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On the structure of a family of probability generating functions induced by shock models
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.
First available in Project Euclid: 1 April 2008
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Roychoudhury, Satrajit; Bhattacharjee, Manish C. On the structure of a family of probability generating functions induced by shock models. Beyond Parametrics in Interdisciplinary Research: Festschrift in Honor of Professor Pranab K. Sen, 78--88, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2008. doi:10.1214/193940307000000536. https://projecteuclid.org/euclid.imsc/1207058265
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