Multivariate divergences with application in multisample density ratio models

28/10/2015
Auteurs : Amor Keziou
Publication GSI2015
OAI : oai:www.see.asso.fr:11784:14347
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We introduce what we will call multivariate divergences between K, K ≥ 1, signed finite measures (Q1, . . . , Q K ) and a given reference probability measure P on a σ-field (X,B), extending the well known divergences between two measures, a signed finite measure Q1 and a given probability distribution P. We investigate the Fenchel duality theory for the introduced multivariate divergences viewed as convex functionals on well chosen topological vector spaces of signed finite measures. We obtain new dual representations of these criteria, which we will use to define new family of estimates and test statistics with multiple samples under multiple semiparametric density ratio models. This family contains the estimate and test statistic obtained through empirical likelihood. Moreover, the present approach allows obtaining the asymptotic properties of the estimates and test statistics both under the model and under misspecification. This leads to accurate approximations of the power function for any used criterion, including the empirical likelihood one, which is of its own interest. Moreover, the proposed multivariate divergences can be used, in the context of multiple samples in density ratio models, to define new criteria for model selection and multi-group classification.

Multivariate divergences with application in multisample density ratio models

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application/pdf Multivariate divergences with application in multisample density ratio models Amor Keziou
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We introduce what we will call multivariate divergences between K, K ≥ 1, signed finite measures (Q1, . . . , Q K ) and a given reference probability measure P on a σ-field (X,B), extending the well known divergences between two measures, a signed finite measure Q1 and a given probability distribution P. We investigate the Fenchel duality theory for the introduced multivariate divergences viewed as convex functionals on well chosen topological vector spaces of signed finite measures. We obtain new dual representations of these criteria, which we will use to define new family of estimates and test statistics with multiple samples under multiple semiparametric density ratio models. This family contains the estimate and test statistic obtained through empirical likelihood. Moreover, the present approach allows obtaining the asymptotic properties of the estimates and test statistics both under the model and under misspecification. This leads to accurate approximations of the power function for any used criterion, including the empirical likelihood one, which is of its own interest. Moreover, the proposed multivariate divergences can be used, in the context of multiple samples in density ratio models, to define new criteria for model selection and multi-group classification.

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The MDR model Examples : LRM, comparaison of exponential distributions Mulivariate divergences : Application in inference for multiple samples under density ratio models Amor Keziou Laboratoire de Math´ematiques de Reims France GSI2015 2015, 30 October Ecole Polytechnique, Paris-Saclay (France) 1/41 The MDR model Examples : LRM, comparaison of exponential distributions Contents • The MDR model • The LR model • Inference and classification in LR and two-sample DRM : parametric conditional likelihood (binomial model), Empirical likelihood and its irregularity • Adjustement of the EL • Duality of Multivariate divergences • Inference in MDRM through dual representation of ψ-divergences • Simulation results 2/41 The MDR model Examples : LRM, comparaison of exponential distributions The MDR model We dispose of 1 + K random samples, (X0,1, . . . , X0,n0 ) , (X1,1, . . . , X1,n1 ) , . . . , (XK,1, . . . , XK,nK ) , of 1 + K random variables (or vectors in Rm) X0, X1, . . . , XK , with distributions P, Q1, . . . , QK . The MDRM postulates that dQk dP (x) = exp θk b(x) =: rk(x, θk), ∀k = 1, . . . , K, for some vector-valued known function b(x) := b0(x), b(x) = (1, b1(x), . . . , bd (x)) , x ∈ Rm , with values in R1+d , and corresponding unknown vector-valued parameters θk := (αk, βk ) ∈ Θk ⊂ R1+d , k = 1, . . . , K. 3/41 The MDR model Examples : LRM, comparaison of exponential distributions We require, for all k = 1, . . . , K, the identifiability of the parameter θk, in the sense that if θk b(x) = θk b(x) ∀x − P-a.s., then θk = θk. We will denote then by θT := θ1T , . . . , θKT ∈ R(1+d)K the (unique) true value of the parameter θ := θ1 , . . . , θK . We aim to • estimate the vector parameter θT ; • testing the null hypothesis H0 : Q1 = . . .