The Annals of Statistics

On the rate of uniform convergence of the product-limit estimator: strong and weak laws

Kani Chen and Shaw-Hwa Lo

Full-text: Open access

Abstract

By approximating the classical product-limit estimator of a distribution function with an average of iid random variables, we derive sufficient and necessary conditions for the rate of (both strong and weak) uniform convergence of the product-limit estimator over the whole line. These findings somehow fill a longstanding gap in the asymptotic theory of survival analysis. The result suggests a natural way of estimating the rate of convergence. We also prove a related conjecture raised by Gill and discuss its application to the construction of a confidence interval for a survival function near the endpoint.

Article information

Source
Ann. Statist., Volume 25, Number 3 (1997), 1050-1087.

Dates
First available in Project Euclid: 20 November 2003

Permanent link to this document
https://projecteuclid.org/euclid.aos/1069362738

Digital Object Identifier
doi:10.1214/aos/1069362738

Mathematical Reviews number (MathSciNet)
MR1447741

Zentralblatt MATH identifier
0880.62056

Subjects
Primary: 62E20: Asymptotic distribution theory 62G30: Order statistics; empirical distribution functions

Keywords
Law of large numbers counting process martingale inequality Kolmogorov zero-one law Feller-Chung lemma slowly varying function stable law domain of attraction

Citation

Chen, Kani; Lo, Shaw-Hwa. On the rate of uniform convergence of the product-limit estimator: strong and weak laws. Ann. Statist. 25 (1997), no. 3, 1050--1087. doi:10.1214/aos/1069362738. https://projecteuclid.org/euclid.aos/1069362738


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