Large Sample Theory for a Bayesian Nonparametric Survival Curve Estimator Based on Censored Samples

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 $P\lbrack X_1 > u\rbrack P\lbrack Y_1 > u\rbrack > 0.$