Parametric estimation and tests through divergences and duality technique

Michel Broniatowski (1) , Amor Keziou (1, 2)

1 LSTA - Laboratoire de Statistique Théorique et Appliquée
2 LMR - Laboratoire de Mathématiques de Reims

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

Mots clés: $\phi$-divergence; Parametric estimation; Parametric test; Maximum likelihood; Mixture; Boundary valued parameter; Power function; Duality; $\phi$-divergence.

Domaines: Statistiques [math.ST]

Michel Broniatowski, Amor Keziou. Parametric estimation and tests through divergences and duality technique. Journal of Multivariate Analysis, 2009, 100 (1), pp.16-36. ⟨hal-00340793⟩