Parametric estimation and tests through divergences and duality technique

TL;DR

This paper develops divergence-based estimation and testing methods for parametric models, providing new dual representations, proving their properties, and addressing irregular cases like boundary parameters and mixture components.

Contribution

It introduces a novel dual representation for divergences, extending maximum likelihood methods, and proposes new tests and confidence regions for complex parametric scenarios.

Findings

Proved existence and consistency of divergence-based estimates.

Derived limit laws for estimates and tests under null and alternative hypotheses.

Proposed new tests for mixture models and boundary parameters.

Abstract

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.