Characterizations of Proportional Hazard and Reversed Hazard Rate Models Based on Symmetric and Asymmetric Kullback-Leibler Divergences

Kullback-Leibler divergence (Kℒ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(\mathcal {K}\mathcal {L})$\end{document} is widely used for selecting the best model from a given set of candidate parametrized probabilistic models as an approximation to the true density function h(·). In this paper, we obtain a necessary and sufficient condition to determine proportional hazard and reversed hazard rate models based on symmetric and asymmetric Kullback-Leibler divergences. Obtained results show that if h belongs to proportional hazard rate (reversed hazard) model, then the Kullback-Leibler divergence between h and baseline density function, f0, is independent of the choice of ξ, the cut point of left (right) truncated distribution.