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

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

2008-05-15

arxiv.org/abs/0805.2246v1

IMS Collections 2008, Vol. 1, 78-88

Mathematics

Statistics Theory

Published in at http://dx.doi.org/10.1214/193940307000000536 the IMS Collections (http://www.imstat.org/publications/imscollec

Scientific paper

10.1214/193940307000000536

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.

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