Parametric estimation and tests through divergences and the duality technique

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 solution to the irregularity problem of the generalized likelihood ratio test, of the number of components in a mixture, is given, and a new test, based on χ-divergence on signed finite measures and duality technique, is proposed.