Lifetime Data Analysis

, Volume 17, Issue 2, pp 234–255 | Cite as

A generalization of Turnbull’s estimator for nonparametric estimation of the conditional survival function with interval-censored data

  • Mohammad Hossein Dehghan
  • Thierry DuchesneEmail author


Simple nonparametric estimates of the conditional distribution of a response variable given a covariate are often useful for data exploration purposes or to help with the specification or validation of a parametric or semi-parametric regression model. In this paper we propose such an estimator in the case where the response variable is interval-censored and the covariate is continuous. Our approach consists in adding weights that depend on the covariate value in the self-consistency equation proposed by Turnbull (J R Stat Soc Ser B 38:290–295, 1976), which results in an estimator that is no more difficult to implement than Turnbull’s estimator itself. We show the convergence of our algorithm and that our estimator reduces to the generalized Kaplan–Meier estimator (Beran, Nonparametric regression with randomly censored survival data, 1981) when the data are either complete or right-censored. We demonstrate by simulation that the estimator, bootstrap variance estimation and bandwidth selection (by rule of thumb or cross-validation) all perform well in finite samples. We illustrate the method by applying it to a dataset from a study on the incidence of HIV in a group of female sex workers from Kinshasa.


EM algorithm Generalized Kaplan–Meier Kernel weights Local likelihood Self-consistent estimator Weighted EM algorithm 


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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of Mathematics & StatisticsSisitan & Blouchestan UniversityZahedanIran
  2. 2.Département de mathématiques et de statistiqueUniversité LavalQuebecCanada

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