Cumulative Residual <i>q</i>-Fisher Information and Jensen-Cumulative Residual <i>χ</i>² Divergence Measures

Abstract

In this work, we define cumulative residual q-Fisher (CRQF) information measures for the survival function (SF) of the underlying random variables as well as for the model parameter. We also propose q-hazard rate (QHR) function via q-logarithmic function as a new extension of hazard rate function. We show that CRQF information measure can be expressed in terms of the QHR function. We define further generalized cumulative residual χ2 divergence measures between two SFs. We then examine the cumulative residual q-Fisher information for two well-known mixture models, and the corresponding results reveal some interesting connections between the cumulative residual q-Fisher information and the generalized cumulative residual χ2 divergence measures. Further, we define Jensen-cumulative residual χ2 (JCR-χ2) measure and a parametric version of the Jensen-cumulative residual Fisher information measure and then discuss their properties and inter-connections. Finally, for illustrative purposes, we examine a real example of image processing and provide some numerical results in terms of the CRQF information measure.

Keywords

• Shannon entropy
• <i>q</i>-Fisher information
• cumulative residual Fisher information
• Jensen inequality
• hazard function
• cumulative residual chi-square divergence