Open Access
Translator Disclaimer
July, 1978 Large Sample Theory for a Bayesian Nonparametric Survival Curve Estimator Based on Censored Samples
V. Susarla, J. Van Ryzin
Ann. Statist. 6(4): 755-768 (July, 1978). DOI: 10.1214/aos/1176344250

Abstract

Let $X_1, \cdots, X_n$ be i.i.d. $F_0$ and let $Y_1, \cdots, Y_n$ be independent (and independent also of $X_1, \cdots, X_n$) random variables. Then assuming that $F$ is distributed according to a Dirichlet process with parameter $\alpha,$ the authors obtained the Bayes estimator $\hat{F}_\alpha$ of $F$ under the loss function $L(F, \hat{F}) = \int (F(u) - \hat{F}(u))^2 dw(u)$ when $X_1, \cdots, X_n$ are censored on the right by $Y_1, \cdots, Y_n,$ respectively, and when it is known whether there is censoring or not. Assuming $X_1, \cdots, X_n$ are i.i.d. $F_0$ and $Y_1, \cdots, Y_n$ are i.i.d. $G,$ this paper shows that $\hat{F}_\alpha$ is mean square consistent with rate $O(n^{-1}),$ almost sure consistent with rate $O(\log n/n^\frac{1}{2}),$ and that $\{\hat{F}_\alpha(u) \mid 0 < u < T\}, T < \infty,$ converges weakly to a Gaussian process whenever $F_0$ and $G$ are continuous and that $P\lbrack X_1 > u\rbrack P\lbrack Y_1 > u\rbrack > 0.$

Citation

Download Citation

V. Susarla. J. Van Ryzin. "Large Sample Theory for a Bayesian Nonparametric Survival Curve Estimator Based on Censored Samples." Ann. Statist. 6 (4) 755 - 768, July, 1978. https://doi.org/10.1214/aos/1176344250

Information

Published: July, 1978
First available in Project Euclid: 12 April 2007

zbMATH: 0378.62019
MathSciNet: MR483136
Digital Object Identifier: 10.1214/aos/1176344250

Subjects:
Primary: 62E20
Secondary: 62G05

Keywords: Censored data , consistency , Dirichlet process , Survival curve estimator , weak convergence

Rights: Copyright © 1978 Institute of Mathematical Statistics

JOURNAL ARTICLE
14 PAGES


SHARE
Vol.6 • No. 4 • July, 1978
Back to Top