The Annals of Statistics

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

Tim Bedford and Isaac Meilijson

Full-text: Open access


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.

Article information

Ann. Statist., Volume 25, Number 4 (1997), 1622-1645.

First available in Project Euclid: 9 September 2002

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62E15: Exact distribution theory 62G15: Tolerance and confidence regions 62N05: Reliability and life testing [See also 90B25] 90C39: Dynamic programming [See also 49L20]

Competing risk dependent censoring identifiability survival analysis Kolmogorov-Smirnov test


Bedford, Tim; Meilijson, Isaac. A characterization of marginal distributions of (possibly dependent) lifetime variables which right censor each other. Ann. Statist. 25 (1997), no. 4, 1622--1645. doi:10.1214/aos/1031594734.

Export citation


  • [1] Cooke, R. M. (1993). The total time on test statistic and age-dependent censoring. Statist. Probab. Lett. 18.
  • [2] Crowder, M. (1991). On the identifiability crisis in competing risks analysis. Scand. J. Statist. 18 223-233.
  • [3] Kaplan, E. L. and Meier, P. (1958). Nonparametric estimation from incomplete observations. J. Amer. Statist. Assoc. 53 457-481.
  • [4] Langberg, N., Proschan, F. and Quinzi, A. J. (1978). Converting dependent models into independent ones, preserving essential features. Ann. Probab. 6 174-181.
  • [5] Langberg, N., Proschan, F. and Quinzi, A. J. (1981). Estimating dependent life lengths, with applications to the theory of competing risks. Ann. Statist. 9 157-167.
  • [6] Miller, D. R. (1977). A note on independence of multivariate lifetimes in competing risk models. Ann. Statist. 5 576-579.
  • [7] N´adas, A. (1970). On estimating the distribution of a random vector when only the smallest coordinate is observable. Technometrics 12 923-924.
  • [8] N´adas, A. (1971). The distribution of the identified minimum of a normal pair determines the distribution of the pair. Technometrics 13 201-202.
  • [9] Peterson, A. V. (1976). Bounds for a joint distribution function with fixed subdistribution functions: application to competing risks. Proc. Nat. Acad. Sci. U.S.A. 73 11-13.
  • [10] Pollard, D. (1984). Convergence of Stochastic Processes. Springer, New York.
  • [11] Tsiatis, A. (1975). A nonidentifiability aspect in the problem of competing risks. Proc. Nat. Acad. Sci. U.S.A. 72 20-22.