The Annals of Statistics

A Note on the Uniform Consistency of the Kaplan-Meier Estimator

Jia-Gang Wang

Full-text: Open access

Abstract

Let $\{X_n, n \geq 1\}$ be i.i.d with $P(X_i \leq u) = F(u)$ and $\{U_n, n \geq 1\}$ be i.i.d. with $P(U_i \leq u) = G(u). \hat{F}_n(t)$ is the Kaplan-Meier estimator based on the censored data $(\tilde{X}_i = X_i \wedge U_i, \delta_i = 1_{(X_i \leq U_i)}, 1 \leq i \leq n)$. In this note, it is shown that for $T_n = \max_{1 \leq i \leq n} \tilde{X}_i$, $\mathrm{pr}-\lim_{n \rightarrow \infty} \sup_{t \leq T_n} |\hat{F}_n(t) - F(t)| = 0.$ Hence, the largest interval on which the Kaplan-Meier estimator is uniformly consistent is found.

Article information

Source
Ann. Statist., Volume 15, Number 3 (1987), 1313-1316.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176350507

Digital Object Identifier
doi:10.1214/aos/1176350507

Mathematical Reviews number (MathSciNet)
MR902260

Zentralblatt MATH identifier
0631.62043

JSTOR
links.jstor.org

Subjects
Primary: 62G05: Estimation
Secondary: 62N05: Reliability and life testing [See also 90B25] 62P10: Applications to biology and medical sciences

Keywords
Product-limit estimator Kaplan-Meier estimator random censoring uniform consistency martingales stochastic integrals

Citation

Wang, Jia-Gang. A Note on the Uniform Consistency of the Kaplan-Meier Estimator. Ann. Statist. 15 (1987), no. 3, 1313--1316. doi:10.1214/aos/1176350507. https://projecteuclid.org/euclid.aos/1176350507


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