The Annals of Statistics

Strong Uniform Consistency for Nonparametric Survival Curve Estimators from Randomly Censored Data

Antonia Foldes and Lidia Rejto

Full-text: Open access

Abstract

Let $X_1, \cdots, X_n$ be i.i.d. $P(X > u) = F(u)$ and $Y_1, \cdots, Y_n$ be i.i.d. $P(Y > u) = G(u)$, where both $F$ and $G$ are unknown continuous distributions. For $i = 1, \cdots, n$ set $\delta_i = 1$ if $X_i \leq Y_i$ and 0 if $X_i > Y_i$ and $Z_i = \min \{X_i, Y_i\}$. One way to estimate $F$ from the observations $(Z_i, \delta_i) i = 1, \cdots, n$ is by means of the product limit (PL) estimator, $F^\ast_n$ (Kaplan-Meier, [8]). In this paper it is shown that $F^\ast_n$ is uniformly almost sure consistent with rate $O(\sqrt{\log n} / \sqrt n)$, that is $P(\sup_{0 \leq u \leq T}|F^\ast_n(u) - F(u)| = O(\sqrt{\log n/n})) = 1.$ Assuming that $F$ is distributed according to a Dirichlet process (Ferguson, [3]) with parameter $\alpha$, Susarla and Van Ryzin ([11]) obtained the Bayes estimator $F^\alpha_n$ of $F$. In the present paper a similar result is established for the Bayes estimator, namely: $P(\sup_{0 \leq u \leq T}| F^\alpha_n(u) - F(u)| = O(\sqrt{(\log n)^{1 + \gamma}} / \sqrt n)) = 1 \quad (\gamma > 0).$

Article information

Source
Ann. Statist., Volume 9, Number 1 (1981), 122-129.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176345337

Digital Object Identifier
doi:10.1214/aos/1176345337

Mathematical Reviews number (MathSciNet)
MR600537

Zentralblatt MATH identifier
0453.62034

JSTOR
links.jstor.org

Subjects
Primary: 60F16
Secondary: 62G05: Estimation

Keywords
Product limit survival distribution strong uniform consistency censored data Bayes estimator

Citation

Foldes, Antonia; Rejto, Lidia. Strong Uniform Consistency for Nonparametric Survival Curve Estimators from Randomly Censored Data. Ann. Statist. 9 (1981), no. 1, 122--129. doi:10.1214/aos/1176345337. https://projecteuclid.org/euclid.aos/1176345337


Export citation