1211.2008 (J. -F. Bercher)

On generalized Cram'er-Rao inequalities and characterizations of
generalized $q$-Gaussian distributions: the multidimensional case    [PDF]

J. -F. Bercher

Extended Cram\'er-Rao inequalities of estimation theory, together with related Cram\'er-Rao inequalities saturated by generalized $q$-Gaussian distributions, were presented in a recent paper. These results introduced an extended version of Fisher information and a new characterization of generalized $q$-Gaussian distributions which are important in several areas of physics and mathematics. In the present work, we extend these results to the mutidimensional case. We show how the generalized Cram\'er-Rao inequality for the estimation of a parameter can be extended to the mutidimensional case, with a formulation that involves a general norm on $\mathbb{R}^{n}$ and its dual norm. As a particular case, we obtain a new multidimensional Cram\'er-Rao inequality which is saturated by generalized $q$-Gaussian distributions. We give another related Cram\'er-Rao inequality, for a general norm, which is also saturated by these distributions. These results yield a new information theoretic characterization of generalized $q$-Gaussian distributions

View original: http://arxiv.org/abs/1211.2008