point x. When P 0 and all Q ∈ M 1 share the same discrete finite support S, then the φ-divergence φ(Q, P 0 ) can be written as

(1.7) φ(Q, P 0 ) = j∈S ϕ Q(j) P 0 (j) P 0 (j).
In this case, φ(Q, P 0 ) can be estimated simply through the plug-in of P n in (1.7), as follows

(1.8) φ(Q, P 0 ) := j∈S ϕ Q(j) P n (j) P n (j).
In the same way, for any θ in Θ, φ M 1 θ , P 0 is estimated by

(1.9) φ M 1 θ , P 0 := inf Q∈M 1 θ j∈S ϕ Q(j) P n (j) P n (j),
and φ M 1 , P 0 = inf θ∈Θ φ M 1 θ , P 0 can be estimated by

(1.10) φ M 1 , P 0 := inf θ∈Θ inf Q∈M 1 θ j∈S ϕ Q(j) P n (j) P n (j).
By uniqueness of inf θ∈Θ φ M 1 θ , P 0 and since this infimum is reached at θ = θ 0 , we estimate θ 0 through (1.11)

θ φ := arg inf θ∈Θ inf Q∈M 1 θ j∈S ϕ Q(j) P n (j) P n (j).
The infimum in (1.9) (i.e., the projection of P n on M 1 θ ) may be achieved on the frontier of M 1 θ . In this case the Lagrange method is not valid. Hence, we endow our statistical approach in the global context of signed finite measures with total mass 1 satisfying the linear constraints.

(1.12)

M θ := Q ∈ M such that dQ = 1 and g(x, θ) dQ(x) = 0 and (1.13) M := θ∈Θ M θ ,
sets of signed finite measures that replace M 1 θ and M 1 . As above, we estimate φ(M θ , P 0 ), φ(M, P 0 ) and θ 0 respectively by

(1.14) φ (M θ , P 0 ) := inf Q∈M θ j∈S ϕ Q(j) P n (j) P n (j), (1.15) φ (M, P 0 ) := inf θ∈Θ inf Q∈M θ j∈S ϕ Q(j) P n (j) P n (j),

and

(1.16)

θ φ := arg inf θ∈Θ inf Q∈M θ j∈S ϕ Q(j) P n (j) P n (j).
Enhancing M 1 to M is motivated by the following arguments -For all θ in Θ, denote Q * 1 and Q * respectively the projection of P n on M 1 θ and on M θ , as defined in (1.9) and in (1.14). If Q * 1 is an interior point of M 1 θ , then, by Proposition 2.5 below, it coincides with Q * , the projection of P n on M θ , i.e., Q * 1 = Q * . Therefore, in this case, both approaches coincide.

-It may occur that for some θ in Θ, Q * 1 , the projection of P n on M 1 θ , is a frontier point of M 1 θ , which makes a real difficulty for the estimation procedure. We will prove in Theorem 3.4 that θ φ , defined in (1.16) and which replaces (1.11), converges to θ 0 . This validates the substitution of the sets M 1 θ by the sets M θ . In the context of a test problem, we will prove that the asymptotic distributions of the test statistics pertaining to Problem 1 and 2 are unaffected by this change. This modification motivates the above extensions in the definitions of the ϕ functions on [-∞, +∞] and of the φ-divergences on the whole space of finite signed measures M .

In the case when Q and P 0 share different discrete finite support or share same or different discrete infinite or continuous support, then formula (1.8) is not defined, due to lack of absolute continuity of Q with respect to P n . Indeed (1.17)

φ(Q, P 0 ) := φ(Q, P n ) = +∞. The plug-in estimate of φ(M θ , P 0 ) is (1.18) φ(M θ , P 0 ) := inf Q∈M θ φ(Q, P n ) = inf Q∈M θ ϕ dQ dP n (x) dP n (x).
If the infimum exists, then it is clear that it is reached at a signed finite measure (or probability measure) which is a.c. w.r.t. P n . So, define the sets

(1.19) M (n) θ := Q ∈ M such that Q ≪ P n , n i=1 Q(X i ) = 1 and n i=1 Q(X i )g(X i , θ) = 0 ,
which may be seen as subsets of R n . Then, the plug-in estimate (1.18) of φ(M θ , P 0 ) can be written as

(1.20) φ(M θ , P 0 ) = inf Q∈M (n) θ 1 n n i=1 ϕ (nQ(X i )) .
In the same way, φ(M, P 0 ) := inf θ∈Θ inf Q∈M θ φ(Q, P 0 ) can be estimated by

(1.21) φ(M, P 0 ) = inf θ∈Θ inf Q∈M (n) θ 1 n n i=1 ϕ (nQ(X i )) .
By uniqueness of inf θ∈Θ φ(M θ , P 0 ) and since this infimum is reached at θ = θ 0 , we estimate θ 0 through

(1.22) θ φ = arg inf θ∈Θ inf Q∈M (n) θ 1 n n i=1 ϕ (nQ(X i )) .
Note that, when P 0 and all Q ∈ M 1 share the same discrete finite support, then the estimates (1.22), (1.21) and (1.20) coincide respectively with (1.16), (1.15) and (1.14). Hence, in the sequel, we study the estimates φ(M θ , P 0 ), φ(M, P 0 ) and θ φ as defined in (1.20), (1.21) and (1.22), respectively. We propose to call the estimates θ φ defined in (1.22) "Minimum Empirical φ-Divergences Estimates" (MEφDE's). As will be noticed later on, the empirical likelihood paradigm (see [START_REF] Owen | Empirical likelihood ratio confidence intervals for a single functional[END_REF] and [START_REF] Owen | Empirical likelihood ratio confidence regions[END_REF]), which is based on this plug-in approach, enters as a special case of the statistical issues related to estimation and tests based on φ-divergences with ϕ(x) = ϕ 0 (x) =log x + x -1, namely on KL m -divergence. The empirical log-likelihood ratio for the model (1.12), in the context of φ-divergences, can be written as -n KL m (M θ , P 0 ). In the case of a single functional, for example when g(x, θ) = x-θ with x and θ belong to R, [START_REF] Owen | Empirical likelihood ratio confidence intervals for a single functional[END_REF] shows that 2n KL m (M θ , P 0 ) has an asymptotic χ 2 (1) distribution when P 0 belongs to M θ . (see [START_REF] Owen | Empirical likelihood ratio confidence intervals for a single functional[END_REF] Theorem 1). This result is a nonparametric version of Wilks's theorem (see [START_REF] Wilks | The large-sample distribution of the likelihood ratio for testing composite hypotheses[END_REF]). In the multivariate case, the same result holds (see [START_REF] Owen | Empirical likelihood ratio confidence regions[END_REF] Theorem 1). When we want to extend the arguments used in [START_REF] Owen | Empirical likelihood ratio confidence intervals for a single functional[END_REF] and [START_REF] Owen | Empirical likelihood ratio confidence regions[END_REF] in order to study the limiting behavior of the statistics φ(M θ , P 0 ), when P 0 ∈ M θ (for example, when θ 0 = θ), most limiting arguments become untractable. We propose to use the so-called "dual representation of φ-divergences"(see [START_REF] Keziou | Dual representation of φ-divergences and applications[END_REF]), a device which is well known for the Kullback-Leibler divergence in the context of large deviations, and which has been used in parametric statistics in [START_REF] Keziou | Dual representation of φ-divergences and applications[END_REF] and [START_REF] Broniatowski | Parametric estimation and testing through divergences[END_REF]. The estimates then turn to be M-estimates whose limiting distributions are obtained through classical methods. On the other hand, the obtention of the limit distributions of the statistics φ(M θ , P 0 ) when P 0 ∈ M θ , requires the study of the existence and the characterization of the projection of the p.m. P 0 on the sets M θ . This paper is organized as follows : In Section 3, we study the asymptotic behavior of the proposed estimates (1.20), (1.21) and (1.22) giving solutions to Problem 2. We then address Problem 1, namely : does there exist some θ 0 in Θ for which P 0 belongs to M θ0 ? In Section 4, extending the result by [START_REF] Qin | Empirical likelihood and general estimating equations[END_REF], we give new estimates for the distribution function using the φ-projections of P n on the model M. We show that the new estimates of the distribution function are generally more efficient than the empirical cumulative distribution function. Section 5 illustrates the concept of empirical likelihood in the context of φ-divergences techniques. In Section 6, we focus on robustness and efficiency of the MEφD estimates. A simulation study aims at emphasizing the specific advantage of the choice of the Hellinger divergence in relation with robustness and efficiency considerations. All proofs are in Section 7.

Estimation for Models satisfying Linear Constraints

At this point, we must introduce some notational convention for sake of brevity and clearness. For any p.m. P on X and any measurable real function f on X , P f denotes f (x) dP (x). For example, P 0 g j (θ) will be used instead of g j (θ, x) dP 0 (x). Hence, we are led to define the following functions : denote g the function defined on X × Θ with values in R l+1 by

g : X × Θ → R (l+1) (x, θ) → g(x, θ) := (½ X (x), g 1 (x, θ), . . . , g l (x, θ)) T ,
and for all θ ∈ Θ, denote also g(θ), g(θ), g j (θ) the functions defined respectively by g(θ) : X → R l+1 x → g(x, θ) := (g 0 (x, θ), g 1 (x, θ), . . . , g l (x, θ))

T , where g 0 (x, θ) := ½ X (x), g(θ) : X → R l x → g(x, θ) := (g 1 (x, θ), . . . , g l (x, θ))

T and g j (θ) : X → R x → g j (x, θ), for all j ∈ {0, 1, . . . , l} .

We now turn back to the setting defined in the Introduction and consider model (1.12). For fixed θ in Θ, define the class of functions

F θ := {g 0 (θ), g 1 (θ), . . . , g l (θ)} ,
and consider the set of finite signed measures M θ defined by (l + 1) linear constraints as defined in (1.12)

M θ := Q ∈ M F θ such that dQ(x) = 1 and g(x, θ) dQ(x) = 0 .
We present explicit tractable conditions for the estimates (1.20), (1.21) and (1.22) to be well defined. This will be done in Propositions 2.1, Remark 2.1, Proposition 2.2 and Remark 2.3 below. First, we present sufficient conditions which assess the existence of the infimum in (1.20), noted Q * θ , the projection of P n on M θ . We also provide conditions under which the Lagrange method can be used to characterize Q * θ . The Fenchel-Legendre transform of ϕ will be denoted ϕ * , i.e., (2.1)

t ∈ R → ϕ * (t) := sup x∈R {tx -ϕ(x)} . Define (2.2) D (n) φ := Q ∈ M such that Q ≪ P n and 1 n n i=1 ϕ (nQ(X i )) < ∞ ,
i.e., the domain of the function

(Q(X 1 ), . . . , Q(X n )) T ∈ R n → 1 n n i=1 ϕ (nQ(X i )) .
We have Proposition 2.1. Assume that there exists some measure R in the interior of

D (n) φ and in M (n) θ such that for all Q in ∂D (n)
φ , the frontier of D

(n) φ , we have

(2.3) 1 n n i=1 ϕ (nR(X i )) < 1 n n i=1 ϕ (nQ(X i )) .
Then the following holds

(i) there exists an unique Q * θ in M (n) θ such that (2.4) inf Q∈M (n) θ 1 n n i=1 ϕ (nQ(X i )) = 1 n n i=1 ϕ n Q * θ (X i ) (ii) Q * θ is an interior point of D (n) φ
and satisfies for all i = 1, . . . , n

(2.5) Q * θ (X i ) = 1 n ← - ϕ ′   l j=0 c j g j (X i , θ)   ,
where ( c 0 , c 1 , . . . , c l ) T := c θ is solution of the system of equations

(2.6)    ← - ϕ ′ c 0 + l i=1 c i g i (x, θ) dP n (x) = 1 g j (x, θ) ← - ϕ ′ c 0 + l i=1 c i g i (x, θ) dP n (x) = 0, j = 1, . . . , l.
Example 2.1. For the χ 2 -divergence, we have D

(n) χ 2 = R n . Hence condition (2.3) holds whenever M (n) θ
is not void. Therefore, the above Proposition holds always independently upon the distribution P 0 . More generally, the above Proposition holds for any φ-divergence which is associated to ϕ function satisfying D ϕ = R. (See (1.6) for the definition of D ϕ ).

Example 2.2. In the case of the modified Kullback-Leibler divergence, which turns to coincide with the empirical likelihood technique (see Section 5), we have D

(n) KLm = (]0, +∞[) n . For α in Θ, define the assertion (2.7)
there exists q = (q 1 , ..., q n ) in R n with 0 < q i < 1 for all i = 1, ..., n and n i=1 q i g j (X i , α) = 0 for all j = 1, ..., l. A sufficient condition, in order to assess that condition (2.3) in the above Proposition holds, is when (2.7) holds for α = θ. In the case when g(x, θ) = xθ, this is precisely what is checked in [START_REF] Owen | Empirical likelihood ratio confidence regions[END_REF]), p. 100, when θ is an interior point of the convex hull of (X 1 , ..., X n ).

Example 2.3. For the modified χ 2 -divergence, we have D

(n) χ 2 m = (]0, ∞[)
n , and therefore, condition

(2.7) for α = θ is sufficient for the condition (2.3) to holds. So, conditions which assess the existence of the projection Q * θ are the same for the modified χ 2 -divergence and the KL m -divergence.

Remark 2.1. If there exists some

Q 0 ∈ M (n) θ such that (2.8) a < inf i nQ 0 (X i ) ≤ sup i nQ 0 (X i ) < b,
then applying Corollary 2.6 in [START_REF] Borwein | Duality relationships for entropy-like minimization problems[END_REF], we get inf

Q∈M (n) θ 1 n n i=1 ϕ (nQ(X i )) = sup t∈R (l+1) t 0 -ψ t T g(x, θ) dP n (x)
with dual attainement. Furthermore, if

ϕ ′ (a) < inf i c T θ g(X i , θ) ≤ sup i c T θ g(X i , θ) < ϕ ′ (b),
with c θ a dual optimal, then the unique projection

Q * θ of P n on M (n) θ
is given by (2.5).

We will make use of the dual representation of φ-divergences (see [START_REF] Broniatowski | Minimization of φ-divergences on sets of signed measures[END_REF] theorem 4.4). So, define

(2.9) C θ := t ∈ R l+1 such that t T g(., θ) belongs to Im ϕ ′ (P 0a.s.) ,

and

(2.10)

C (n) θ := t ∈ R l+1 such t T g(X i , θ) belongs to Im ϕ ′ for all i = 1, . . . , n .
We omit the subscript θ when unnecessary. Note that both C θ and C

(n) θ depend upon the function ϕ but, for simplicity, we omit the subscript ϕ.

If P 0 admits a projection Q * θ on M θ with the same support as P 0 , using the second part in Corollary 3.5 in [START_REF] Broniatowski | Minimization of φ-divergences on sets of signed measures[END_REF], there exist constants c 0 , . . . , c l , obviously depending on θ, such that

ϕ ′ dQ * θ dP 0 (x) = c 0 + l j=1
c j g j (x, θ), for all x (P 0a.s.).

Since Q * θ belongs to M θ , the real numbers c 0 , c 1 , . . . , c l are solutions of (2.11)

   ϕ ′ -1 c 0 + l j=1 c j g j (x, θ) dP 0 (x) = 1 g j (x, θ)ϕ ′ -1 c 0 + l j=1 c j g j (x, θ) dP 0 (x) = 0, j = 1, . . . , l.
Since Q → φ(Q, P 0 ) is strictly convex, the projection Q * θ of P 0 on the convex set M θ is unique. This implies, by [START_REF] Broniatowski | Minimization of φ-divergences on sets of signed measures[END_REF] Corollary part 1, that the solution c θ := (c 0 , c 1 , . . . , c l ) T of the system (2.11) is unique provided that the functions g i (θ) are linearly independent. Further, using the dual representation of φ-divergences (see [START_REF] Broniatowski | Minimization of φ-divergences on sets of signed measures[END_REF] Theorem 4.4), we get

φ(M θ , P 0 ) := φ(Q * θ , P 0 ) = sup f ∈F f dQ * θ -ϕ * (f ) dP 0 ,
and the sup is unique and is reached at

f = ϕ ′ (dQ * θ /dP 0 ) = c 0 + l j=1 c j g j (., θ)
, if it belongs to F . This motivates the choice of the class F through

F := x → t T g(x, θ) for t in C θ .
It is the smallest class of functions that contains ϕ ′ (dQ * θ /dP 0 ) and which does not presume any knowledge on Q * θ . We thus obtain

φ(M θ , P 0 ) = sup t∈C θ m(x, θ, t) dP 0 (x),
where m(θ, t) is the function defined on X by

x ∈ X → m(x, θ, t) := t 0 -ϕ * t T g(x, θ) = t 0 -t T g(x, θ) ϕ ′ -1 t T g(x, θ) + ϕ ϕ ′ -1 t T g(x, θ) .
With the above notation, we state (2.12) φ(M θ , P 0 ) = sup

t∈C θ P 0 m(θ, t).
So, a natural estimate of φ(M θ , P 0 ) is

(2.13) sup t∈C (n) θ P n m(θ, t)
which coincides with the estimate defined in (1.20). Hence, we can write

(2.14) φ(M θ , P 0 ) = sup t∈C (n) θ P n m(θ, t).
which transforms the constrained optimization in (1.20) into the above unconstrained one.

On the other hand, the sup in (2.12) is reached at t 0 = c 0 , . . . , t l = c l which are solutions of the system of equations (2.11), i.e.,

(2.15) c θ = arg sup

t∈C θ P 0 m(θ, t).
So, a natural estimate of c θ in (2.15) is therefore defined through

(2.16) arg sup t∈C (n) θ P n m(θ, t).
This coincides with c θ , the solution of the system of equations (2.6). So, we can write

(2.17)

c θ = arg sup t∈C (n) θ P n m(θ, t).
Using (2.14), we obtain the following representations for the estimates φ(M, P 0 ) in (1.21) and θ φ in (1.22)

(2.18) φ(M, P 0 ) = inf θ∈Θ sup t∈C (n) θ P n m(θ, t) and (2.19) θ φ = arg inf θ∈Θ sup t∈C (n) θ P n m(θ, t),
respectively.

Formula (2.12) also has the following basic interest : Consider the function

(2.20) t ∈ C θ → P 0 m(θ, t),
In order for integral (2.20) to be properly defined, we assume that (2.21)

|g i (x, θ)| dP 0 (x) < ∞, for all i ∈ {1, . . . , l} . The domain of the function (2.20) is (2.22) D φ (θ) := {t ∈ C θ such that P 0 m(θ, t) > -∞} .
The function t → P 0 m(θ, t) is strictly concave on the convex set D φ (θ). Whenever it has a maximum t * , then it is unique, and if it belongs to the interior of D φ (θ), then it satisfies the first order condition. Therefore t * satisfies system (2.11). In turn, this implies that the measure Q * defined through dQ * := ϕ ′ -1 t * T g(θ) dP 0 is the projection of P 0 on Ω, by Theorem 3.4 part 1 in [START_REF] Broniatowski | Minimization of φ-divergences on sets of signed measures[END_REF]. This implies that Q * and P 0 share the same support. We summarize the above arguments as follows Proposition 2.2. Assume that (2.21) holds and that (i) there exists some s in the interior of D φ (θ) such that for all t in ∂D φ (θ), the frontier of D φ (θ), it holds P 0 m(θ, t) < P 0 m(θ, s); (ii) for all t in the interior of D φ (θ), there exists a neighborhood V (t) of t, such that the classes of functions x → ∂ ∂ri m(x, θ, r), r ∈ V (t) are dominated (P 0 -a.s.) by some P 0 -integrable function x → H(x, θ). Then P 0 admits an unique projection Q * θ on M θ having the same support as P 0 and (2.23)

dQ * θ = ϕ ′-1 c θ T g(θ) dP 0 ,
where c θ is the unique solution of the system of equations (2.11).

Remark 2.2. In the case of KL-divergence, comparing this Proposition with Theorem 3.3 in [START_REF] Csiszár | I-divergence geometry of probability distributions and minimization problems[END_REF], we observe that the dual formula (2.12) provides weaker conditions on the class of functions {g(θ), θ ∈ Θ} than the geometric approach.

Remark 2.3. The result of [START_REF] Borwein | Duality relationships for entropy-like minimization problems[END_REF], with some additional conditions, provides more practical tools for obtaining the results in Proposition 2.2. Assume that the functions g j (θ) belongs to the space L p (X , P 0 ) with 1 ≤ p ≤ ∞ and that the following "constraint qualification" holds

(2.24) there exists some

Q 0 in M θ such that : a < inf dQ 0 dP 0 ≤ sup dQ 0 dP 0 < b,
with (a, b) is the domain D ϕ of the divergence function ϕ and M θ is the set of all signed measures Q a.c. w.r.t. P 0 , satisfying the linear constraints and such that dQ dP0 belong to L q (X , P 0 ), (1 ≤ q ≤ ∞ and 1/p + 1/q = 1). In this case, applying Corollary 2 in [START_REF] Borwein | Duality relationships for entropy-like minimization problems[END_REF], we obtain

φ (M θ , P 0 ) = sup t∈R (l+1) t 0 -ϕ * t T g(x, θ) dP 0 (x)
(with dual attainement). Furthermore, if for a dual optimal c θ , it holds

lim y↓-∞ ϕ(y) y < inf x c T θ g(x, θ) ≤ sup x c T θ g(x, θ) < lim y↑+∞ ϕ(y) y
for all x (P 0 a.s.),

then the unique projection Q * θ of P 0 on M θ is given by (2.25) dQ * θ = ϕ * ′ c θ T g(θ) dP 0 .
Note that if ϕ * is strictly convex, then c θ is unique and

sup t∈R (l+1) t 0 -ϕ * t T g(x, θ) dP 0 (x) = sup t∈C θ t 0 -ϕ * t T g(x, θ) dP 0 (x) , and
ϕ * ′ c θ T g(θ) = ϕ ′ -1 c θ T g(θ) .
Léonard (2001a) and [START_REF] Léonard | Minimizers of energy functionals[END_REF] gives, under minimal conditions, duality theorems of minimum φ-divergences and characterization of projections under linear constraints, which generalize the results given by [START_REF] Borwein | Duality relationships for entropy-like minimization problems[END_REF] and [START_REF] Borwein | Partially-finite programming in L 1 and the existence of maximum entropy estimates[END_REF]. These results are used recently by [START_REF] Bertail | Empirical likelihood in nonparametric and semiparametric models. In Parametric and semiparametric models with applications to reliability, survival analysis, and quality of life[END_REF] and [START_REF] Bertail | Empirical likelihood in some semiparametric models Bernoulli[END_REF] in empirical likelihood.

Asymptotic properties and Statistical Tests

In the sequel, we assume that the conditions in Proposition 2. [START_REF] Sen | Large sample methods in statistics[END_REF]. We first consider the case when θ is fixed, and we study the asymptotic behavior of the estimate φ(M θ , P 0 ) (see (1.20)) of φ(M θ , P 0 ) := inf Q∈M θ φ(Q, P 0 ) both when P 0 ∈ M θ and when P 0 ∈ M θ . This is done in the first Subsection.

In the second Subsection, we study the asymptotic behavior of the EMφD estimates θ φ and the estimates φ (M, P 0 ) both in the two cases when P 0 belongs to M and when P 0 does not belong to M. The solution of Problem 1 is given in Subsection 3.3 while Problem 2 is treated in Subsections 3.1, 3.2, 3.3 and 3.4.

3.1. Asymptotic properties of the estimates for a given θ ∈ Θ. First we state consistency.

Consistency. We state both weak and strong consistency of the estimates c θ and φ(M θ , P 0 ) using their representations (2.17) and (2.14), respectively. Denote . the Euclidian norm defined on R d or on R l+1 . In order to state consistency, we need to define

T θ := {t ∈ C θ such that P 0 m(θ, t) > -∞} ,
and denote T c θ the complementary of the set T θ in the set C θ , namely T c θ := {t ∈ C θ such that P 0 m(θ, t) = -∞} . Note that, by Proposition 2.2, the set T θ contains c θ .

We will consider the following condition (C.1) sup t∈T θ |P n m(θ, t) -P 0 m(θ, t)| converges to 0 a.s. (resp. in probability);

(C.2) there exists M < 0 and n 0 > 0, such that, for all n > n 0 , it holds sup t∈T c θ P n m(θ, t) ≤ M a.s. (resp. in probability).

The condition (C.2) makes sense, since for all t ∈ T c θ we have P 0 m(θ, t) = -∞.

Since the function t ∈ T θ → P 0 m(θ, t) is strictly concave, the maximum c θ is isolated, that is (3.1) for any positive ǫ, we have sup

{t∈C θ : t-c θ ≥ǫ} P 0 m(θ, t) < P 0 m(θ, c θ ).
Proposition 3.1. Assume that conditions (C.1) and (C.2) hold. Then (i) the estimates φ(M θ , P 0 ) converge to φ(M θ , P 0 ) a.s. (resp. in probability).

(ii) the estimates c θ converge to c θ a.s. (resp. in probability).

Asymptotic distributions. Denote m ′ (θ, t) the (l+1)-dimensional vector with entries ∂ ∂ti m(θ, t), m ′′ (θ, t) the (l + 1) × (l + 1)-matrix with entries ∂ 2 ∂ti∂tj m(θ, t), 0 l := (0, . . . , 0) T ∈ R l , 0 d := (0, . . . , 0) T ∈ R d , c the (l + 1)-vector defined by c := 0, 0 T l T , and P 0 g(θ)g(θ) T the l × l-matrix defined by

P 0 g(θ)g(θ) T := [P 0 g i (θ)g j (θ)] i,j=1,...,l .
We will consider the following assumptions (A.1) c θ converges in probability to c θ ; (A.2) the function t → m(x, θ, t) is C 3 on a neighborhood V (c θ ) of c θ for all x (P 0 -a.s.), and all partial derivatives of order 3 of the function {t → m(x, θ, t), t ∈ V (c θ )} are dominated by some P 0 -integrable function x → H(x); (A.3) P 0 m ′ (θ, c θ ) 2 is finite, and the matrix P 0 m ′′ (θ, c θ ) exists and is invertible.

Theorem 3.2. Assume that assumptions (A.1-3) hold. Then

(1) √ n ( c θc θ ) converges to a centered normal multivariate variable with covariance matrix

(3.2) V = [-P 0 m ′′ (θ, c θ )] -1 P 0 m ′ (θ, c θ )m ′ (θ, c θ ) T [-P 0 m ′′ (θ, c θ )] -1 .
In the special case, when P 0 belongs to M θ , then c θ = c and

(3.3) V = ϕ ′′ (1) 2 0 0 T l 0 l P 0 g(θ)g(θ) T -1 .
(2) If P 0 belongs to M θ , then the statistics

2n ϕ ′′ (1) φ (M θ , P 0 )
converge in distribution to a χ 2 variable with l degrees of freedom.

(3) If P 0 does not belong to M θ , then √ n φ (M θ , P 0 ) -φ(M θ , P 0 )
converges to a centered normal variable with variance

σ 2 := P 0 m(θ, c θ ) 2 -(P 0 m(θ, c θ )) 2 .
Remark 3.1.

(a) When specialized to the modified Kullback-Leibler divergence, Theorem 3.2 part (2) gives the limiting distribution of the empirical log-likelihood ratio 2n KL m (M θ , P 0 ) which is the result in [START_REF] Owen | Empirical likelihood ratio confidence regions[END_REF] Theorem 1. Part (3) gives its limiting distribution when P 0 does not belong to M θ . (b) Nonparametric confidence regions (CR φ ) for θ 0 of asymptotic level (1-ǫ) can be constructed using the statistics

2n ϕ ′′ (1) φ (M θ , P 0 ) , through CR φ := θ ∈ Θ such that 2n ϕ ′′ (1) φ (M θ , P 0 ) ≤ q (1-ǫ) ,
where (1ǫ) is the (1ǫ)-quantile of a χ 2 (l) distribution. It would be interesting to obtain the divergence leading to optimal confidence regions in the sense of [START_REF] Neyman | Outline of a theory of statistical estimation based on the classical theory of probability[END_REF] (see [START_REF] Takagi | A new criterion of confidence set estimation: improvement of the Neyman shortness[END_REF]), or the optimal divergence leading to confidence regions with small length (volume, area or diameter) and covering the true value θ 0 with large enough probability.

3.2. Asymptotic properties of the estimates θ φ and φ(M, P 0 ). First we state consistency.

Consistency. We assume that when P 0 does not belong to the model M, the minimum, say θ * , of the function θ ∈ Θ → inf Q∈M θ φ(Q, P 0 ) exists and is unique. Hence P 0 admits a projection on M which we denote Q * θ * . Obviously when P 0 belongs to the model M, then θ * = θ 0 and Q * θ * = P 0 . We will consider the following conditions (C.3) sup {θ∈Θ,t∈T θ } |P n m(θ, t) -P 0 m(θ, t)| tends to 0 a.s. (resp. in probability); (C.4) there exists a neighborhood V (c θ * ) of c θ * such that (a) for any positive ǫ, there exists some positive η such that for all t ∈ V (c θ * ) and all θ ∈ Θ satisfying θθ * ≥ ǫ, it holds P 0 m(θ * , t) < P 0 m(θ, t)η; (b) there exists some function H such that for all t in V (c θ * ), we have |m(t, θ 0 )| ≤ H(x) (P 0 -a.s.) with P 0 H < ∞; (C.5) there exits M < 0 and n 0 > 0 such that for all n ≥ n 0 , we have (ii) sup θ∈Θ c θc θ converge to 0 a.s. (resp. in probability).

(iii) The MEφD estimates θ φ converge to θ * a.s. (resp. in probability).

Asymptotic distributions. When P 0 ∈ M, then by assumption, there exists unique θ 0 ∈ Θ such that P 0 ∈ M θ0 . Hence θ * = θ 0 and c θ * = c θ0 = c. We state the limit distributions of the estimates θ φ and c θ φ when P 0 ∈ M and when P 0 ∈ M. We will make use of the following assumptions (A.4) Both estimates θ φ and c θ φ converge in probability respectively to θ * and c θ * ; (A.5) the function (θ, t) → m(x, θ, t) is C 3 on some neighborhood V (θ * , c θ * ) for all x (P 0 -a.s.), and the partial derivatives of order 3 of the functions {(θ, t) → m(x, θ, t), (θ, t) ∈ V (θ * , c θ * )} are dominated by some P 0 -integrable function H(x); Theorem 3.4. Let P 0 belongs to M and assumptions (A.4-6) hold. Then, both √ n θ φθ 0 and √ n c θ φc converge in distribution to a centered multivariate normal variable with covariance matrix, respectively

(3.5) V = P 0 ∂ ∂θ g(θ 0 ) P 0 g(θ 0 )g(θ 0 ) T -1 P 0 ∂ ∂θ g(θ 0 ) T -1
, and

U = ϕ ′′ (1) 2 0 0 T l 0 l P 0 g(θ 0 )g(θ 0 ) T -1 -ϕ ′′ (1) 2 0 0 T l 0 l P 0 g(θ 0 )g(θ 0 ) T -1 × × 0 d , P 0 ∂ ∂θ g(θ 0 ) T V 0 d , P 0 ∂ ∂θ g(θ 0 ) 0 0 T l 0 l P 0 g(θ 0 )g(θ 0 ) T -1 ,
and the estimates θ φ and c θ φ are asymptotically uncorrelated.

Remark 3.2. When specialized to the modified Kullback-Leibler divergence, the estimate θ KLm is the empirical likelihood estimate (ELE) (noted θ in Qin and Lawless (1994)), and the above result gives the limiting distribution of √ n( θ KLmθ 0 ) which coincides with the result in Theorem 1 in [START_REF] Qin | Empirical likelihood and general estimating equations[END_REF]. Note also that all MEφDE's including ELE have the same limiting distribution with the same variance when P 0 belongs to M. Hence they are all equally first order efficient.

Theorem 3.5. Assume that P 0 does not belong to M and that assumptions (A.4-6) hold. Then

√ n c θ φ -c θ * θ φ -θ *
converges in distribution to a centered multivariate normal variable with covariance matrix

W = S -1 M S -1
where

M := P 0 ∂ ∂t m (θ * , c θ * ) ∂ ∂θ m (θ * , c θ * ) ∂ ∂t m (θ * , c θ * ) ∂ ∂θ m (θ * , c θ * ) T .
θ * and c θ * are characterized by

θ * := arg inf θ∈Θ φ (M θ , P 0 ) , dQ * θ * = ϕ ′ -1 c T θ * g(θ) dP 0 and Q * θ * ∈ M θ * . 3.
3. Tests of model. In order to test the hypothesis H 0 : P 0 belongs to M against the alternative H 1 : P 0 does not belong to M, we can use the estimates φ(M, P 0 ) of φ(M, P 0 ), the φ-divergences between the model M and the distribution P 0 . Since φ(M, P 0 ) is nonnegative and take value 0 only when P 0 belongs to M (provided that P 0 admits a projection on M), we reject the hypothesis H 0 when the estimates take large values. In the following Corollary, we give the asymptotic law of the estimates φ(M, P 0 ) both under H 0 and under H 1 .

Corollary 3.6.

(i) Assume that the assumptions of Theorem 3.4 hold and that l > d. Then, under H 0 , the statistics 2n ϕ ′′ (1) φ(M, P 0 ) converge in distribution to a χ 2 variable with (ld) degrees of freedom.

(ii) Assume that the assumptions of Theorem 3.5 hold. Then, under H 1 , we have :

(3.6) √ n φ(M, P 0 ) -φ(M, P 0 )
converges to centered normal variable with variance

σ 2 = P 0 m(θ * , c θ * ) 2 -(P 0 m(θ * , c θ * )) 2
where θ * and c θ * satisfy

θ * := arg inf θ∈Θ φ (M θ , P 0 ) , ϕ ′ dQ * θ * dP 0 (x) = c T θ * g(x, θ * ) and Q * θ * ∈ M θ * .
Remark 3.3. This Theorem allows to perform tests of model of asymptotic level α; the critical regions are

(3.7) C φ := 2n ϕ ′′ (1) φ(M, P 0 ) > q (1-α) ,
where q (1-α) is the (1-α)-quantile of the χ 2 distribution with (l-d) degrees of freedom. Also these tests are all asymptotically powerful, since the estimates φ(M, P 0 ) are n-consistent estimates of φ(M, P 0 ) = 0 under H 0 and √ n-consistent estimates of φ(M, P 0 ) under H 1 .

We assume now that the p.m. P 0 belongs to M. We will perform simple and composite tests on the parameter θ 0 taking into account of the information P 0 ∈ M.

3.4.

Simple tests on the parameter. Let (3.8)

H 0 : θ 0 = θ 1 versus H 1 : θ 0 ∈ Θ \ {θ 1 },
where θ 1 is a given known value. We can use the following statistics to perform tests pertaining to (3.8)

S φ n := φ(M θ1 , P 0 ) -inf θ∈Θ φ(M θ , P 0 ). Since φ(M θ1 , P 0 ) -inf θ∈Θ φ(M θ , P 0 ) = φ(M θ1 , P 0 )
are nonnegative and take value 0 only when θ 0 = θ 1 , we reject the hypothesis H 0 when the statistics S φ n take large values.

We give the limit distributions of the statistics S φ n in the following Corollary which we can prove using some algebra and arguments used in the proof of Theorem 3.4 and Theorem 3.5.

Corollary 3.7.

(i) Assume that assumptions of Theorem 3.4 hold. Then under H 0 , the statistics 2n ϕ ′′ (1) S φ n converge in distribution to χ 2 variable with d degrees of freedom. (ii) Assume that assumptions of Theorem 3.4 hold. Then under H 1 , √ n S φ nφ (M θ1 , P 0 ) converges to a centered normal variable with variance

σ 2 = P 0 m(θ 1 , c θ1 ) 2 -(P 0 m(θ 1 , c θ1 )) 2 .
Remark 3.4. When specialized to the KL m -divergence, the statistic 2nS KLm n is the empirical likelihood ratio statistic (see [START_REF] Qin | Empirical likelihood and general estimating equations[END_REF] Theorem 2). We consider the composite test

(3.11) H 0 : θ 0 ∈ Θ 0 versus H 1 : θ 0 ∈ Θ \ Θ 0 ,
i.e., the test (3.12)

H 0 : P 0 ∈ θ∈Θ0 M θ versus H 1 : P 0 ∈ θ∈Θ\Θ0 M θ .
This test is equivalent to the following one

(3.13) H 0 : θ 0 ∈ f (B 0 ) versus H 1 : θ 0 ∈ f (B 0 ),
where f : R (d-k) → R d is a function such that the matrix G(β) := ∂ ∂β g(β) exists and has rank (dk), and B 0 := β ∈ R (d-k) such that f (β) ∈ Θ 0 . Therefore θ 0 ∈ Θ 0 is an equivalent statement for θ 0 = f (β 0 ), β 0 ∈ B 0 .

The following statistics are used to perform tests pertaining to (3.13) :

T φ n := inf β∈B0 φ M f (β) , P 0 -inf θ∈Θ φ (M θ , P 0 ) . Since inf β∈B0 φ(M f (β) , P 0 ) -inf θ∈Θ φ(M θ , P 0 ) = inf β∈B0 φ(M f (β) , P 0 )
are nonnegative and take value 0 only when H 0 holds, we reject the hypothesis H 0 when the statistics T φ n take large values.

We give the limit distributions of the statistics T φ n in the following Corollary.

Corollary 3.8.

(i) Assume that assumptions of Theorem 3.4 hold. Under H 0 , the statistics T φ n converge in distribution to a χ 2 variable with (dk) degrees of freedom.

(ii) Assume that there exists β * ∈ B 0 , such that β * = arg inf β∈B0 φ M f (β) , P 0 . If the assumptions of Theorem 3.5 hold for θ

* = f (β * ), then √ n T φ n -φ (M θ * , P 0 )
converges to a centered normal variable with variance

σ 2 = P 0 m(θ * , c θ * ) 2 -(P 0 m(θ * , c θ * )) 2 .

Estimates of the distribution function through projected distributions

In this Subsection, the measurable space (X , B) is (R, B R ). For all φ-divergence, by (1.21), we have

φ (M, P 0 ) = φ (M, P n ) = φ Q * θ φ , P n .
Proposition 2.4 above provides the description of Q * θ φ .

So, for all φ-divergence, we estimate the distribution function

F using Q * θ φ the φ-projection of P n on M, through F n (x) := n i=1 Q * θ φ (X i )½ (-∞,x] (X i ) = 1 n n i=1 ← - ϕ ′ c θ φ T g(X i , θ φ ) ½ (-∞,x] (X i ).
* θ φ (X i ) = 1 n ,
and F n (x), in this case, is the empirical cumulative distribution function, i.e.,

F n (x) = F n (x) := 1 n n i=1 ½ (-∞,x] (X i ).
So, the main interest is in the case where (4.2) does not admit a solution, that is in general when l > d.

Remark 4.2. The φ-projections Q * θ φ of P n on M may be signed measures. For all φ-divergence

satisfying D ϕ = R * + , the φ-projection Q * θ φ
is a p.m. if it exists. (for example, KL m , KL, Hellinger, and χ 2 m divergences all provide p.m.'s).

We give the limit law of the estimates F n of the distribution function F in the following Theorem. We will see that the estimate F n (x) is generally more efficient than the empirical cumulative distribution function F n (x).

Theorem 4.1. Under the assumptions of Theorem 3.4, √ n F n (x) -F (x) converges in distribution to a centered normal variable with variance

(4.3) W (x) = F (x) (1 -F (x)) -P 0 g(θ 0 )½ (-∞,x] T Γ P 0 g(θ 0 )½ (-∞,x] ,
with

Γ = P 0 g(θ 0 )g(θ 0 ) T -1 -P 0 g(θ 0 )g(θ 0 ) T -1 P 0 ∂ ∂θ g(θ 0 ) T V × × P 0 ∂ ∂θ g(θ 0 ) P 0 g(θ 0 )g(θ 0 ) T -1 ,
and

V = P 0 ∂ ∂θ g(θ 0 ) P 0 g(θ 0 )g(θ 0 ) T -1 P 0 ∂ ∂θ g(θ 0 ) T -1
.

Empirical likelihood and related methods

In the present setting, the empirical likelihood (EL) approach for the estimation of the parameter θ 0 can be summarized as follows. For any θ in Θ, define the profile likelihood ratio of the sample

X := (X 1 , ..., X n ) through L n (θ) := sup n i=1 nQ(X i ) where Q(X i ) ≥ 0, n i=1 Q(X i ) = 1, n i=1 g(X i , θ)Q(X i ) = 0 .
The estimate of θ 0 through empirical likelihood (EL) approach is then defined by (5.1)

θ EL := arg sup θ∈Θ L n (θ).
The paper by [START_REF] Qin | Empirical likelihood and general estimating equations[END_REF] introduces θ EL and presents its properties. In this Section, we show that θ EL belongs to the family of MEφD estimates for the specific choice ϕ(x) = log x + x -1. We also discuss the problem of the existence of the solution of (5.1) for all n.

When ϕ(x) =log x+x-1, formula (1.22) clearly coincides with θ EL . For test of hypotheses given by H 0 : P 0 ∈ M θ against H 1 : P 0 ∈ M θ or for construction of nonparametric confidence regions for θ 0 , the statistic 2n KL m (M θ , P 0 ) coincides with the empirical log-likelihood ratio introduced in Owen (1988), [START_REF] Owen | Empirical likelihood ratio confidence regions[END_REF] and [START_REF] Qin | Empirical likelihood and general estimating equations[END_REF]. We state the results of Section 3 in the present context. We will see that the approach of empirical likelihood by divergence minimization, using the dual representation of the KL m -divergence and the explicit form of the KL m -projection of P 0 , yields to the limit distribution of the statistic 2n KL m (M θ , P 0 ) under H 1 , which can not be achieved using the approach in [START_REF] Owen | Empirical likelihood ratio confidence regions[END_REF] and [START_REF] Qin | Empirical likelihood and general estimating equations[END_REF]. Consider

θ KLm = arg inf θ∈Θ KL m (M θ , P 0 ) where (5.2) KL m (M θ , P 0 ) = sup t∈C θ P n m(θ, t) with ϕ(x) = ϕ 0 (x) = -log x + x -1. The explicit form of m(θ, t) in this case is x → m(x, θ, t) = t 0 -t T g(x, θ) 1 1 -t T g(x, θ) + log 1 -t T g(x, θ) + 1 1 -t T g(x, θ) -1. = t 0 + log 1 -t T g(x, θ) . (5.3)
For fixed θ ∈ Θ, the sup in (5.2), which we have noted c θ , satisfies the following system (5.4)

   1 1-c0-l j=1 cj gj (x,θ) dP n (x) = 1 gj (x,θ) 1-c0-l
j=1 cj gj (x,θ) dP n (x) = 0, for all j = 1, ..., l a system of (l + 1) equations and (l + 1) variables. The projection Q * θ is then obtained using Proposition 2.1 part (ii). We have for all i ∈ {1, . . . , n} 1

Q * θ (X i ) = n   1 -c 0 - n j=1 c j g j (X i , θ)  
which, multiplying by Q * θ (X i ) and summing upon i yields c 0 = 0. Therefore the system (5.4) reduces to the system (3.3) in [START_REF] Qin | Empirical likelihood and general estimating equations[END_REF] replacing c 1 , . . . , c l by -t 1 , . . . , -t l . Simplify (5.3) plugging t 0 = 0. Notice that 2n KL m (M θ , P 0 ) = l E (θ 0 ) in the notation of [START_REF] Qin | Empirical likelihood and general estimating equations[END_REF], and that the function of t = (0, -τ 1 , . . . , -τ l ) defined by t → P n m(θ, t) coincide with the function τ → P n log 1 + τ T g(., θ)

used in [START_REF] Qin | Empirical likelihood and general estimating equations[END_REF]. The interest in formula (5.2) lays in the obtention of the limit distributions of 2n KL m (M θ , P 0 ) under H 1 . By Theorem 3.2, we have

√ n KL m (M θ , P 0 ) -KL m (M θ , P 0 )
converges to a normal distribution variable, which proves consistency of the test; this results cannot be obtained by the Qin and Lawless (1994)'s approach.

and φ( Q * θ , P n ) is finite. The estimation of θ 0 is achieved minimizing φ(M θ , P 0 ) on the sets

Θ φ n := θ ∈ Θ such that M (n) θ ∩ D (n) φ is not void .
Clearly the description of Θ φ n depends on the divergence φ. Consider the following example, with n = 2, X = (X 1 , X 2 ) and g(x, θ) = xθ. Then M θ = (q 1 , q 2 ) T such that q 1 + q 2 = 1 and q 1 (X 1θ) + q 2 (X 2θ) = 0 and

D (2) φ = (q 1 , q 2 ) such that 1 2 2 i=1 ϕ(2q i ) < ∞ . When φ = KL m , then D (2) KLm = R * + × R * + . So, according to the value of θ, M (n) θ ∩ D (n) KLm
may be void and therefore Θ KLm n has a complex structure. At the opposite, for example when φ = χ 2 , then D

(2)

χ 2 = R 2 . Hence M (n) θ ∩D (n) φ = M (n) θ
which is not void for all θ and hence Θ χ 2 n = Θ.

On the other hand, we have for any φ-divergence

θ φ := arg inf θ∈Θ inf Q∈M (n) θ φ(Q, P 0 ) = arg inf θ∈Θ inf Q∈M (n) θ ∩D (n) φ φ(Q, P 0 ).
When D

(n) φ = R n , the infimum in θ above should be taken upon Θ φ n which might be quite cumbersome. [START_REF] Owen | Empirical Likelihood[END_REF] indeed mentions such a difficulty.

In relation to this problem, [START_REF] Qin | Empirical likelihood and general estimating equations[END_REF] bring some asymptotic arguments in the case of the empirical likelihood. They show that there exists a sequence of neighborhoods

V n (θ 0 ) := θ such that θ -θ 0 ≤ n -1/3
on which, with probability one as n tends to infinity, L n (θ) has a maximum. This turns out, in the context of φ-divergences, to write that the mapping

θ → inf Q∈M (n) θ KL m (Q, P n )
has a minimum when θ belongs to V n (θ 0 ). This interesting result does not solve the problem for fixed n, as θ 0 is unknown. For such problem, the use of φ-divergences, satisfying D (n) φ = R n (for example χ 2 -divergence), might give information about θ 0 and localizes it through φ-divergence confidence regions (CR φ 's).

The choice of the divergence φ also depends upon some knowledge on the support of the unknown p.m. P 0 . When P 0 has a projection on M with same support as P 0 , Proposition 2.2 yields its description and its explicit calculation. A necessary condition for this is that C θ , as defined in (2.9), has non void interior in R (l+1) . Consider the case of the empirical likelihood, that is when

ϕ(x) = -log x + x -1; then Im ϕ ′ =] -∞, 1[. Consider g(x, θ) = x -θ, i.
e., a constraint on the mean. Assume that the support of P 0 is unbounded. Then

C θ = t ∈ R 2 such that for all x (P 0 -a.s.) , t 0 + t 1 (x -θ) ∈] -∞, 1[ . Therefore, t 1 = 0 and C θ =] -∞, 1[×{0}
which implies that the interior of C θ is void. This results indicates that the support of Q * is not the same as the support of P 0 . Hence in this case we cannot use the dual representation of KL m (M θ , P 0 ). The arguments used in Section 3 for the obtention of limiting distributions cannot be used, if the support of P 0 is unbounded, in order to obtain the limiting distribution of the estimates KL m (M θ , P 0 ) under H 1 (i.e., when P 0 does not belong to M θ ). We thus cannot conclude in this case that the tests pertaining to θ 0 are consistent.

Robustness and Efficiency of MEφD estimates and Simulation Results

Lindsay (1994) introduced a general instrument for the study of the asymptotic properties of parametric estimates by minimum φ-divergences, called Residual Adjustment Function (RAF). We first recall its definition. Let {P θ ; θ ∈ Θ} be some parametric model defined on a finite set X . Let X 1 , . . . , X n a sample with distribution P θ0 . A minimum φ-divergence estimate (MφDE) (called also minimum disparity estimator) of θ 0 is given by (6.1)

θ φ := arg inf θ∈Θ x∈X ϕ P θ (x) P n (x) P n (x),
where P n (x) is the proportion of the sample point that take value x. When the parametric model

{P θ : θ ∈ Θ} is regular, then θ φ is solution of the equation (6.2) x∈X ϕ ′ P θ (x) P n (x) Ṗθ (x) = 0,
which can be written as

(6.3) x∈X A ϕ (δ(x)) Ṗθ (x) = 0.
In this display, A ϕ (u) := ϕ ′ 1 u+1 depends only upon the divergence function ϕ and δ(x) := P n (x) P θ (x) -1 is the "Pearson Residual" at x which belongs to ] -1, +∞[. The function A ϕ (.) is the RAF.

The points x for which δ(x) is close to -1 are called "inliers", whereas points x such that δ(x) is large are called "outliers". Efficiency properties are linked with the behavior of A ϕ (.) in the neighborhood of 0 (see [START_REF] Lindsay | Efficiency versus robustness: the case for minimum Hellinger distance and related methods[END_REF] Proposition 3 and [START_REF] Basu | Minimum disparity estimation for continuous models: efficiency, distributions and robustness[END_REF]) : the smaller the value of A ′′ ϕ (0) , the more second efficient the estimate θ φ in the sense of [START_REF] Rao | Asymptotic efficiency and limiting information[END_REF].

It is easy to verify that the RAF's of the power divergences φ γ , defined by the divergence functions in (1.4), have the form (6.4)

A γ (δ) = (δ + 1) 1-γ -1 (γ -1) .
In particular, the Mφ γ DE of (6.2) with the RAF in (6.4) corresponds to the maximum likelihood when γ = 0, minimum Hellinger distance when γ = 0.5, minimum χ 2 divergence when γ = 2, minimum modified χ 2 divergence when γ = -1 and minimum KL divergence when γ = 1.

From (6.4), we see that A ′′ γ (0) = γ. Hence for the maximum likelihood estimate, we have

A ′′ γ (0) = |A ′′ 0 (0)| = 0 which is the smallest value of A ′′ γ (0) , γ ∈ R.
Therefore, according to Proposition 3 in [START_REF] Lindsay | Efficiency versus robustness: the case for minimum Hellinger distance and related methods[END_REF], the maximum likelihood estimate is the most second-order efficient estimate (in the sense of [START_REF] Rao | Asymptotic efficiency and limiting information[END_REF]) among all minimum power divergences estimates.

Robustness features of θ φ against inliers and outliers are related to the variations of A ϕ (u) or ϕ(x) when u or x close to -1 and +∞, respectively as seen through the following heuristic arguments. Let φ 1 and φ 2 two divergences associated to the functions ϕ 1 and ϕ 2 . If

lim x↓0 ϕ 1 (x) ϕ 2 (x)
= +∞, then the estimating equation (6.2) corresponding to ϕ 1 in not as stable as that corresponding to ϕ 2 , and hence the MEφ 2 DE is more robust than MEφ 1 DE against outliers. If

lim x↑+∞ ϕ 1 (x) ϕ 2 (x) = +∞,
then the estimating equation (6.2) corresponding to ϕ 1 is not as stable as that corresponding to ϕ 2 , and hence the MEφ 2 DE is more robust than MEφ 1 DE against inliers.

In all cases, the divergence associated to the divergence function having the smallest variations on its domain leads to the most robust estimate against both outliers and inliers.

It is shown also in [START_REF] Jiménez | On robustness and efficiency of minimum divergence estimators[END_REF] that no minimum power divergence estimate (including the maximum likelihood one) is better than the minimum Hellinger divergence in terms of both second-order efficiency and robustness.

In the examples below, we compare by simulations the efficiency and robustness properties of some MEφDE's for some models satisfying linear constraints. We will see that the minimum empirical Hellinger divergence estimate represents a suitable compromise between efficiency and robustness.

A theoretical study of efficiency and robustness properties of MEφDE's is necessary and should envolve second-order efficiency versus robustness since all MEφDE's are all equally first-order efficient (see Remark 3.2 and Theorem 3.4).

Numerical Results. We consider for illustration the same model as in [START_REF] Qin | Empirical likelihood and general estimating equations[END_REF] Section 5 Example 1. The model M θ (see 1.12) here is the set of all signed finite measures Q satisfying (6.5) dQ = 1 and g(x, θ) dQ(x) = 0, with g(x, θ) = (xθ), (x 2 -2θ 2 -1) T and θ, the parameter of interest, belongs to R.

In Examples 1.a and 1.b below, we compare the efficiency property of various estimates : we generate 1000 pseudorandom samples of sizes 25, 50, 75 and 100 from a normal distribution with mean θ 0 and variance θ 2 0 + 1 (i.e., P 0 = N (θ 0 , θ 2 0 + 1)) for two values of θ 0 : θ 0 = 0 in Example 1.a and θ 0 = 1 in Example 1.b. Note that P 0 satisfies (6.5).

For each sample, we consider various estimates of θ 0 : the sample mean estimate (SME), the parametric ML estimate (MLE) based on the normal distribution N (θ, θ 2 + 1) and MEφD estimates θ φ associated to the divergences : φ = χ 2 m , H, KL, χ 2 and KL m -divergence (which coincides with the MEL one, i.e., MEKL m E=MELE).

For all divergence φ considered, in order to calculate the MEφDE θ φ , we first calculate φ(M θ , P 0 ) for all given θ (using the representation (2.14)) by Newton's method, and then minimize it to obtain θ φ .

The results of Theorem 3.4 show that for all φ-divergence

√ n θ φ -θ 0 → N (0, V )
where V is independent of the divergence φ; it is given in Theorem 3.4. For the present model, following [START_REF] Qin | Empirical likelihood and general estimating equations[END_REF], V writes

(6.6) V = V ar(X) -△ -1 m ′ (θ 0 )V ar(X) + θ 0 m(θ 0 ) -E(X 3 ) ) 2
where △ = E m ′ (θ 0 )(Xθ 0 ) + m(θ 0 ) -X 2 2 and m(θ) := 2θ 2 + 1. Thus V ≤ V ar(X) which is the variance of √ n X nθ 0 with X n := 1 n n i=1 X i , the sample mean estimate (SME) of θ 0 . So, EMφD estimates are all asymptotically at least as efficient as X n .

6.1. Example 1.a. In this example the true value of the parameter is θ 0 = 0. We can see from Table 1 that all the estimates converge in a satisfactory way. The estimated variances are almost the same for all estimates. This is not surprising since the limit variance of all estimates in this Example (when θ 0 = 0) is close to V (X).

6.2. Example 1.b. In this example the true value of the parameter is θ 0 = 1.

MEχ We can see from Table 2 and Figure 1 that the estimated bias of EφDE's are all smaller than the SME one for moderate and large sample sizes. Furthermore, from Figure 2, we observe that the estimated variances of EφDE's are all less than the SME one. They lie between that of the sample mean and that of the parametric maximum likelihood estimate. We observe also that the estimated variances of the MELE and MEHDE are equal and are the smallest among the variances of all MEφDE's considered. It should be emphasized that even for small sample sizes, the MSE of the SM is larger than any of MEφDE's.

In Examples 2.a and 2.b below, we compare robustness property of the estimates considered above for contaminated data : we consider the same model M θ as in (6.5).

6.3. Example 2.a. In this Example, we generate 1000 pseudo-random samples of sizes 25, 50, 75 and 100 from a distribution P 0 = (1ǫ)P 0 + ǫδ 5

where P 0 = N (θ 0 , θ 2 0 + 1), ǫ = 0.15 and θ 0 = 2. We consider the same estimates as in the above examples.

In this Example, we can see from Table 3 and Figure 3 that the MEχ 2 D estimate is the most robust and MEχ 2 m estimate is the least robust. We observe also that the MELE which is the MEKL m DE is less robust than the MEKLDE and that the MEHD estimate is more robust than MEL one. 6.4. Example 2.b. In this Example, we generate 1000 pseudo-random samples of sizes 50, 100, 150 and 200 from a distribution P 0 = N (θ 0 , θ 2 0 + 1) with θ 0 = 2 and we cancel the observations in the interval [4,5] . We consider the same estimates as in the above examples.

In this example, in contrast with Example 2.b, we observe that the MEχ if a MEφDE is more robust than its adjoint 1 (i.e., MEφ ∼ DE) against "outliers", then it is less

1 For all divergence φ associated to a convex function ϕ, its adjoint, noted φ ∼ , is the divergence associated to the convex function, noted ϕ ∼ , defined by : ϕ ∼ (x) = xϕ(1/x), for all x. is closed in R n . Hence, by condition (2.3), the infimum of the function

(Q(X 1 ), . . . , Q(X n )) T ∈ R n → 1 n n i=1 ϕ (nQ(X i ))
(Q(X 1 ), . . . , Q(X n )) T ∈ R n → 1 n n i=1 ϕ (nQ(X i )) on the set D (n) φ ∩ M (n) θ
exists as an interior point of D

(n)

φ . Since the above function is strictly convex and the set D

(n) φ ∩ M (n) θ is convex, then this infimum is unique. It is noted Q * θ . This concludes the proof of part (i). Proof of part (ii). Since (Q(X 1 ), . . . , Q(X n )) T ∈ R n → 1 n n i=1 ϕ (nQ(X i )) is C 1 on the interior of D (n)
φ , and since Q * θ is in the interior of D

(n) φ , we can use the Lagrange method. This yields the explicit form (2.5) of the projection Q * θ in which c 0 is the Lagrange multiplier associated to the constraint n i=1 Q(X i ) = 1 and c j to the constraint l i=1 Q(X i )g j (X i , θ) = 0, for all j = 1, . . . , l. This concludes the proof of Proposition 2.1. We prove that φ (M θ , P 0 ) and c θ converge to φ (M θ , P 0 ) and c θ respectively. Since c θ is isolated, then consistency of c θ holds as a consequence of Theorem 5.7 in [START_REF] Van Der Vaart | Asymptotic statistics[END_REF]. For the estimate φ (M θ , P 0 ), we have

φ (M θ , P 0 ) -φ (M θ , P 0 ) = |P n m(θ, c θ ) -P 0 m(θ, c θ )| := |A|,

which implies

P n m(θ, c θ ) -P 0 m(θ, c θ ) < A < P n m(θ, c θ ) -P 0 m(θ, c θ ). Both the RHS and the LHS terms in the above display go to 0, under condition (C.1). This implies that A tends to 0, which concludes the proof of Proposition 3.1. 7.3. Proof of Theorem 3.2. . Proof of part (1). Some calculus yield

P 0 m ′ (θ, c θ ) = P 0 1 - ← - ϕ ′ c T θ g(θ) , -g 1 (θ) ← - ϕ ′ c T θ g(θ) , . . . , -g l (θ) ← - ϕ ′ c T θ g(θ)
T = 0 T l . (7.1) and (7.2)

P 0 m ′′ (θ, c θ ) = P 0   - g i g j ϕ ′′ ← - ϕ ′ c T θ g(θ)   i,j=0,...,l
, which implies that the matrix P 0 m ′′ (θ, c θ ) is symmetric. Under assumption (A.2), by Taylor expansion, there exists t n ∈ R l+1 inside the segment that links c θ and c θ with (7.3)

0 = P n m ′ (θ, c θ ) = P n m ′ (θ, c θ ) + (P n m ′′ (θ, c θ )) T ( c θ -c θ ) + 1 2 ( c θ -c θ ) T P n m ′′′ (θ, t n ) ( c θ -c θ ) ,
in which, P n m ′′′ (θ, t n ) is a (l + 1)-vector whose entries are (l + 1) × (l + 1)-matrices. By (A.2), we have for the sup-norm of vectors and matrices

P n m ′′′ (θ, t n ) := 1 n n i=1 m ′′′ (X i , θ, t n ) ≤ 1 n n i=1 |H(X i )|.
By the Law of Large Numbers (LLN), P n m ′′′ (θ, t n ) = O P (1). So using (A.1), we can write the last term in the right hand side of (7.3) as o P (1) ( c θc θ ). On the other hand by (A.3),

P n m ′′ (θ, c θ ) := 1 n n i=1 m ′′ (X i , θ, c θ ) converges to the matrix P 0 m ′′ (θ, c θ ). Write P n m ′′ (θ, c θ ) as P 0 m ′′ (θ, c θ ) + o P (1) to obtain from (7.3) (7.4) -P n m ′ (θ, c θ ) = (P 0 m ′′ (θ, c θ ) + o P (1)) ( c θ -c θ ) .
Under (A.3), by the Central Limit Theorem, we have √ nP n m ′ (θ, c θ ) = O P (1), which by (7.4) implies that

√ n ( c θc θ ) = O P (1). Hence, from (7.4), we get

(7.5) √ n ( c θ -c θ ) = [-P 0 m ′′ (θ, c θ )] -1 √ nP n m ′ (θ, c θ ) + o P (1).
Under (A.3), the Central Limit Theorem concludes the proof of part 1. In the case when P 0 belongs to M θ , then c T θ = (ϕ ′ (1), 0 T ) := c and calculation yields

P 0 m ′ (θ, c)m ′ (θ, c) T = 0 0 T l 0 l P 0 g(θ)g(θ) T and -ϕ ′′ (1)P 0 m ′′ (θ, c) = 1 0 T l 0 l P 0 g(θ)g(θ) T .
A simple calculation yields (3.3).

Proof of part (2). By Taylor expansion, there exists t n inside the segment that links c θ and c θ with

φ n (M θ , P 0 ) = P n m(θ, c θ ) = P n m(θ, c θ ) + (P n m ′ (θ, c θ )) T ( c θ -c θ ) + 1 2 ( c θ -c θ ) T [P n m ′′ (θ, c θ )] ( c θ -c θ ) + 1 3! 1≤i,j,k≤d ( c θ -c θ ) i ( c θ -c θ ) j × ×( c θ -c θ ) k P n ∂ 3 ∂t i ∂t j ∂t k m(θ, t n ). (7.6) When P 0 belongs to M θ , then c T θ = c. Hence P n m(θ, c θ ) = P n m(θ, c) = P n 0 = 0. Furthermore, by part (1) in Theorem 3.2, it holds √ n( c θ -c θ ) = O p (1)
. Hence, by (A.1), (A.2) and (A.3), we get

φ n (M θ , P 0 ) = (P n m ′ (θ, c θ )) T ( c θ -c θ ) + 1 2 ( c θ -c θ ) T [P 0 m ′′ (θ, c θ )] ( c θ -c θ ) + o P (1/n),
which by (7.5), implies

φ n (M θ , P 0 ) = [P n m ′ (θ, c θ )] T [-P 0 m ′′ (θ, c θ )] -1 [P n m ′ (θ, c θ )] + 1 2 [P n m ′ (θ, c θ )] T [P 0 m ′′ (θ, c θ )] -1 [P n m ′ (θ, c θ )] + o P (1/n) = 1 2 [P n m ′ (θ, c θ )] T [-P 0 m ′′ (θ, c θ )] -1 [P n m ′ (θ, c)] + o P (1/n).
This yields to

(7.7) 2n ϕ ′′ (1) φ n (M θ , P 0 ) = √ nP n m ′ (θ, c θ ) T [-ϕ ′′ (1)P 0 m ′′ (θ, c θ )] -1 √ nP n m ′ (θ, c θ ) + o P (1).
Note that when P 0 belongs to M θ , then c T θ = c and calculation yields

P 0 m ′ (θ, c)m ′ (θ, c) T = 0 0 T l 0 l P 0 g(θ)g(θ) T and -ϕ ′′ (1)P 0 m ′′ (θ, c) = 1 0 T l 0 l P 0 g(θ)g(θ) T .
Combining this with (7.7), we conclude the proof of part (2).

Proof of part (3). Since ( c θc θ ) = O P (1/ √ n) and P n m ′ (θ, c θ ) = P 0 m ′ (θ, c θ )+o P (1) = 0+o P (1) = o P (1), then, using (7.6), we obtain

√ n φ n (M θ , P 0 ) -φ(M θ , P 0 ) = √ n φ n (M θ , P 0 ) -P 0 m(θ, c θ ) = √ n (P n m(θ, c θ ) -P 0 m(θ, c θ )) + o P (1),
and the Central Limit Theorem yields to the conclusion of the proof of Theorem 3.2. By condition (C.5), for all n sufficiently large, it holds θ φ = θ φ and φ (M, P 0 ) = φ (M, P 0 ) .

We prove that θ φ and φ (M, P 0 ) are consistent. First, we prove the consistency of φ (M, P 0 ). We have φ (M, P 0 )φ (M, P 0 ) = P n m θ φ , c θ φ -P 0 m(θ * , c θ * ) =: |A|.

This implies

P n m θ φ , c θ * -P 0 m θ φ , c θ * ≤ A ≤ P n m θ * , c θ φ -P 0 m θ * , c θ φ .
By condition (C.3), both the RHS and LHS terms in the above display go to 0. This implies that A tends to 0 which concludes the proof of part (i).

Proof of part (ii). Since for sufficiently large n, by condition (C.5), we have c θ = c θ for all θ ∈ Θ, the convergence of sup θ∈Θ c θc θ to 0 implies (ii). We prove now that sup θ∈Θ c θc θ tends to 0. By the very definition of c θ and condition (C.3), we have (7.8) where o P (1) does not depends upon θ (due to condition (C.3)). Hence, we have for all θ ∈ Θ,

P n m (θ, c θ ) ≥ P n m (θ, c θ ) ≥ P 0 m (θ, c θ ) -o P (1),
P 0 m (θ, c θ ) -P 0 m (θ, c θ ) ≤ P n m (θ, c θ ) -P 0 m (θ, c θ ) + o P (1). (7.9)
The term in the RHS of the above display is less than

sup θ∈Θ,t∈T θ |P n m(θ, t) -P 0 m(θ, t)| + o P (1)
which by (C.3), tends to 0. Let ǫ > 0 be such that sup θ∈Θ c θc θ > ǫ. There exists some a n ∈ Θ such that c anc an > ǫ. Together with the strict concavity of the function t ∈ T θ → P 0 m(θ, t) for all θ ∈ Θ, there exists η > 0 such that P 0 m (a n , c an ) -P 0 m (a n , c an ) > η.

We then conclude that

P sup θ∈Θ c θ -c θ > ǫ ≤ P {P 0 m (a n , c an ) -P 0 m (a n , c an ) > η} ,
and the RHS term tends to 0 by (7.9). This concludes the proof part (ii).

Proof of part (iii). We prove that θ φ converges to θ * . By the very definition of θ φ , condition (C.4.b) and part (ii), we obtain

P n m θ φ , c θ φ ≤ P n m (θ * , c θ * ) ≤ P 0 m θ * , c θ φ -o P (1), from which P 0 m θ φ , c θ φ -P 0 m θ * , c θ φ ≤ P 0 m θ φ , c θ φ -P n m θ φ , c θ φ + o P (1) ≤ sup {θ∈Θ,t∈T θ }
|P n m(θ, t) -P 0 m(θ, t)| + o P (1). (7.10) Further, by part (ii) and condition (C.4.a), for any positive ǫ, there exists η > 0 such that

P θ φ -θ * > ǫ ≤ P P 0 m θ φ , c θ φ -P 0 m θ * , c θ φ > η .
The RHS term, under condition (C.3), tends to 0 by (7.10). This concludes the proof of Proposition 3.3.

7.5. Proof of Theorem 3.4.

Since P 0 ∈ M, then c θ = c. Some calculus yield ∂ ∂t m(θ 0 , c) = [0, -g 1 (θ 0 ), . . . , -g l (θ 0 )] T = -0, g(θ 0 ) T T , (7.11) ∂ ∂θ m(θ, t) = - l j=0 t i ← - ϕ ′ (t T g(θ)) ∂ ∂θ g j (θ), ∂ ∂θ m(θ 0 , c) = 0 d , ∂ 2 ∂θ∂t m(θ 0 , c) = 0 d , - ∂ ∂θ g 1 (θ 0 ), . . . , - ∂ ∂θ g l (θ 0 ) = -0 d , ∂ ∂θ g(θ) ∂ 2 ∂t∂θ m(θ 0 , c) = ∂ 2 ∂θ∂t m(θ 0 , c) T , ∂ 2 ∂θ 2 m(θ 0 , c) = [0 d , . . . , 0 d ] , and ∂ 2 ∂t 2 m(θ 0 , c) = 1 ϕ ′′ (1)
[-g i (θ 0 )g j (θ 0 )] i,j=0,1,...,l := -1 ϕ ′′ (1) g(θ 0 )g(θ 0 ) T .

Integrating w.r.t. P 0 , we obtain (7.12)

P 0 ∂ ∂t m(θ 0 , c) = 0 l , P 0 ∂ ∂θ m(θ 0 , c) = 0 d , P 0 ∂ 2 ∂θ 2 m(θ 0 , c) = [0 d , . . . , 0 d ] , (7.13) P 0 ∂ 2 ∂θ∂t m(θ 0 , c) = -0 d , P 0 ∂ ∂θ g(θ 0 ) , (7.14) P 0 ∂ 2 ∂t∂θ m(θ 0 , c) = P 0 ∂ 2 ∂θ∂t m(θ 0 , c) T = -0 d , P 0 ∂ ∂θ g(θ 0 ) T , and
P 0 ∂ 2 ∂t 2 m(θ 0 , c) = -1 ϕ ′′ (1) [P 0 g i (θ 0 )g j (θ 0 )] i,j=0,1,...,l . = -1 ϕ ′′ (1) 1 0 T l 0 l P 0 g(θ 0 )g(θ 0 ) T (7.15)
By the very definition of θ φ and c θ φ , they both obey . By (A.5), the LLN implies that A n = O P (1). So using (A.4), we can write the last term in right hand side of (7.16) as o P (1)a n . On the other hand by (A.6), we can write also In the same way, using a Taylor expansion in (E2), there exists (θ n , t n ) inside the segment that links θ φ , c θ φ and (θ 0 , c) such that 0 = P n ∂ ∂θ m(θ 0 , c) + P n ∂ 2 ∂t∂θ m(θ 0 , c) This concludes the proof of Theorem 4.1.

P n ∂ ∂t m (θ, t) = 0 P n ∂ ∂θ m (θ, t(θ)) = 0, i.e.,    P n ∂ ∂t m θ φ , c θ φ = 0 P n ∂ ∂θ m θ φ , c θ φ + P n ∂ ∂t m θ φ , c θ φ ∂ ∂θ c θ φ = 0.

1. 1 .

 1 Statistical examples and motivations.

  an interval which may be open or not, bounded or unbounded. Hence, write D ϕ := (a, b) in which a and b may be finite or infinite. In this paper, we will only consider ϕ functions defined on [-∞, +∞] with values in [0, +∞] such that a < 1 < b, and which satisfy ϕ(1) = 0, are strictly convex and are C 2 on the interior of its domain D ϕ ; we define ϕ(a), ϕ ′ (a), ϕ ′′ (a), ϕ(b), ϕ ′ (b) and ϕ ′′ (b) respectively by ϕ(a) := lim x↓a ϕ(x), ϕ ′ (a) := lim x↓a ϕ ′ (x), ϕ ′′ (a) := lim x↓a ϕ ′′ (x), ϕ(b) := lim x↑b ϕ(x), ϕ ′ (b) := lim x↑b ϕ ′ (x) and ϕ ′′ (b) := lim x↑b ϕ ′′ (x). These quantities may be finite or infinite. All the functions ϕ γ (see (1.5)) satisfy these conditions. Definition 1.1.

P

  n m(θ, t) ≤ M a.s. (resp. in probability).Proposition 3.3. Assume that conditions (C.3-5) hold. Then (i) the estimates φ(M, P 0 ) converge to φ(M, P 0 ) a.s. (resp. in probability).

  θ * , c θ * ) 2 and P 0 ∂ ∂θ m(θ * , c θ * ) 2 are finite, and the matrix S := S 11 S 12 S 21 S 22 , with S 11 := P 0 ∂ 2 ∂t 2 m(θ * , c θ * ), S 12 = S 21 T := P 0 ∂ 2 ∂t∂θ m(θ * , c θ * ) and S 22 := P 0 ∂ 2 ∂θ 2 m(θ * , c θ * ), exists and is invertible.

3. 5 .

 5 Composite tests on the parameter. Let (3.9) h : R d → R k be some function such that the (d × k)-matrix H(θ) := ∂ ∂θ h(θ) exists, is continuous and has rank k with 0 < k < d. Let us define the composite null hypothesis (3.10) Θ 0 := {θ ∈ Θ such that h(θ) = 0} .

  i , θ) = 0 d admits a solution θ n , then P n belongs to M. If the solution is unique then θ φ = θ n . Hence by Proposition 2.1 for all i ∈ {1, 2, . . . , n} , we have Q

Figure 2 .

 2 Figure 2. Estimated variance of the estimates of θ 0 in Example 1.b.

Figure 4 .

 4 Figure 4. Estimated mean of the estimates of θ 0 in Example 2.b.

7. 2 .

 2 Proof of Proposition 3.1. Define the estimates c θ = arg inf t∈T θ P n m(θ, t) and φ (M θ , P 0 ) = sup t∈T θ P n m(θ, t). By condition (C.2), for all n sufficiently large, we have c θ = c θ and φ (M θ , P 0 ) = φ (M θ , P 0 ) .

7. 4 .

 4 Proof of Proposition 3.3. Define the estimates θ φ := arg inf

  The second term in the left hand side of the second equation is equal to 0, due to the first equation. Hence c θ φ and θ φ are solutions of the somehow simpler system  expansion in (E1); there exists θ n , t n inside the segment that links ( θ φ , c θ φ ) and(θ 0 , c) m( θ, c n )

  0 , c) = P 0 ∂ 2 ∂t 2 m(θ 0 , c) + o P (1), P 0 ∂ 2 ∂θ∂t m(θ 0 , c) T + o P (1) a n .

F

  (θ 0 , c) = [0 d , . . . , 0 d ] .The inverse matrix S -1 of the matrix S writes (0 g(θ 0 )g(θ 0 ) T -1 0 d , P 0 ∂ ∂θ g(θ 0 ) T = ϕ ′′ (1) P 0 ∂ ∂θ g(θ 0 ) P 0 g(θ 0 )g(θ 0 ) T -1 P 0 assumption (A.6), by the Central Limit Theorem, converges in distribution to a centered multivariate normal variable with covariance matrix (7.31) M = M 11 M 12 M 21 M 22 to a centered multivariate normal variable with covariance matrix (7.34)C = S -1 M S -1 T := C 11 C 12 C 21 C 22 ,and using (7.32) and some algebra, we getC 11 = ϕ ′′ (1) 0 g(θ 0 )g(θ 0 ) T -1ϕ ′′ (1) C 12 = [0 l , . . . , 0 l ] , C 21 = [0 d , . . . ,33), we deduce that C 11 and C 22 are respectively the limit covariance matrix of √ n c θ φc and √ n θ φθ 0 , i.e., U = C 11 and V = C 22 . (7.36) implies that √ n c θ φc and √ n θ φθ 0 are asymptotically uncorrelated. This concludes the Proof of Theorem 3.4. 7.6. Proof of Theorem 3.5. Under assumptions (A.4-6), as in the proof of Theorem 3θ * , c θ * ) + o P (1), and the CLT concludes the proof. 7.7. Proof of Theorem 4.1. Using Taylor expansion at (c, θ 0 ), we get θφ T g(X i , θ φ ) ½ (-∞,x] (X i ) = F n (x) + 1 n n i=1 g(X i , θ 0 )½ (-∞,x] (X i ) T 1 ϕ ′′ (1) c θ φc + o P (δ n ),(7.38) where δ n := c θ φc + θ φθ 0 , which by Theorem 3.4, is equal to O P 41)H = S -1 11 + S -1 11 S 12 S -1 22.1 S 21 S -1 11 .We will use f (.) to denote the function ½ (-∞,x] (.) -F (x), for all x ∈ R. Substituting (7.40) in (7.39), we get √ n F n (x) -F (xto a centered multivariate normal variable which implies that √ n F n (x) -F (x) is asymptotically centered normal variable. We calculate now its limit variance, noted W (x).W (x) = F (x)(1 -F (x)) + 1 ϕ ′′ (1) 2 P 0 g(θ 0 )½ (-∞,x] T U P 0 g(θ 0 )½ (-∞,x] + +2 1 ϕ ′′ (1) P 0 g(θ 0 )½ (-∞,x] T H P 0 -∂ ∂t m(θ 0 , c)½ (-∞,x] . (7.43)Use the explicit forms of ∂ ∂t m(θ 0 , c), the matrices U and V and some algebra to obtain W (x) = F (x) (1 -F (x)) -P 0 g(θ 0 )½ (-∞,x] T Γ P 0 g(θ 0 )½ (-∞,x] ,with Γ = P 0 g(θ 0 )g(θ 0 ) T -1 -P 0 g(θ 0 )g(θ 0 ) T -1 P 0 0 ) P 0 g(θ 0 )g(θ 0 ) T -1 .

Table 1 .

 1 Estimated mean and variance of the estimates of θ 0 in Example 1.a.

		MEχ 2 m DE	MEKL m DE=MELE	MEHDE	MEKLDE
	n	mean	var	mean	var	mean	var	mean	var
	25	0.0089 0.0314 0.0086	0.0315	0.0084 0.0315 0.0082 0.0314
	50 -0.0116 0.0209 -0.0118	0.0210	-0.0119 0.0210 -0.0120 0.0210
	75 -0.0025 0.0171 -0.0024	0.0170	-0.0023 0.0170 -0.0022 0.0169
	100 -0.0172 0.0112 -0.0174	0.0111	-0.0174 0.0111 -0.0175 0.0112
		MEχ 2 DE	PMLE	SME
	n	mean	var	mean	var	mean	var
	25	0.0077 0.0313 0.0026	0.0318	0.0081 0.0394
	50 -0.0125 0.0212 -0.0063	0.0196	-0.0040 0.0200
	75 -0.0019 0.0167 -0.0011	0.0170	0.0013 0.0164
	100 -0.0177 0.0112 -0.0158	0.0108	-0.0149 0.0102

Table 2 .

 2 Estimated mean and variance of the estimates of θ 0 in Example 1.b.

			2 m DE	MEKL m DE=MELE	MEHDE	MEKLDE
	n	mean	var	mean	var	mean	var	mean	var
	25 0.9394 0.0310 0.9387	0.0312	0.9385 0.0313 0.9378 0.0316
	50 0.9994 0.0186 0.9967	0.0186	0.9954 0.0186 0.9941 0.0187
	75 1.0009 0.0156 0.9988	0.0154	0.9975 0.0154 0.9966 0.0153
	100 0.9984 0.0113 0.9959	0.0112	0.9945 0.0112 0.99315 0.0112
		MEχ 2 DE		PMLE	SME
	n	mean	var	mean	var	mean	var
	25 0.9350 0.0322 0.9540	0.0325	1.0033 0.0810
	50 0.9909 0.0190 1.0036	0.0174	1.0021 0.0407
	75 0.9940 0.0152 1.0003	0.0149	0.9912 0.0288
	100 0.9900 0.0113 0.9970	0.0107	0.9851 0.0262

  2 m DE is the most robust, MEχ 2 DE is the least robust and MEKLDE is less robust than MEKL m DE (=MELE). Generally, Estimated mean of the estimates of θ 0 in Example 1.b.

		1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00	*	*
						*
								*
	Estimation of theta	0.96 0.98 0.98 0.98 0.98 0.98 0.98 0.98 0.98 0.96 0.96 0.96 0.96 0.96 0.96 0.96						chi2m
								H
								KL
								chi2
								MEL
		0.94 0.94 0.94 0.94 0.94 0.94 0.94 0.94						*	ML SM
			40 40 40 40 40 40 40 40		60 60 60 60 60 60 60 60		80 80 80 80 80 80 80 80	100 100 100 100 100 100 100 100
					Samples sizes
	0.01 0.02 0.03 0.04 0.05 0.07 0.08 0.01 0.02 0.03 0.04 0.05 0.07 0.08 0.01 0.02 0.03 0.04 0.05 0.07 0.08 0.01 0.02 0.03 0.04 0.05 0.07 0.08 0.01 0.02 0.03 0.04 0.05 0.07 0.08 0.01 0.02 0.03 0.04 0.05 0.07 0.08 0.01 0.02 0.03 0.04 0.05 0.07 0.08 Figure 1. 40 variance 40 40 40 40 40 * 40 0.06 0.06 0.06 0.06 0.06 0.06 0.06	*	60 60 60 60 60 60 60	*	80 80 80 80 80 80 80	*	100 100 100 100 100 100 * 100 chi2m H KL chi2 MEL ML SM
					Samples sizes

Table 3 .

 3 Estimated mean and variance of the estimates of θ 0 in Example 2.a.

		MEχ 2 m DE	MEKL m DE=MELE	MEHDE	MEKLDE
	n	mean	var	mean	var	mean	var	mean	var
	25 2.1609 0.0654 2.1513	0.0653	2.1453 0.0653 2.1396 0.0652
	50 2.2087 0.0303 2.1975	0.0304	2.1912 0.0307 2.1848 0.0309
	75 2.2218 0.0214 2.2106	0.0213	2.2046 0.0213 2.1987 0.0215
	100 2.2283 0.0151 2.2169	0.0149	2.2110 0.0148 2.2052 0.0149
		MEχ 2 DE		PMLE	SME
	n	mean	var	mean	var	mean	var
	25 2.1278 0.0646 2.2088	0.0581	2.4265 0.2178
	50 2.1729 0.0316 2.2296	0.0280	2.4535 0.1076
	75 2.1877 0.0219 2.2337	0.0197	2.4545 0.0721
	100 2.1947 0.0151 2.2352	0.0139	2.4572 0.0543

Table 4 .

 4 Estimated mean and variance of the estimates of θ 0 in Example 2.b.

		MEχ 2 m DE	MEKL m DE=MELE	MEHDE	MEKLDE
	n	mean	var	mean	var	mean	var	mean	var
	50 1.9917 0.0451 1.9784	0.0431	1.9721 0.0426 1.9659 0.0423
	100 1.9962 0.0362 1.9844	0.0346	1.9787 0.0341 1.9729 0.0336
	150 2.0011 0.0150 1.9903	0.0142	1.9849 0.0139 1.9795 0.0137
	200 1.9602 0.0162 1.9516	0.0158	1.9473 0.0157 1.9430 0.0156
		MEχ 2 DE		PMLE	SME
	n	mean	var	mean	var	mean	var
	50 1.9522 0.0428 1.9705	0.0358	1.7750 0.1039
	100 1.9590 0.0329 1.9687	0.0298	1.7365 0.0576
	150 1.9671 0.0135 1.9781	0.0121	1.7456 0.0283
	200 1.9325 0.0155 1.9420	0.0146	1.7247 0.0317


K Grenier

W Chen

D Dubuc

Non-invasive real time monitoring of cellular modifications with microwave dielectric spectroscopy at the single cell level

Microwave dielectric spectroscopy constitutes an attractive sensing technique for cellular analysis. One key feature is its ability to non-invasively and dielectrically monitor cellular modifications at the single cell level. The proposed microwave sensor dedicated to single cell measurement includes a mechanical trap, which permits to apply on individual cells whatever liquid-based environment. The dielectric consequences on the cell due to the application of saponine at 0.02% is recorded every two minutes, demonstrating the possibility to reach dielectric transformation kinetic of cells. This proof of concept opens the door to the non-invasive and nondestructive evaluation of biological reactions at the cellular level.

Introduction

Microwave dielectric spectroscopy is very attractive as it presents many advantages suitable for performing nondestructive and non-invasive sensing [START_REF] Kupfer | Electromagnetic aquametry[END_REF]. Solid, liquid or gaseous materials may be characterized by their dielectric properties, with a frequency-dependent response. The technique was firstly developed with large and cumbersome materials detection due to a lack of miniature equipment for various applicative domains such as the moisture sensing in agriculture, wood industry or food quality assessments [START_REF] Trabelsi | New calibration technique for microwave moisture sensors[END_REF][START_REF] Nelson | Sensing quality of watermelons through dielectric permittivity[END_REF].

Since the advent of the microtechnologies and more recently microfluidic, it is possible to envision new applications for biology and medicine with molecular and cellular sensing notably. The intrinsic features of the microwave sensing naturally fulfil interesting abilities for cellular analyzes. It does not require any direct contact or labels. The microwave frequency range also gives access to internal modifications of cells as electromagnetic waves may bypass the bi-lipidic cytoplasmic membrane. Additionally, cells may be measured directly within their rich and salty traditional culture medium.

Firstly demonstrated with tissues, microwave spectroscopy was demonstrated for cell quantification, pathological state discrimination and even cell identification [START_REF] Grenier | Recent Advances in Microwave-based Dielectric Spectroscopy at the Cellular Level for Cancer Investigations[END_REF][START_REF] Grenier | Label-free discrimination of human lymphoma cell subpopulations with microwave dielectric spectroscopy[END_REF], for cells suspensions and also at the single cell level [START_REF] Ma | Sensitivity Analysis for Ultra-wideband 2-port Impedance Spectroscopy of a Live Cell[END_REF][START_REF] Chen | Microwave biosensor dedicated to the dielectric spectroscopy of a single alive biological cell in its culture medium[END_REF].

One attractive ability of the technique consists in monitoring the evolution of the material under test. This paper therefore focuses on the demonstration of monitoring the reaction of a cell to a chemical stress.

Description of the microwave sensor dedicated to single cell measurements

The microwave sensor includes a coplanar waveguide with two tapers and a capacitive gap located at its center. A photography is given in Fig. 1. This configuration permits to focalize the electromagnetic field, where the cell is placed for analysis.

Perpendicularly is crossing a microfluidic channel with a mechanical trap in its middle. This latter presents a small interstice below, in order to attract any cell arriving in front of it. Once a cell is trapped, the hydrodynamic flow rejects the other cells apart the blocker, leaving only one cell in the sensor for RF measurement. The main advantage of this trap is that it works regardless of the liquid used, whereas dielectrophoresis requires to fulfil specific rules between the dielectric properties of the element to trap compare to the host medium.

Investigated cells and preparation protocol

Fluidic walls

Bead Trap Fluid THP1 cell line is selected. These cells are derived from a human with an acute monocytic leukemia. They are grown in the RPMI culture medium. These circulating cells present a round shape. For measurement, a cell suspension exhibiting a concentration of 1 million cells/ml is prepared. Cells are removed from their culture plate with trypsine and placed in an Eppendorf. After counting, cells are centrifugated, allowing the replacement of the supernatant by a fresh culture medium, as indicated in Fig. 2.

To enable the monitoring of a short duration stress, a detergent, called saponine, is used. This chemical agent creates pores in the cytoplasmic membrane while maintaining the integrity of the rest of the cells. This chemical is used at 0.02% in combination to the culture medium.

Evaluation of the time stability of cellular dielectric response

Before monitoring a chemical stress, the stability of the test setup and the dielectric response of different models are evaluated. The sensor is consequently measured every 3 minutes for 15 minutes, while unloaded, followed by loaded with de-ionized water, RPMI culture medium, a polystyrene bead and finally a cell. The capacitive response of the sensor is given in Fig. 3 for the different configurations. The low measured standard deviations for each case enables the use of the sensor for monitoring cellular modification.

Monitoring of a chemical stress on a single cell

In a second step, a THP1 cell is trapped in the sensor. After 10 minutes, the culture medium is replaced by the saponine based one. Dielectric measurements are performed from 40 MHz to 40 GHz, while the cell reaction is recorded for 10 additional minutes. As expected, the dielectric response of the cell changes, as visible on the fig. 4. The capacitive contrast of the sensor loaded by the cell in liquid, compare to the sensor with only the culture medium starts to decrease in absolute value. The zero level on the ordinate axis corresponds to the culture medium employed as the reference level. As soon as the chemical is inserted in the vicinity of the cell, poration of the cell occurs, leading to its artificial and rapid death. This translates into the decrease of the capacitive contrast toward the reference level, as an exchange between the extra and intracellular liquids happens, progressively reaching an equilibrium between both compartments. The reference liquid level may not be reached as all organelles are still maintained within the cell.

To better highlight the cellular reaction, the dielectric kinetic is plotted at 5 GHz in fig. 5. The cell exhibits a stable capacitive contrast, while maintained in its culture medium during the first 10 minutes. Once the chemical agent is added to the culture medium, the dielectric response of the cell is abrupt, reaching in only 4 minutes the smallest difference, compare to the culture medium followed by strong internal modifications.

Conclusions

This paper deals with the possible monitoring of cellular modification with microwave dielectric spectroscopy. A microwave-based sensor suitable for the analysis of single cells is described. Its cell trapping mechanism based on a mechanical blocking with hydrodynamic flow enables to modify the host medium, whatever the dielectric properties of the liquids and the cells. The proof of concept is obtained with the monitoring of a cancer cell submitted to a chemical agent, which creates rapidly holes in the cytoplasmic membrane. The dielectric response is recorded and reflects the cell modifications, leading to the death of the cell. In the proposed example, the time resolution is of 2 minutes. This value may be both increased or decreased depending of the targeted biological reactions to evaluate. Kinetic of biological processes may therefore be monitored, which may be of high interest in personalized medicine notably or to further understand both rapid or slow transformations of the living.

Figure 1 .

 1 Figure 1. Photography of the microwave sensor dedicated to individual cell measurement, which includes a coplanar waveguide surmounted by a microfluidic channel with a cell trap. As an example, this photography includes a bead solution passing through the fluidic channel before trapping.

Figure 2 .

 2 Figure 2. Protocol used to prepare the cell suspension samples.

Figure 3 .

 3 Figure 3. Time stability of the extracted capacitor of the sensor for different conditions : unloaded, loaded with DI water, with the RPMI culture medium, with one bead in culture medium and finally with a cell in culture medium.

Figure 4 .

 4 Figure 4. Monitoring of the capacitive contrast of a cell submitted to a chemical treatment.

Figure 5 .

 5 Figure 5. Evolution of the extracted capacitive contrast at 5 GHz of an individual cell in its culture medium, submitted to a chemical treatment.

Acknowledgements

This work was partly supported by the LAAS-CNRS micro and nanotechnologies platform of the French RENATECH network.