Fisher information and its extensions based on infinite mixture density functions

In this work, we consider the Fisher information and some of its well-known extended versions and then establish some results based on infinite mixture density functions for the proposed information measures. Specifically we introduce the Jensen-type of Fisher information (parametric-type and density-based) and generalized Fisher information measures based on infinite mixture density functions. We then show that the proposed Jensen–Fisher information measure have two representations in terms of Fisher information distance in the both parametric-type and density-based cases. We have further shown that the generalized Fisher information of an infinite mixture density can be stated based on Pearson–Vajda χ^k divergence. Finally, we examine the convexity property of q−Fisher information measure based on finite mixture density functions under a mild condition. Some examples related to scale mixture of skew-normal family of distributions, with emphasis on skew-normal density, are illustrated within the paper.