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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Submitted on : Sunday, February 24, 2013 - 11:35:28 PM
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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, IOP Publishing, 2013, 46 (9), pp.095303.1-095303.18. ⟨10.1088/1751-8113/46/9/095303⟩. ⟨hal-00766695v2⟩

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