Abdellah Ogbi
email: abdellah.ogbi@ec-lyon.fr

Laurent Nicolas

Ronan Perrussel

Sheppard J Salon

Damien Voyer

Numerical identification of effective multipole moments of polarizable particles

Keywords: Electromechanical effects, multipole moments, polarization

In electromechanics of particles, the effective moment method relies on the knowledge of the induced multipole moments. For arbitrary shaped particles, only linear multipole moments are usually considered in the literature. On cylindrically symmetric particles, we show that the neglected multipole moments can be computed, and that they should be taken into consideration.

I. INTRODUCTION

W HEN a dielectric particle is subjected to an electro- static field, interfacial polarization mechanisms cause a charge accumulation on the surface between the particle and the medium in which it is suspended [START_REF] Pohl | Dielectrophoresis of cells[END_REF], [START_REF] Morgan | AC electrokinetics: colloids and nanoparticles, ser. Microtechnologies and microsystems series[END_REF]. Usually this polarization gives rise to an induced dipole that will interact with the external field to create forces and torques [START_REF] Pohl | Dielectrophoresis of cells[END_REF], [START_REF] Jones | Electromechanics of particles[END_REF]. This interaction is exploited in the dielectrophoretic phenomena such as electrorotation and dielectrophoresis for the manipulation and the characterization of biological particles [START_REF] Pohl | Dielectrophoresis of cells[END_REF], [START_REF] Jones | Electromechanics of particles[END_REF].

In the conventional dielectrophoresis theory, it is usually assumed that only a dipole is induced [START_REF] Cao | A numerical analysis of forces imposed on particles in conventional dielectrophoresis in microchannels with interdigitated electrodes[END_REF], [START_REF] Zimmermann | Electromanipulation of Cells[END_REF]. Although this approximation seems to be adequate in many circumstances, exceptions exist and higher-order multipole corrections are required to predict the particle behavior. In particular, these higher-order corrections can influence forces and torques exerted on cells in planar quadrupolar traps and multilayer structures when the particle is at a zero field, or when the particle size becomes comparable to the dimensions of the electrode structure used to generate the electric field [START_REF] Masao | Higher-order dielectrophoretic effects: levitation at a field null[END_REF]- [START_REF] Schnelle | Combined dielectrophoretic field cages and laser tweezers for electrorotation[END_REF]. The effective moment method developed by Jones and Washizu, use higher-order multipole moments to provide a more accurate estimation of the force and the torque exerted upon the manipulated particle [START_REF] Jones | Multipolar dielectrophoretic and electrorotation theory[END_REF]- [START_REF] Washizu | Equivalent multipole-moment theory for dielectrophoresis and electrorotation in electromagnetic field[END_REF]. In this method, the particle under consideration is substituted by a set of multipoles (dipole, quadrupole, octupole,...) producing the same field distortion caused by the presence of the particle; the force exerted on multipoles should be equal to the force exerted on the particle by the external field [START_REF] Jones | Electromechanics of particles[END_REF], [START_REF] Wang | General expressions for dielectrophoretic force and electrorotational torque derived using the Maxwell stress tensor method[END_REF]. The simplest multipole model considers only the linear multipoles. Green and Jones use this model when determining the dipole moment and higher-order moments [START_REF] Green | Numerical determination of the effective moments of non-spherical particles[END_REF]. The linear multipoles can be represented by an arrangement of equally spaced positive and negative charge centers along the axis of the dipole (Fig. 1). However, this linear multipole model cannot be used when the external field is not aligned with the symmetry axis of a cylindrically symmetric particle. It is unfortunately the most frequent situation in biological dielectrophoretic applications where, for instance, the radial component of the field is higher than the z-component (Fig. 2).

Here, we also take into account the neglected multipole moments to identify an equivalent model of the polarized particle (Fig. 3) from the induced potential. The multipole expansion is first recalled. On cylindrically symmetric particles, a procedure is then proposed to compute all the multipole moments. Numerical results show the relevance of the approach.

II. MODEL AND METHOD

A. Induced potential

Consider a particle immersed in a homogeneous medium and exposed to an electric field generated by an external system. The particle and the surrounding medium are assumed to have linear dielectric properties. The electrostatic potential in the system containing the different materials, is obtained by solving a Laplace type equation. The total electric potential U total outside the particle is the sum of both applied potential U ext and induced potential U ind caused by the polarization of the particle, i.e.

U total = U ext + U ind = n,m A n,m r n Y m n (θ, ϕ) + n,m B n,m r n+1 Y m n (θ, ϕ), (1)
where Y m n are the complex spherical harmonics, and (r, θ, ϕ) a spherical coordinate system. The magnitudes A n,m and B n,m can be determined numerically or analytically using proper boundary conditions.

The multipole polarization coefficients p m n are related to the induced potential constants B n,m by

p m n = N m n B n,m , with N m n = (2n + 1) 4π (n -m)! (n + m)! . (2)
The linear multipoles correspond to the case m = 0 in the expression (2). For a spherical particle with radius a and permittivity ε 2 suspended in a medium with permittivity ε 1 , the B n,m can be expressed in terms of the A n,m [START_REF] Jones | Electromechanics of particles[END_REF]:

B n,m = K n (2n + 1) a 2n+1 A n,m , (3)
where

K n = n(2n + 1)(ε 1 -ε 2 ) (n + 1)ε 2 + nε 1 , (4)
is the generalized Clausius-Mossotti factor.

B. Calculation of the effective multipole moments

The constants A n,m extracted from the calculation of the potential in the system without the particle are used as source terms for a local problem including the particle. Then, based on the spherical harmonics representation for the induced potential, we identify the effective moments.

Using the orthogonality property of the spherical harmonics the individual moments are determined according to

p m n = N m n R n+1 S Y m n (θ, ϕ) U ind (R, θ, ϕ)ds, (5)
where S is a spherical surface centered on the particle with a radius R larger than the largest particle dimension (Fig. 4).

C. Solution of the problem

The particle of interest in our problem is cylindrically symmetric with permittivity ε 2 , immersed in a containing medium with permittivity ε 1 . The particle is assumed to be sufficiently small to consider the external electric field constant in its neighborhood. Consequently, the 3D problem is transformed into two axisymmetric problems (Fig. 5). The electric potential in the whole space is obtained using both solutions:

U (r, z, θ) = U z + U r cos(ϕ). (6)
Two linear problems for U z and U r are then required to be solved in the computational domain Ω:

III. NUMERICAL RESULTS

The numerical solution of problems ( 7) and ( 8) is performed by using the finite element method and the implementation is based on the getfem++ finite element library [START_REF] Renard | getfem++ finite element library[END_REF] (Fig. 6). Fig. 7. Radial and z-axis components of the dipole moment for a spheroidal particle. The numerical results are plotted as symbols, while the corresponding analytical values pr and pz, given by ( 9) and [START_REF] Washizu | Multipolar dielectrophoretic force calculation[END_REF], are plotted as dotted lines.

The geometries and the adaptive mesh of the computational domain, were built using the three-dimensional finite element mesh generator Gmsh, and the numerical integration required in [START_REF] Zimmermann | Electromanipulation of Cells[END_REF], was carried out using Gaussian quadrature. The method described in the previous section enables to handle nonspherical particles, for which there is no analytical method for identifying the higher-order moments. A particular shape frequently invoked in biological cells modeling [START_REF] Gao | Theory of ac electrokinetic behavior of spheroidal cell suspensions with an intrinsic dispersion[END_REF], is the spheroidal shape. By changing the eccentricity ratio (Fig. 8), the shape dependence of the particle polarization can be identified (Fig. 9).

A. Dipole moment

An analytical representation of the dipole moment for the ellipsoidal shape exists. In particular for a spheroid with eccentricity ratio α = c/a (Fig. 8), the dipole polarization factors are given by:

p r = ca 2 3 (ε 2 -ε 1 ) [ε 2 + L r (ε 1 -ε 2 )] (9)
p z = ca 2 3 (ε 2 -ε 1 ) [ε 2 + L z (ε 1 -ε 2 )] (10)
where the magnitudes L r and L z are the depolarization factors, given for the case of an oblate spheroid by [START_REF] Rivette | A note on the electrostatic force and torque acting on an isolated body in an electric field[END_REF]:

L r = 1 2(1 -α -2 ) - α -2 4(1 -α -2 ) 3/2 ln 1 + √ 1 -α -2 1 - √ 1 -α -2 , (11)
L z = α -2 1 -α -2 + α -2 2(1 -α -2 ) 3/2 ln 1 + √ 1 -α -2 1 - √ 1 -α -2 . ( 12
)
For the spherical case, the depolarization factors verify L r = L z = 1/3, and the polarization coefficients are given by Clausius-Mossotti factor K 1 . In Fig. 7, these analytical values are compared with the computed values of the dipole moment and it shows a good agreement.

B. Higher-order moments

The higher-order moments are obtained only numerically using the proposed approach. The first case considered is a homogeneous spherical particle (α = 1). In this case, only the dipole moment is non-zero and matches well with the analytical values given by the Clausius-Mossotti factor K 1 . As it was expected from the isotropy of the sphere, note that the radial and the z-axis dipole moments have the same magnitude (see upper row in Fig. 9). For lower eccentricities i.e α < 1 (oblate spheroids), the numerical results in Fig. 9 (middle and lower rows) indicate that the dipole component decreases when α decreases, particularly in the z-direction. This behavior is also predictable with the analytical description of the dipole moments with the depolarization factors. Besides, the magnitudes of the identified higher-order components, become significant as α decreases (with α < 1), in particular, the octupole moments has important magnitudes in both the radial and the z-axis directions, with larger values when the particle permittivity is greater than the surrounding medium permittivity.

IV. CONCLUSION

In this paper, a complete multipole model was exploited in a numerical approach for the identification of the effective multipole moments. This complete multipole model enables to identify multipole components (terms) that are displaced from the central axis (m = 0). Only such terms can describe the observed interaction of a particle placed in an external field that is not aligned with its central axis, such as in the center of the quadrupolar electrode structure [START_REF] Schnelle | The influence of higher moments on particle behaviour in dielectrophoretic field cages[END_REF]. The next step in this work is to extend this approach to layered particles model invoked in complex biological cells modeling. Fig. 9. Numerical results following the radial axis (column left) and the z-axis (column right), for spheroidal (oblate) particles of different aspect ratios (eccentricity) c/a. The five first orders are plotted as symbols against the ratio of particle permittivity to fluid medium permittivity ε 2 /ε 1 .

Fig. 1 .

 1 Fig. 1. Stratton's representation of the first four linear multipoles.

Fig. 2 .

 2 Fig.2. Example of a 3D system of electrodes and a particle.

Fig. 3 .

 3 Fig. 3. A 3D representation of general multipoles with the corresponding orders (n, m).

Fig. 4 .

 4 Fig. 4. Integration surface around the particle.

d/ 2 Fig. 5 .

 25 Fig. 5. Boundary conditions for the radial and z-axis potential calculations.

Fig. 6 .

 6 Fig.6. Details of the solutions Ur (left) and Uz (right). The white line is the boundary of the particle. ε 2 /ε 1 = 10 4 .

Fig. 8 .

 8 Fig. 8. Oblate spheroids with eccentricities 5:1 and 2:1.


Paul Rochet

Bayesian interpretation of generalized empirical likelihood by maximum entropy

We study a parametric estimation problem related to moment condition models. As an alternative to the generalized empirical likelihood (GEL) and the generalized method of moments (GMM), a Bayesian approach to the problem can be adopted, extending the MEM procedure to parametric moment conditions. We show in particular that a large number of GEL estimators can be interpreted as a maximum entropy solution. Moreover, we provide a more general field of applications by proving the method to be robust to approximate moment conditions.

Introduction

We consider a parametric estimation problem in a moment condition model. Assume we observe an i.i.d. sample X 1 , ..., X n drawn from an unknown probability measure µ 0 , we are interested in recovering a parameter θ 0 ∈ Θ ⊂ R d , defined by a set of moment conditions Φ(θ 0 , x)dµ 0 (x) = 0,

where Φ : θ × X → R k is a known map. This model is involved in many problems in Econometry, notably when dealing with instrumental variables. We refer to [START_REF] Chamberlain | Asymptotic efficiency in estimation with conditional moment restrictions[END_REF], [START_REF] Hansen | Large sample properties of generalized method of moments estimators[END_REF], [START_REF] Qin | Empirical likelihood and general estimating equations[END_REF], [START_REF] Owen | Empirical likelihood for linear models[END_REF] and [START_REF] Donald | Choosing instrumental variables in conditional moment restriction models[END_REF]. Two main approaches to the problem have been studied in the literature, namely the generalized method of moments (GMM) and the generalized empirical likelihood (GEL). While the main advantage of GMM relies in its computational feasibility, likelihood-related methods have appeared to be the most efficient in term of small-sample properties. In its original form, the empirical likelihood (EL) of Owen [START_REF] Owen | Empirical likelihood for linear models[END_REF] defines an estimator by a maximum likelihood 1 procedure on a discretized version of the model. As an alternative, GEL replaces the Kullback criterion relative to EL by a f -divergence, thus providing a large choice of solutions. A number of estimators corresponding to particular choices of f -divergences have emerged in the literature over the last decades, such as the exponential tilting (ET) of Kitamura and Stutzer [START_REF] Kitamura | An information-theoretic alternative to generalized method of moments estimation[END_REF] and the continuous updating estimator (CUE) of Hansen, Yeaton and Yaron [START_REF] Hansen | Finite-sample properties of some alternative gmm estimators[END_REF].

While an attractive feature of GEL is its wide range of solutions, a number of f -divergence used in the computation of the GEL estimators are mainly justified by empirical studies and lack a probabilistic interpretation. This issue can be solved by incorporating some prior information to the problem using a Bayesian point of view, as made in [START_REF] Florens | Bayes, bootstrap, moments[END_REF]. In this paper, we investigate a different Bayesian approach to the inverse problem, known as maximum entropy on the mean (MEM). Although the method was originally introduced in the frame of exact moment condition models (as opposed here to parametric moment conditions), it appears to provide a natural solution to the problem, expressed as the minimizer of a convex functional on a set of discrete measures and subject to linear constraints. When applied in a particular setting, we show that the MEM approach leads to a GEL solution for which the f -divergence is determined by the choice of the prior. As a result, the method gives an alternate point of view on some widely spread estimators such as EL, ET or CUE, as well as a general Bayesian background to GEL.

In many actual situations, the true moment condition is not exactly known to the statistician and only an approximation is available. It occurs for instance when Φ has a complicated form that must be evaluated numerically. Simulation-based methods have been implemented to deal with approximate constraints in [START_REF] Carrasco | Generalization of GMM to a continuum of moment conditions[END_REF] and [START_REF] Mcfadden | A method of simulated moments for estimation of discrete response models without numerical integration[END_REF], in the frame of the generalized method of moments. To our knowledge, the efficiency of GEL in a similar situation has not been studied. In [START_REF] Loubes | Maximum entropy solution to illposed inverse problems with approximately known operator[END_REF], the MEM procedure is shown to be robust to approximate moment conditions, introducing the approximate maximum entropy on the mean estimator. Seeing GEL as a particular case of MEM, we extend the model in a situation where only an approximation Φ m of the true constraint function Φ is available. We provide sufficient conditions under which the GEL method remains efficient asymptotically when replacing Φ by its approximation. This paper falls into the following parts. Section 2 is devoted to the position of the problem. We introduce the maximum entropy method for parametric moment condition models and discuss its close relationship with generalized empirical likelihood in Section 2.2. In Section 3, we discuss the asymptotic efficiency of the method when dealing with an approximate constraint. Proofs are postponed to the Appendix.

Estimation of the parameter

Let X be an open subset of R q , endowed with its Borel field B(X ) and let P(X ) denote the set of probability measures on X . We observe an i.i.d. sample X 1 , ..., X n drawn from the unknown distribution µ 0 . We want to estimate the parameter θ 0 ∈ Θ ⊂ R d defined by the moment condition

X Φ(θ 0 , x)dµ 0 (x) = 0, (2)
where

Φ : Θ × X → R k (k ≥ d) is a known map.
To avoid a problem of identifiability, we assume that θ 0 is the unique solution to (2). This problem has many applications in Econometry, see for instance [START_REF] Chamberlain | Asymptotic efficiency in estimation with conditional moment restrictions[END_REF], [START_REF] Hansen | Large sample properties of generalized method of moments estimators[END_REF] and [START_REF] Qin | Empirical likelihood and general estimating equations[END_REF]. The information given by the moment condition (2) can be interpreted to determine the set M of possible values for µ 0 (the model). The true value of the parameter being unknown, the distribution of the observations can be any probability measure µ for which the map θ → Φ(θ, .)dµ is null for a unique θ = θ(µ) ∈ Θ. The model is therefore defined as

M = µ ∈ P(X ) : ∃! θ = θ(µ) ∈ Θ, Φ(θ, .)dµ = 0 ,
where the map µ → θ(µ), defined on M, is the parameter of interest. Let us introduce some notations and assumptions. For µ a measure and g a function, we shall note µ[g] = gdµ. Let E be an Euclidean space and let . denote an Euclidean norm in E. For a function f : Θ → E and a set S ⊆ Θ, we note

f S = sup θ∈S f (θ) .
We assume that the following conditions are fulfilled.

A.1. Θ is a compact subset of R d .

A.2. The true value θ 0 of the parameter lies in the interior of Θ.

A.3. For all x ∈ X , θ → Φ(θ, x) is continuous on Θ and the map x → Φ(., x) Θ is dominated by a µ 0 -integrable function.

A.4. For all x ∈ X , θ → Φ(θ, x) is twice continuously differentiable in a neighborhood N of θ 0 and we note ∇Φ(θ, x) = ∂Φ(θ, x)/∂θ ∈ R d×k and Ψ(θ, x) = ∂ 2 Φ(θ, x)/∂θ∂θ t ∈ R d×d×k (where a t stands for the transpose of a). Moreover, we assume that x → ∇Φ(., x) N and x → Ψ(., x) N are dominated by a µ 0 -integrable function.

A.5. The matrices

D := X ∇Φ(θ 0 , .)dµ 0 ∈ R d×k and V := X Φ(θ 0 , .)Φ t (θ 0 , )dµ 0 ∈ R k×k are of full rank.
Some issues for estimating θ 0 may be due to the indirect definition of the parameter and these assumptions ensure that the map θ(.) is sufficiently smooth in a neighborhood of µ 0 for the total variation topology, which will make the asymptotic properties of the GEL estimator easily tractable (see for instance [START_REF] Newey | Higher order properties of GMM and generalized empirical likelihood estimators[END_REF]).

Generalized empirical likelihood

Generalized empirical likelihood (GEL) was first applied to this problem in [START_REF] Qin | Empirical likelihood and general estimating equations[END_REF], generalizing an idea of [START_REF] Owen | Empirical likelihood for linear models[END_REF]. An estimate μ of µ is obtained as an entropic projection of the empirical measure P n onto the model M. Precisely, for two probability measures µ and ν and f a convex function such that f

(1) = f ′ (1) = 0, define D f (ν|µ) = X f dν dµ dµ if ν ≪ µ, D f (ν|µ) = +∞ otherwise.
Moreover, we define for A ⊂ P(X ), D f (A|µ) = inf ν∈A D f (ν|µ). The GEL estimator μ of µ 0 is the element of the model that minimizes a given fdivergence D f (., P n ) with respect to the empirical distribution. Noticing that M = ∪ θ∈Θ M θ where M θ := {µ ∈ P(X ) : µ[Φ(θ, .)] = 0}, the GEL estimator θ = θ(μ) follows by θ = arg min

θ∈Θ D f (M θ , P n ).
Since the set of discrete measures in M θ is closed and convex, the entropy

D f (M θ , P n ) is reached for a unique measure μ(θ) in M θ , provided that D f (M θ , P n ) is finite.
Then, it appears that computing the GEL estimator involves a two-step procedure. First, build for each θ ∈ Θ, the entropic projection μ(θ) of P n onto M θ . Then, minimize D f (μ(θ), P n ) with respect to θ. Since μ(θ) is absolutely continuous w.r.t. P n by construction, minimizing D f (., P n ) reduces to finding the proper weights p 1 , ..., p n to allocate to the observations X 1 , ..., X n . This turns into a finite dimensional problem, which can be solved by classical convex optimization tools (see for instance [START_REF] Kitamura | Empirical likelihood methods in econometrics: Theory and practice[END_REF]).

In fact, the GEL estimator θ can be expressed as the solution to the saddle point problem

θ = arg min θ∈Θ sup (γ,λ)∈R×R k γ -P n f * (γ + λ t Φ(θ, .)) ,
where f * (x) = sup y {xyf (y)} denotes the convex conjugate of f .

Note that if the choice of the f -divergence plays a key role in the construction of the estimator, it has no influence on its asymptotic efficiency. Indeed, it is shown in [START_REF] Qin | Empirical likelihood and general estimating equations[END_REF] that all GEL estimators are asymptotically efficient, regardless of the f -divergence used for their computation. Nevertheless, some situations justify the use of specific f -divergences. The empirical likelihood estimator introduced by Owen in [START_REF] Owen | Empirical likelihood for linear models[END_REF] uses the Kullback entropy K(., .) as f -divergence, pointing out that minimizing K(., P n ) reduces to maximizing likelihood among multinomial distributions. Newey and Smith [START_REF] Newey | Higher order properties of GMM and generalized empirical likelihood estimators[END_REF] remark that a quadratic f -divergence leads to the CUE estimator of Hansen Heaton and Yaron [START_REF] Hansen | Finite-sample properties of some alternative gmm estimators[END_REF].

Maximum entropy on the mean

In this section, we study a Bayesian approach to the inverse problem, known as maximum entropy on the mean (MEM) [START_REF] Gamboa | Bayesian methods and maximum entropy for ill-posed inverse problems[END_REF]. The method was developed to estimate a measure µ 0 based the observation of some of its moments. In this framework, it turns out that the MEM estimator of µ 0 can be used to estimate efficiently the parameter θ 0 . We shall briefly recall the MEM procedure. Consider an estimator of µ 0 in the form of a weighted version of the empirical measure P n ,

P n (w) = 1 n n i=1 w i δ X i ,
for w = (w 1 , ..., w n ) ′ ∈ R n a collection of weights. Then, fix a prior distribution ν 0 on the vector of weight w so that each solution P n (w) can be viewed as a realization of the random measure P n (W ), where W is drawn from ν 0 . This setting enables to incorporate some prior knowledge on the shape or support of µ 0 through the choice of the prior ν 0 , as discussed in [START_REF] Gamboa | Bayesian methods and maximum entropy for ill-posed inverse problems[END_REF].

Here, the observations X 1 , ..., X n are considered fixed. Actually, it is the moment condition that is used to built the estimator a posteriori. In this framework where the true value θ 0 of the parameter is unknown, the information provided by the moment condition reduces to the statement µ 0 ∈ M. So, in order to take this information into consideration, the underlying idea of MEM is to build the estimator μ as the expectation of P n (W ) conditionally to the event {P n (W ) ∈ M}. However, we may encounter some difficulties if this conditional expectation is not properly defined. To deal with this issue, the MEM method replaces the possibly ill-defined conditional expectation by a well-defined estimator, whose construction is motivated by large deviation principles. Precisely, construct the posterior distribution ν * as the entropic projection of ν 0 onto the set

Π(M) = {µ ∈ P(R n ), E µ [P n (W )] ∈ M} ,
where E µ [P n (W )] denotes the expectation of P n (W ) when W has distribution µ. The MEM solution to the inverse problem is defined as the expectation of P n (W ) under the posterior distribution ν * ,

μ = E ν * [P n (W )] = P n (E ν * (W )).
This construction is justified by the large deviation principle stated in Theorem 2.3 in [START_REF] Gamboa | Bayesian methods and maximum entropy for ill-posed inverse problems[END_REF], which establishes the asymptotic equivalence between μ and the conditional expectation E ν 0 (P n (W )| P n (W ) ∈ M), whenever it is well defined. The existence of the MEM estimator requires the problem to be feasible in the sense that there exists at least one solution δ in the interior of the convex hull of the support of ν 0 , such that P n (δ) ∈ M. This assumption warrants that the set Π(M) is non-empty and therefore allows the construction of the posterior distribution ν * .

The MEM estimator μ lies in the model M by construction. As a result, there exists a solution θ to the moment condition μ[Φ(θ, .)] = 0. this solution is precisely the MEM estimator of θ 0 . In Theorem 2.1 below, we give an explicit expression for the MEM estimator θ. We note 1 = (1, ..., 1) t ∈ R n , Φ(θ, X) = (Φ(θ, X 1 ), ..., Φ(θ, X n )) t ∈ R n×k and as previously, Λ ν denotes the log-Laplace transform of ν.

Theorem 2.1 If the problem is feasible, the MEM estimator θ is given by

θ = arg min θ∈Θ sup (γ,λ)∈R×R k {nγ -Λ ν 0 (γ1 + Φ(θ, X)λ)} .
In particular, if ν 0 has equal orthogonal marginals, i.e. ν 0 = ν ⊗n for some probability measure ν on R, then

θ = arg min θ∈Θ sup (γ,λ)∈R×R k γ -P n Λ ν (γ + λ t Φ(θ, .)) .
The MEM estimator θ can be expressed as the solution to a saddle point problem, specific to generalized empirical likelihood. Actually, this result points out that maximum entropy on the mean with a particular form of prior ν 0 = ν ⊗n leads to a GEL procedure, for which the criterion is the log-Laplace transform of ν. This approach provides a general Bayesian interpretation of GEL. Regularity conditions on the criterion Λ ν in the GEL framework are reflected through conditions on the prior ν. Indeed, the usual normalization conditions Λ ′ ν (0) = Λ ′′ ν (0) = 1 corresponds to taking a prior ν with mean and variance equal to one, while the normalization Λ ν (0) = 0 is imposed by the condition ν ∈ P(R).

An interesting choice of the prior is the exponential distribution dν(x) = e -x dx for x > 0. Indeed, observe that if the W i are i.i.d. with exponential distribution, the likelihood of P n (W ) is constant over the set of probability discrete measures {P n (w) : n i=1 w i = n}. Hence, an exponential prior can be roughly interpreted as a non-informative prior in this framework. The discrepancy associated to this prior is Λ ν (s) =log(1s), s < 1, which corresponds to the empirical likelihood estimator of Owen [START_REF] Owen | Empirical likelihood for linear models[END_REF].

The MEM approach also provides a new probabilistic interpretation of some commonly used specific GEL estimators. The exponential tilting of Kitamura and Stutzer [START_REF] Kitamura | An information-theoretic alternative to generalized method of moments estimation[END_REF] is obtained for a Poisson prior of parameter 1, for which we have Λ ν (s) = e s -1. Another example is the Gaussian prior ν ∼ N (1, 1), leading to the continuous updating estimator of Hansen, Yeaton and Yaron [START_REF] Hansen | Finite-sample properties of some alternative gmm estimators[END_REF], as we have in this case Λ ν (s) = 1 2 (s -1) 2 . The Gaussian prior allows the discrete measure P n (W ) to have negative weights w i and must be handled with care. Remark however that this is generally not an issue in practice since the solution μ is implicitly chosen close to the empirical distribution P n and will have all its weights w i positive with high probability. More examples of classical priors leading to usual discrepancies can be found in [START_REF] Gamboa | Bayesian methods and maximum entropy for ill-posed inverse problems[END_REF].

Dealing with an approximate operator

In many actual applications, only an approximation of the constraint function Φ is available to the practitioner. This occurs for instance if the moment condition takes a complicated form that can only be evaluated numerically. In [START_REF] Mcfadden | A method of simulated moments for estimation of discrete response models without numerical integration[END_REF], McFadden suggested a method dealing with approximate constraint in a similar situation, introducing the method of simulated moments (see also [START_REF] Carrasco | Generalization of GMM to a continuum of moment conditions[END_REF]). In [START_REF] Loubes | Maximum entropy solution to illposed inverse problems with approximately known operator[END_REF] and [START_REF] Loubes | Regularization with approximated l 2 maximum entropy method[END_REF], the authors study a MEM procedure for linear inverse problems with approximate constraints. Here, we propose to extend the results of [START_REF] Loubes | Maximum entropy solution to illposed inverse problems with approximately known operator[END_REF] and [START_REF] Loubes | Regularization with approximated l 2 maximum entropy method[END_REF] to the GEL framework, using the connections between GEL and MEM.

We assume that we observe a sequence {Φ m } m∈N of approximate constraints, independent with the original sample X 1 , ..., X n and converging toward the true function Φ at a rate ϕ m . We are interested in exhibiting sufficient conditions on the sequence {Φ m } m∈N under which estimating θ 0 by the GEL procedure remains efficient when the constraint is replaced by its approximation. We discuss the asymptotic properties of the resulting estimates in a framework where both indices n and m simultaneously grow to infinity.

The approximate estimator is obtained by the GEL methodology, replacing the constraint function Φ by its approximation Φ m , θm = arg min

θ∈Θ sup (γ,λ)∈R×R k γ -P n Λ(γ + λ t Φ m (θ, .)) , (3)
where Λ : R → R is a strictly convex, twice differentiable function such that Λ ′ (0) = Λ ′′ (0) = 1 and Λ(0) = 0. As previously, the existence of θm requires the feasibility condition that the supremum of γ -P n [Λ(γ + λ t Φ m (θ, .))] is reached for a finite value of (γ, λ) ∈ R × R k , for at least one value of θ ∈ Θ. This condition relies essentially on the domain of Λ being sufficiently widespread. We make the following additional assumptions.

A.6. The functions x → Φ(., x) Θ , x → ∇Φ(., x) N and x → Ψ(., x) N are dominated by a function κ such that κ 4 (x)dµ 0 (x) < ∞.

A.7. For all x ∈ X and for sufficiently large m, the map θ → Φ m (θ, .) is twice continuously differentiable in N and we note ∇Φ m (θ, .) = ∂Φ m (θ, .)/∂θ and Ψ m (θ, .) = ∂ 2 Φ m (θ, .)/∂θ∂θ t .

A.8. The functions x → Φ m (., x)-Φ(., x) Θ , x → ∇Φ m (., x)-∇Φ(., x) N and x → Ψ m (., x) -Ψ(., x) N are dominated by a function κ m such that κ 4 m (x)dµ 0 (x) = O(ϕ -4 m ). A.9. The function Λ ′′ is bounded by a constant K < ∞.

Assumptions A.6 to A.8 are made to obtain a uniform control over θmθ for all n ∈ N. The condition A.9 implies that Λ is dominated by a quadratic function. In the MEM point of view, this condition is fulfilled for the log-Laplace transform Λ ν of sub-Gaussian priors ν. Moreover, θm is √ n-consistent and asymptotically equivalent to the GEL estimator computed with exact constraint function Φ whenever nϕ -2 m tends to zero.

By considering a situation with approximate operator, we extend the GEL model to a more general framework that gives a more realistic formulation of actual problems. The previous theorem gives an upper bound of the error caused by the use of the approximation Φ m in place of the true function Φ. By this result, we aim to provide an insight on convergence conditions that are necessary for asymptotic efficiency when dealing with an approximate operator.

Proofs

4.1 Proof of Theorem 2.1 The feasibility assumption warrants that the extrema are reached. Hence, using Sion's minimax Theorem, we find θ = arg min

Let S θ = {w ∈ R n : P n (w) ∈ M θ } and F w = {µ ∈ P(R n ) : E µ (W ) = w}. We use that inf µ∈Fw K(µ, ν 0 ) = Λ * ν 0 (w) (see [GG97]). Let Π(M θ ) = {µ ∈ P(R n ), E µ [P n (W )] ∈ M θ },
θ∈Θ sup τ ∈R n inf w∈S θ {τ t w -Λ ν 0 (τ )},
We know that w = (w 1 , ..., w n ) t ∈ S θ if and only if n i=1 w i = n and n i=1 w i Φ(θ, X i ) = 0. Thus, for a fixed value of τ , the map w → τ t w -Λ ν 0 (τ ) can be arbitrarily close to -∞ on S θ whenever τ is not orthogonal to 1 and Φ(θ, X). As a result, we may assume that τ = γ1 + Φ(θ, X)λ for some (γ, λ) ∈ R × R k without loss of generality. In this case, the map w → τ t w -Λ ν 0 (τ ) is constant over S θ , equal to nγ -Λ ν 0 (γ1 + Φ(θ, X)λ), which ends the proof. If ν 0 = ν ⊗n , then Λ ν 0 (w) = n i=1 Λ ν (w i ) and we conclude easily.

Proof of Theorem 3.1

The proof of the results relies mainly on the uniform law of large numbers, using that the set { Φ m (θ, .) , ∇Φ m (θ, .) , Ψ m (θ, .) , θ ∈ Θ, m ∈ N} is a Glivenko-Cantelli class of functions, consequently to A.6 and A.8. For all

θ ∈ Θ, v ∈ R k , x ∈ X , let h n (θ, v) = P n [Φ(θ, .)Λ ′ (v t Φ(θ, .))] P n [v t ∇Φ t (θ, .)Λ ′ (v t Φ(θ, .))] h m,n (θ, v) = P n [Φ m (θ, .)Λ ′ (v t Φ m (θ, .))] P n [v t ∇Φ t m (θ, .)Λ ′ (v t Φ m (θ, .))]
.

The pair ( θm , vm ) (resp. ( θ, v)) is defined as the unique zero over Θ × R k of h m,n (resp. h n ). The condition A.9 implies that there exists a constant K > 0 such that Λ ′ (s) ≤ Ks + 1 for all s ∈ R. Hence, using successively the mean value theorem and Cauchy-Schwarz's inequality, we show that the contrast function h m,n converges uniformly on every compact set toward h n as m → ∞, which warrants the convergence of ( θm , vm ) toward ( θ, v). For all v ∈ R k , the application θ → ∇h m,n (θ, v) is continuous in a neighborhood on θ * m for sufficiently large values of m by the condition A.7, as explicit calculation gives

∇h m,n (θ, v) = A m,n (θ, v) D m,n (θ, v) D t m,n (θ, v) V m,n (θ, v)
, where A m,n (θ, v) = P n Ψ m (θ, .)vΛ ′ (v t Φ m (θ, .)) +∇Φ m (θ, .)v v t ∇Φ t m (θ, .)Λ ′′ (v t Φ m (θ, .)) D m,n (θ, v) = P n ∇Φ m (θ, .)Λ ′ (v t Φ m (θ, .)) +∇Φ m (θ, .)vΦ t m (θ, .)Λ ′′ (v t Φ m (θ, .)) V m,n (θ, v) = P n Φ m (θ, .)Φ t m (θ, .)Λ ′′ (v t Φ m (θ, .)) . We define in the same way A n (θ, v), D n (θ, v) and V n (θ, v) by replacing Φ m by Φ in the expressions above. Using Cauchy-Schwarz's inequality, A.8 ensures the uniform convergence of ∇h m,n toward ∇h n on every compact set at the rate ϕ m . Note ρ n the smallest eigenvalue of ∇h n ( θ, v), we know from Theorem 3.2 in [START_REF] Newey | Higher order properties of GMM and generalized empirical likelihood estimators[END_REF] that P(ρ n > η) = O(n -1 ) for sufficiently small η > 0, since A.5 ensures that the limit of ∇h n ( θ, v) as n → ∞ is positive definite. Thus, for c > 0 sufficiently small, consider the event Ω = {ρ n > c}. Writing the Taylor expansion

Theorem 3. 1 (

 1 Robustness of GEL) If Assumptions 1 to 9 hold, n θmθ 2 = O P (nϕ -2 m ) + o P (1).

  {τ t w -Λ ν 0 (τ )}.

  h m,n ( θ, v) = ∇h m,n ( θm , vm ) θθm vvm + o( θmθ ),we deduce that on Ω,θmθ vmv = -∇h n ( θ, v) -1 h m,n ( θ, v) + O P (ϕ -1 m ).

The Schur complement formula gives in particular

where Dn = D n ( θ, v) and Vn = V n ( θ, v) and where we used that v = O P (n -1 ) (see for instance Theorem 3.2 in [START_REF] Newey | Higher order properties of GMM and generalized empirical likelihood estimators[END_REF]). Thus, on the event Ω,

By construction, P n [Φ( θ, .)Λ ′ (v t Φ( θ, .

))] = 0, which yields

as a consequence of A.9. We conclude that θmθ 2 1 Ω = O P (ϕ -2 m )+o P (n -1 ) by the condition A.8. On the complement of Ω, θmθ can be bounded by the diameter δ of Θ, yielding θmθ 1 Ω c = o P (n -1 ), which ends the proof.