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 $X_1$ and $X_2$ which right censor each other cannot be specified in terms of the subsurvival functions $$P(X_2 > X_1 > x),\quad P(X_1 > X_2 > x)\text{and} P(X_1 = X_2 > x)$$ without additional assumptions su...

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Bibliographic Details
Published in:The Annals of statistics 1997-08, Vol.25 (4), p.1622-1645
Main Authors: Bedford, Tim, Meilijson, Isaac
Format: Article
Language:English
Subjects:
62E15
62G15
62N05
90C39
Censored data
Censoring
Censorship
Competing risk
Confidence interval
Continuous functions
Critical points
dependent censoring
Distribution functions
Distribution theory
Exact sciences and technology
identifiability
Kolmogorov Smirnov test
Mathematical functions
Mathematics
Nonparametric inference
Probability and statistics
Random variables
Sciences and techniques of general use
Standard deviation
Statistics
survival analysis
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Summary: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 $$P(X_2 > X_1 > x),\quad P(X_1 > X_2 > x)\text{and} P(X_1 = X_2 > x)$$ 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.
ISSN:0090-5364
2168-8966
DOI:10.1214/aos/1031594734