Characterizations of Probability Distributions with Completely Monotone Hazard Functions

Mohammed Albassam

doi:10.5539/ijsp.v4n3p55

Abstract

In this article, we characterize the classes of absolutelycontinuous distributions concentrated on $(0, \infty)$ anddiscrete distributions concentrated on $\{0,1,2, ...\}$, with(non-vanishing survivor functions having) completely monotonehazard functions; in the latter case, we refer to the hazardfunctions also as the hazard sequences. These provide us withcharacterizations of the certain specialized versions of mixturesof exponential and geometric distributions with mixingdistributions, satisfying some further criteria, which by theGoldie-Steutel theorem and a result of Kaluza are seen to bespecialized versions of infinitely divisible distributions. Weshed light on the implications of our findings, giving somepertinent examples and remarks.

Highlights

By Kaluza (1928, Proposition 1), it follows that all logconvex, and completely monotone, sequences {c(n) : n = 0, 1, ...}, with c(0) = 1, are renewal
We characterize the classes of absolutely continuous distributions concentrated on (0, ∞) and discrete distributions concentrated on {0, 1, 2, ...}, with completely monotone hazard functions; in the latter case, we refer to the hazard functions 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

Summary

Introduction

We give below our main results; for the relevant definitions of completely monotone sequences and functions that we have followed in our analysis, we refer the reader to Feller (1966, pages 224 and 415), respectively.

Results

Conclusion