The Annals of Probability

Multivariate Distributions with Increasing Hazard Rate Average

J. D. Esary and A. W. Marshall

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Several conditions are considered that extend to a multivariate setting the univariate concept of an increasing hazard rate average. The relationships between the various conditions are established. In particular it is shown that if for some independent random variables $X_1, \cdots, X_k$ with increasing hazard rate average and some coherent life functions $\tau_1, \cdots, \tau_n$ of order $k, T_i = \tau_i(X_1, \cdots, X_k)$, then the joint survival function $\bar{F}(\mathbf{t}) = P(T_1 > t_1, \cdots, T_n > t_n)$ has the property that $\alpha^{-1} \log \bar{F}(\alpha\mathbf{t})$ is decreasing in $\alpha > 0$ whenever each $t_i \geqslant 0$. Various other properties of the multivariate conditions are given. The conditions can all be stated in terms of inequalities in which equality implies that the one dimensional marginal distributions are exponential. For most of the conditions, the form of the multivariate exponential distributions that satisfy the equality is exhibited.

Article information

Ann. Probab., Volume 7, Number 2 (1979), 359-370.

First available in Project Euclid: 19 April 2007

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Primary: 62H05: Characterization and structure theory
Secondary: 62N05: Reliability and life testing [See also 90B25]

Multivariate life distributions reliability increasing hazard rate average hazard gradient


Esary, J. D.; Marshall, A. W. Multivariate Distributions with Increasing Hazard Rate Average. Ann. Probab. 7 (1979), no. 2, 359--370. doi:10.1214/aop/1176995093.

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