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Keywords:
$(h,\Phi )$-entropy measures; information metric; geodesic distance between probability distributions; maximum likelihood estimators; asymptotic distributions; Cramér-Rao inequality.; generalized entropies
$(h,\Phi )$-entropy measures; information metric; geodesic distance between probability distributions; maximum likelihood estimators; asymptotic distributions; Cramér-Rao inequality.; generalized entropies
Summary:
Burbea and Rao (1982a, 1982b) gave some general methods for constructing quadratic differential metrics on probability spaces. Using these methods, they obtained the Fisher information metric as a particular case. In this paper we apply the method based on entropy measures to obtain a Riemannian metric based on $(h,\Phi )$-entropy measures (Salicrú et al., 1993). The geodesic distances based on that information metric have been computed for a number of parametric families of distributions. The use of geodesic distances in testing statistical hypotheses is illustrated by an example within the Pareto family. We obtain the asymptotic distribution of the information matrices associated with the metric when the parameter is replaced by its maximum likelihood estimator. The relation between the information matrices and the Cramér-Rao inequality is also obtained.
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