Generalization of Cramér-Rao and Bhattacharyya inequalities for the weighted covariance matrix

Abstract

The paper considers a family of probability distribution depending on a parameter. The goal is to derive the generalized versions of Rao-Cramer, Bhattacharyya inequalities for the weighted covariance matrix and of the Kullback inequality for the weighted Kullback distance which are important objects themselves. The asymptotic forms of these inequalities for a particular family of probability distribution and for a particular class of weight functions are given.