On the structure of a family of probability generating functions induced by shock models

TL;DR

This paper investigates the conditions under which a class of integral-defined functions serve as probability generating functions for positive integer-valued variables, revealing connections to shock models and aging properties.

Contribution

It establishes new criteria for these functions to be valid probability generating functions and links shock models to the concept of strongly decreasing failure rate (SDFR).

Findings

Identifies conditions for integral-defined functions to be probability generating functions.

Shows a connection between shock models and SDFR aging properties.

Provides a counterexample illustrating limitations of the integral representation.

Abstract

We explore conditions for a class of functions defined via an integral representation to be a probability generating function of some positive integer valued random variable. Interest in and research on this question is motivated by an apparently surprising connection between a family of classic shock models due to Esary et. al. (1973) and the negatively aging nonparametric notion of ``strongly decreasing failure rate'' (SDFR) introduced by Bhattacharjee (2005). A counterexample shows that there exist probability generating functions with our integral representation which are not discrete SDFR, but when used as shock resistance probabilities can give rise to a SDFR survival distribution in continuous time.