On multidimensional generalized Cramér-Rao inequalities, uncertainty relations and characterizations of generalized $q$-Gaussian distributions - Archive ouverte HAL
Journal Articles Journal of Physics A: Mathematical and Theoretical Year : 2013

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.
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Dates and versions

hal-00766695 , version 1 (18-12-2012)
hal-00766695 , version 2 (24-02-2013)

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Jean-François Bercher. On multidimensional generalized Cramér-Rao inequalities, uncertainty relations and characterizations of generalized $q$-Gaussian distributions. Journal of Physics A: Mathematical and Theoretical, 2013, 46 (9), pp.095303.1-095303.18. ⟨10.1088/1751-8113/46/9/095303⟩. ⟨hal-00766695v2⟩
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