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April, 1979 Multivariate Shock Models for Distributions with Increasing Hazard Rate Average
Albert W. Marshall, Moshe Shaked
Ann. Probab. 7(2): 343-358 (April, 1979). DOI: 10.1214/aop/1176995092


Suppose that $n$ devices are subjected to shocks occurring randomly in time as events in a Poisson process. Upon occurrence of the $i$th shock the devices suffer nonnegative random damages with joint distribution $F_i$. Damages from successive shocks are independent and accumulate additively. Failure of the $j$th device occurs at the time $T_j$ when its accumulated damage first exceeds its breaking threshold $x_j$. If $\tau$ is the life function of a coherent system, then the system life length $\tau(T_1, \cdots, T_n)$ has a distribution with increasing hazard rate average providing that $F_1, F_2, \cdots$ satisfy a multivariate stochastic ordering condition that depends upon $\tau$. If $F_1 = F_2 = \cdots$ and $\bar{H}$ is the joint survival function of $T_1, \cdots, T_n$, then $\lbrack\bar{H}(\alpha\mathbf{t})\rbrack^{1/\alpha}$ is decreasing in $\alpha$ for all $\mathbf{t} \geqslant 0. \bar{H}$ also satisfies a multivariate "new better than used" property. Moreover $T_1, \cdots, T_n$ are associated when $F_1 = F_2 = \cdots$. Examples of specific distributions are given which arise from the shock model, including a new bivariate gamma distribution which reduces to the bivariate exponential distribution of Marshall and Olkin as a special case.


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Albert W. Marshall. Moshe Shaked. "Multivariate Shock Models for Distributions with Increasing Hazard Rate Average." Ann. Probab. 7 (2) 343 - 358, April, 1979.


Published: April, 1979
First available in Project Euclid: 19 April 2007

zbMATH: 0394.62033
MathSciNet: MR525058
Digital Object Identifier: 10.1214/aop/1176995092

Primary: 62H05
Secondary: 60K10 , 62N05

Keywords: increasing hazard rate average , multivariate life distributions , Poisson process , reliability , shock models

Rights: Copyright © 1979 Institute of Mathematical Statistics


Vol.7 • No. 2 • April, 1979
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