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 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.