Characterizations of Probability Distributions with Completely Monotone Hazard Functions

Abstract

In this article, we characterize the classes of absolutely continuous distributions concentrated on (0, ∞) and discrete distributions concentrated on {0, 1, 2, ...}, with (non-vanishing survivor functions having) completely monotone hazard functions; in the latter case, we refer to the hazard functions also as the hazard sequences. These provide us with characterizations of the certain specialized versions of mixtures of exponential and geometric distributions with mixing distributions, satisfying some further criteria, which by the Goldie-Steutel theorem and a result of Kaluza are seen to be specialized versions of infinitely divisible distributions. We shed light on the implications of our findings, giving some pertinent examples and remarks.