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Biometrika 1998 85(4):785-798; doi:10.1093/biomet/85.4.785
© 1998 by Biometrika Trust
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Nonparametric estimation of the joint distribution of survival time and mark variables

YIJIAN HUANG and THOMAS A. LOUIS

Division of Public Health Sciences, Fred Hutchinson Cancer Research Center Seattle, Washington 98109, U.S.A.eugene{at}hivnet.fhcrc.org
Division of Biostatistics, School of Public Health, University of Minnesota Minneapolis, Minnesota 55455, U.S.A.tom{at}biostat.umn.edu

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 markspecilk 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 variance ropose sample-based variance estimates. Numerical studies dem stimators and their variance estimates perform well for practical sample sizes. We outline an application strategy.

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


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