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

Satrajit Roychoudhury, Manish C. Bhattacharjee
Source: N. Balakrishnan, Edsel A. Peña and Mervyn J. Silvapulle, eds., Beyond Parametrics in Interdisciplinary Research: Festschrift in Honor of Professor Pranab K. Sen (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2008), 78-88.

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.

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Primary Subjects: 60K10
Secondary Subjects: 90B25
Keywords: Esary-Marshall-Proschan shock model; strong DFR
Full-text: Open access
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Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.imsc/1207058265
Digital Object Identifier: doi:10.1214/193940307000000536

References

[1] Bhattacharjee, M. C. (2005). Strong versions of the DFR property. CAMS Research Report 0405-18. Available at http://m.njit.edu/CAMS/ Technical_Reports/CAMS04_05/report18.pdf.
[2] Esary, J. D. and Marshall, A. W. (1973). Shock models and wear processes. Ann. Probab. 1 624–649.
Mathematical Reviews (MathSciNet): MR350893
Digital Object Identifier: doi:10.1214/aop/1176996891
[3] Feller, W. (1939). Completely montone functions and sequences. Duke J. Math. 5 661–674.
[4] Feller, W. (1960). An Introduction to Probability Theory and Its Application. II. Wiley, New York.
[5] Müller, A. and Stoyan, D. (2002). Comparison Methods for Stochastic Models and Risk. Wiley, New York.
[6] Pakes, A. G. (1997). On the recognition and structure of probabilitygenerating functions. In Classical and Modern Branching Processes (K. B. Athreya and P. Jagers, eds.) 263–284. Springer, New York.
Mathematical Reviews (MathSciNet): MR1601681
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2012 © Institute of Mathematical Statistics

Institute of Mathematical Statistics Collections

Institute of Mathematical Statistics Collections