On generalized Cramér–Rao inequalities, generalized Fisher information and characterizations of generalized q-Gaussian distributions

Abstract

This paper deals with Cramér–Rao inequalities in the context of nonextensive statistics and in estimation theory. It gives characterizations of generalized q-Gaussian distributions and introduces generalized versions of Fisher information. The contributions of this paper are (i) the derivation of new extended Cramér–Rao inequalities for the estimation of a parameter, involving general q-moments of the estimation error, (ii) the derivation of Cramér–Rao inequalities saturated by generalized q-Gaussian distributions, (iii) the definition of generalized Fisher information, (iv) the identification and interpretation of some prior results and finally (v) the suggestion of new estimation methods.