The Annals of Statistics

Estimating a Distribution Function with Truncated and Censored Data

Tze Leung Lai and Zhiliang Ying

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A minor modification of the product-limit estimator is proposed for estimating a distribution function (not necessarily continuous) when the data are subject to either truncation or censoring, or to both, by independent but not necessarily identically distributed truncation-censoring variables. Making use of martingale integral representations and empirical process theory, uniform strong consistency of the estimator is established and weak convergence results are proved for the entire observable range of the function. Numerical results are also given to illustrate the usefulness of the modification, particularly in the context of truncated data.

Article information

Ann. Statist., Volume 19, Number 1 (1991), 417-442.

First available in Project Euclid: 12 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62E20: Asymptotic distribution theory
Secondary: 62G05: Estimation 60F05: Central limit and other weak theorems

Product-limit estimator truncated data censoring strong consistency weak convergence empirical process stochastic integral martingales


Lai, Tze Leung; Ying, Zhiliang. Estimating a Distribution Function with Truncated and Censored Data. Ann. Statist. 19 (1991), no. 1, 417--442. doi:10.1214/aos/1176347991.

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