The ϕ-divergence family of measures based on quantile function

Abstract

The quantile function can be considered an efficient alternative to the distribution function in cases where the quantile functions are tractable while the distribution functions do not have explicit forms. In this paper, we introduce a quantile-based definition of the ϕ-divergence family of measures to assess the discrepancy between two random variables. The proposed measure defines a rich family of divergence measures and includes various common divergences as special cases. Some properties of these measures of divergence are studied and several examples are also provided. Dynamic versions of the quantile-based ϕ-divergence family of measures between residual and past lifetimes are developed and their properties are studied. The suggested divergence measures are also examined in a general class of transformed models which results in some well-known models in the lifetime studies and survival analysis. Furthermore, the non-parametric estimations of the proposed divergence measures are discussed, and the performance of the resulting estimators is evaluated via simulation.

Keywords:

• Divergence measure
• Csiszár's family of measures
• quantile function
• transformation models

Acknowledgments

The author would like to thank the Associate Editor and two anonymous reviewers for their constructive comments that greatly improved the paper. The author would also thank Professor Majid Asadi of the Department of Statistics, University of Isfahan for the fruitful discussion.

Disclosure statement

No potential conflict of interest was reported by the author(s).