Quantile-based cumulative Kullback-Leibler divergence in past lifetime: Some properties and applications

Abstract

This article introduces a quantile-based cumulative Kullback-Leibler (KL) divergence measure for past lifetimes (QPCKL divergence) and studies its various properties. Unlike the cumulative KL divergence using the distribution function, the proposed measure offers an alternative method for computing the divergence between two random variables, especially when one lacks a closed-form distribution function but possesses a closed-form quantile function. A characterization property of a distribution is derived using the constant QPCKL divergence model. Subsequently, a comparison of some important reliability characteristics using the proposed measure is discussed. The QPCKL divergence for a general family of transformation models is derived. We also propose a non parametric estimator for the measure, and simulation studies are carried out for validation. Finally, the importance of the estimator is also illustrated through real data sets useful in survival studies.