The Annals of Statistics

A characterization of marginal distributions of (possibly dependent) lifetime variables which right censor each other

Tim Bedford and Isaac Meilijson

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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.

Article information

Ann. Statist. Volume 25, Number 4 (1997), 1622-1645.

First available in Project Euclid: 9 September 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62E15: Exact distribution theory 62G15: Tolerance and confidence regions 62N05: Reliability and life testing [See also 90B25] 90C39: Dynamic programming [See also 49L20]

Competing risk dependent censoring identifiability survival analysis Kolmogorov-Smirnov test


Bedford, Tim; Meilijson, Isaac. A characterization of marginal distributions of (possibly dependent) lifetime variables which right censor each other. Ann. Statist. 25 (1997), no. 4, 1622--1645. doi:10.1214/aos/1031594734.

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