Some results on a $χ$-divergence, an~extended~Fisher information and~generalized~Cramér-Rao inequalities
This work provides incremental theoretical extensions in information theory and statistics, with potential applications in estimation and uncertainty relations.
The authors introduced a modified χ-divergence and derived a generalized Fisher information, leading to new Cramér-Rao inequalities and characterizations of generalized q-Gaussians as minimizers of this information.
We propose a modified $χ^β$-divergence, give some of its properties, and show that this leads to the definition of a generalized Fisher information. We give generalized Cramér-Rao inequalities, involving this Fisher information, an extension of the Fisher information matrix, and arbitrary norms and power of the estimation error. In the case of a location parameter, we obtain new characterizations of the generalized $q$-Gaussians, for instance as the distribution with a given moment that minimizes the generalized Fisher information. Finally we indicate how the generalized Fisher information can lead to new uncertainty relations.