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\((h,\Psi)\)-entropy differential metric. (English) Zbl 0898.62005
Considered are monotone functions of integrals of concave or convex functions of probability densities, called generalized entropies. An example is the class of Rényi entropies. For families of parametrized densities, the Hessians along the directions of tangents of the parameter space define differential metrics of a Riemannian geometry. The corresponding second order covariant tensors depend on the two functions figuring in the definition of entropy. For the class of Rényi entropies these tensors are explicitly calculated when the family of densities is exponential, e.g. Bernoulli, geometric, Pareto, Erlang, etc. Asymptotic normality of the tensors is established under appropriate regularity of the entropy and the family of densities.
Reviewer: I.Vajda (Praha)

MSC:
62B10 Statistical aspects of information-theoretic topics
62E20 Asymptotic distribution theory in statistics
53B20 Local Riemannian geometry
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