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

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.