The Annals of Statistics

Asymptotic Optimality of the Product Limit Estimator

Jon A. Wellner

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Abstract

The product limit estimator due to Kaplan and Meier (1958) is well-known to be the nonparametric maximum likelihood estimator of a distribution function based on censored data. It is shown here that the product limit estimator is an asymptotically optimal estimator in two senses: in the sense of a Hajek-Beran type representation theorem for regular estimators; and in an asymptotic minimax sense similar to the classical result for the uncensored case due to Dvoretzky, Kiefer, and Wolfowitz (1956). The proofs rely on the methods of Beran (1977) and Millar (1979).

Article information

Source
Ann. Statist., Volume 10, Number 2 (1982), 595-602.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176345800

Digital Object Identifier
doi:10.1214/aos/1176345800

Mathematical Reviews number (MathSciNet)
MR653534

Zentralblatt MATH identifier
0489.62036

JSTOR
links.jstor.org

Subjects
Primary: 62G05: Estimation
Secondary: 62G20: Asymptotic properties 62E20: Asymptotic distribution theory

Keywords
Asymptotic minimax censored data convolution representation distribution function regular estimation

Citation

Wellner, Jon A. Asymptotic Optimality of the Product Limit Estimator. Ann. Statist. 10 (1982), no. 2, 595--602. doi:10.1214/aos/1176345800. https://projecteuclid.org/euclid.aos/1176345800


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