Nonparametric estimation of the joint distribution of survival time and mark variables

Yijian Huang, Thomas A. Louis

Research output: Contribution to journalArticlepeer-review

72 Scopus citations


In many applications, variables of interest are marks of the endpoint which are not observed when the survival time is censored. This paper focuses on nonparametric estimation of the joint distribution and summaries of survival time and mark variables. We establish a representation of the joint distribution function through the cumulative markspecific hazard function, which is analogous to the product integral representation of univariate survival function. We identify a basic data structure common to various applications, propose nonparametric estimators and show that they maximise the likelihood. We formulate the problem in the marked point process framework and study both finite and large-sample properties of the estimators. We show that the joint distribution function estimator is nearly unbiased, uniformly strongly consistent and asymptotically normal. We also derive asymptotic variances for the estimators and propose sample-based variance estimates. Numerical studies demonstrate that both the estimators and their variance estimates perform well for practical sample sizes. We outline an application strategy.

Original languageEnglish (US)
Pages (from-to)785-798
Number of pages14
Issue number4
StatePublished - 1998
Externally publishedYes


  • Dependent censoring
  • Identifiability
  • Marked point process
  • Medical cost
  • Multivariate distribution
  • Multivariate failure time
  • Nonparametric maximum likelihood
  • Quality adjusted survival time

ASJC Scopus subject areas

  • Statistics and Probability
  • Mathematics(all)
  • Agricultural and Biological Sciences (miscellaneous)
  • Agricultural and Biological Sciences(all)
  • Statistics, Probability and Uncertainty
  • Applied Mathematics


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