## 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), P(X_{1} > X_{2} > x) and 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 subsurvival 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.

Original language | English |
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Pages (from-to) | 1622-1645 |

Number of pages | 24 |

Journal | Annals of Statistics |

Volume | 25 |

Issue number | 4 |

DOIs | |

State | Published - Aug 1997 |

## Keywords

- Competing risk
- Dependent censoring
- Identifiability
- Kolmogorov-Smirnov test
- Survival analysis