On multidimensional generalized Cramér-Rao inequalities, uncertainty relations and characterizations of generalized $q$-Gaussian distributions

Abstract

In the present work, we show how the generalized Cramér-Rao inequality for the estimation of a parameter, presented in a recent paper, can be extended to the mutidimensional case with general norms on ${\mathbb{R}}^n$, and to a wider context. As a particular case, we obtain a new multidimensional Cramér-Rao inequality which is saturated by generalized $q$-Gaussian distributions. We also give another related Cramér-Rao inequality, for a general norm, which is saturated as well by these distributions. Finally, we derive uncertainty relations from these Cramér-Rao inequalities. These uncertainty relations involve moments computed with respect to escort distributions, and we show that some of these relations are saturated by generalized $q$-Gaussian distributions. These results introduce extended versions of Fisher information, new Cramér-Rao inequalities, and new characterizations of generalized $q$-Gaussian distributions which are important in several areas of physics and mathematics.

References in corpus (3)

• Formulation of the uncertainty relations in terms of the Renyi entropies
• The q-exponential family in statistical physics
• q-Gaussians in the porous-medium equation: stability and time evolution