Open Access
August, 1973 Shock Models and Wear Processes
J. D. Esary, A. W. Marshall
Ann. Probab. 1(4): 627-649 (August, 1973). DOI: 10.1214/aop/1176996891


The life distribution $H(t)$ of a device subject to shocks governed by a Poisson process is considered as a function of the probabilities $P_k$ of not surviving the first $k$ shocks. Various properties of the discrete failure distribution $P_k$ are shown to be reflected in corresponding properties of the continuous life distribution $H(t)$. As an example, if $P_k$ has discrete increasing hazard rate, then $H(t)$ has continuous increasing hazard rate. Properties of $P_k$ are obtained from various physically motivated models, including that in which damage resulting from shocks accumulates until exceedance of a threshold results in failure. We extend our results to continuous wear processes. Applications of interest in renewal theory are obtained. Total positivity theory is used in deriving many of the results.


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J. D. Esary. A. W. Marshall. "Shock Models and Wear Processes." Ann. Probab. 1 (4) 627 - 649, August, 1973.


Published: August, 1973
First available in Project Euclid: 19 April 2007

zbMATH: 0262.60067
MathSciNet: MR350893
Digital Object Identifier: 10.1214/aop/1176996891

Primary: 60K10

Keywords: life distributions , Poisson process , reliability , shock models , total positivity , wear

Rights: Copyright © 1973 Institute of Mathematical Statistics

Vol.1 • No. 4 • August, 1973
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