Bivariate extension of (dynamic) cumulative past entropy

ABSTRACT

Recently, the concept of cumulative residual entropy (CRE) has been studied by many researchers in higher dimensions. In this article, we extend the definition of (dynamic) cumulative past entropy (DCPE), a dual measure of (dynamic) CRE, to bivariate setup and obtain some of its properties including bounds. We also look into the problem of extending DCPE for conditionally specified models. Several properties, including monotonicity, and bounds of DCPE are obtained for conditional distributions. It is shown that the proposed measure uniquely determines the distribution function. Moreover, we also propose a stochastic order based on this measure.

KEYWORDS:

• Bivariate reversed hazard rate and expected inactivity time
• cumulative past entropy
• stochastic ordering

MATHEMATICS SUBJECT CLASSIFICATION:

• Primary 62G30
• Secondary 62E10, 62B10, 94A17

Acknowledgments

The authors thank the anonymous reviewers for their constructive comments which have considerably improved the content of the paper. We also thank Prof. Asok K. Nanda, IISER Kolkata for helpful discussions while revising the manuscript. The financial support (vide no. PSW-103/13-14 (ERO), ID No. WK4-031 dated 18.03.2014) from the University Grants Commission, Government of India, is acknowledged with thanks by Amarjit Kundu. Also Chanchal Kundu acknowledges with thanks the financial support (Ref. No. SR/FTP/MS-016/2012) rendered by the Department of Science and Technology, Government of India for doing this research.