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

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 Rn, 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. 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. 46, 095303, 2013 doi:10.1088/1751-8113/46/9/095303, in minor revisions, pagination and typographics details. ha l-0 07 66 69 5, v er si on 2 24 F eb 2 01 3 Author manuscript, published in "Journal of Physics A: Mathematical and Theoretical 46, 9 (2013) 095303" DOI : 10.1088/1751-8113/46/9/095303