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

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 de nition of generalized Fisher informations, (iv) the identi cation and interpretation of some prior results, and nally, (v) the suggestion of new estimation methods. PACS numbers: 02.50.-r, 05.90.+m, 89.70.-a AMS classi cation scheme numbers: 28D20, 94A17, 62B10, 39B62 ‡ This is a preprint version that di ers from the published version, J. Phys. A: Math. Theor. 45 255303 2012, doi:10.1088/1751-8113/45/25/255303, in minor revisions, pagination and typographics details. ha l-0 07 33 75 0, v er si on 1 19 S ep 2 01 2 Author manuscript, published in "J. Phys. A: Math. Theor. 45 (2012) 255303" DOI : 10.1088/1751-8113/45/25/255303