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August 1997 A characterization of marginal distributions of (possibly dependent) lifetime variables which right censor each other
Tim Bedford, Isaac Meilijson
Ann. Statist. 25(4): 1622-1645 (August 1997). DOI: 10.1214/aos/1031594734

Abstract

It is well known that the joint distribution of a pair of lifetime variables $X_1$ and $X_2$ which right censor each other cannot be specified in terms of the subsurvival functions $$P(X_2 > X_1 > x), \quad P(X_1 > X_2 > x)$ \quad \text{and} \quad $P(X_1 = X_2 > x)$$ without additional assumptions such as independence of $X_1$ and $X_2$. For many practical applications independence is an unacceptable assumption, for example, when $X_1$ is the lifetime of a component subjected to maintenance and $X_2$ is the inspection time. Peterson presented lower and upper bounds for the marginal distributions of $X_1$ and $X_2$, for given subsurvival functions. These bounds are sharp under nonatomicity conditions. Surprisingly, not every pair of distribution functions between these bounds provides a feasible pair of marginals. Crowder recognized that these bounds are not functionally sharp and restricted the class of functions containing all feasible marginals. In this paper we give a complete characterization of the possible marginal distributions of these variables with given sub-survival functions, without any assumptions on the underlying joint distribution of $X_1, X_2$. Furthermore, a statistical test for an hypothesized marginal distribution of $(X_1$ based on the empirical subsurvival functions is developed.

The characterization is generalized from two to any number of variables.

Citation

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Tim Bedford. Isaac Meilijson. "A characterization of marginal distributions of (possibly dependent) lifetime variables which right censor each other." Ann. Statist. 25 (4) 1622 - 1645, August 1997. https://doi.org/10.1214/aos/1031594734

Information

Published: August 1997
First available in Project Euclid: 9 September 2002

zbMATH: 0936.62014
MathSciNet: MR1463567
Digital Object Identifier: 10.1214/aos/1031594734

Subjects:
Primary: 62E15, 62G15, 62N05, 90C39

Rights: Copyright © 1997 Institute of Mathematical Statistics

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Vol.25 • No. 4 • August 1997
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