The Annals of Probability

Multivariate Shock Models for Distributions with Increasing Hazard Rate Average

Abstract

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

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

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176995092

Digital Object Identifier
doi:10.1214/aop/1176995092

Mathematical Reviews number (MathSciNet)
MR525058

Zentralblatt MATH identifier
0394.62033

JSTOR