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

It is well known that the joint distribution of a pair of lifetime variables X₁ and X₂ which right censor each other cannot be specified in terms of the subsurvival functions P(X₂ > X₁ > x), P(X₁ > X₂ > x) and P(X₁ = X₂ > x) without additional assumptions such as independence of X₁ and X₂. For many practical applications independence is an unacceptable assumption, for example, when X₁ is the lifetime of a component subjected to maintenance and X₂ is the inspection time. Peterson presented lower and upper bounds for the marginal distributions of X₁ and X₂, 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 subsurvival functions, without any assumptions on the underlying joint distribution of (X₁, X₂). Furthermore, a statistical test for an hypothesized marginal distribution of X₁ based on the empirical subsurvival functions is developed. The characterization is generalized from two to any number of variables.