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Parametric estimation and tests through divergences and the duality technique. (English) Zbl 1151.62023

Summary: We introduce estimation and test procedures through divergence optimization for discrete or continuous parametric models. The approach is based on a new dual representation for divergences. We treat point estimation and tests for simple and composite hypotheses, extending the maximum likelihood technique. Another view of the maximum likelihood approach, for estimation and tests, 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 under both the null and the alternative hypotheses, and approximations of the power functions are 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 the duality technique.

MSC:

62F12 Asymptotic properties of parametric estimators
62F25 Parametric tolerance and confidence regions
62F03 Parametric hypothesis testing
62F10 Point estimation
62E20 Asymptotic distribution theory in statistics
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