The Annals of Probability

Multivariate Shock Models for Distributions with Increasing Hazard Rate Average

Albert W. Marshall and Moshe Shaked

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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.

Article information

Ann. Probab., Volume 7, Number 2 (1979), 343-358.

First available in Project Euclid: 19 April 2007

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Zentralblatt MATH identifier


Primary: 62H05: Characterization and structure theory
Secondary: 60K10: Applications (reliability, demand theory, etc.) 62N05: Reliability and life testing [See also 90B25]

Shock models Poisson process reliability multivariate life distributions increasing hazard rate average


Marshall, Albert W.; Shaked, Moshe. Multivariate Shock Models for Distributions with Increasing Hazard Rate Average. Ann. Probab. 7 (1979), no. 2, 343--358. doi:10.1214/aop/1176995092.

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