Nonparametric Bayesian Estimation Research Articles

WEAK CONVERGENCE OF A BAYESIAN NONPARAMETRIC ESTIMATOR OF THE SURVIVAL FUNCTION UNDER PROGRESSIVE CENSORING

Abstract : A nonparametric Bayesian estimator F (circumflex) of the survival function F constructed from time-sequential progressively censored observations is found to subsume several estimators of F utilized in practice. Weak convergence of F (circumflex) is developed and the limiting process is found to coincide with that obtained when complete response profiles of the sample are available, leading to suitable application of F (circumflex) with consequent reductions in costs and time and without loss of asymptotic accuracy. (Author)

A Note on a Paper by Ferguson and Phadia

Ferguson and Phadia have recently discussed the nonparametric Bayesian estimation of a distribution function from a right-censored random sample using process priors that are neutral to the right. The more general problem of estimating the baseline distribution function with right censored data from the proportional hazards model has been studied by Kalbfleisch who uses the more restrictive class of gamma process priors. This note shows that, with a simple modification, the analysis of Ferguson and Phadia can be extended to deal with the proportional hazards situation with constant covariates. Extensions to time dependent covariates and other regression models are also considered.

A bayesian nonparametric estimator of survival probability assuming increasing failure rate

Bayesian nonparametric estimators of the survival function, the failure rate function, and the density function are obtainedusing jump processes as prior distributions on the space of increasing failure rate functions. The jump processes are intui-tively appealing and have a meaningful physical interpretation. Examples are given and the estimates are compared with the maxi-mum likelihood estimates. In addition, the Bayesian nonpara-metric estimators are presented for arbitrarily right-censored observations.

Addendum to "Large Sample Theory for a Bayesian Nonparametric Survival Curve Estimator Based on Censored Data"

Addendum to "Large Sample Theory for a Bayesian Nonparametric Survival Curve Estimator Based on Censored Data"

Shrinkage estimation in nonparametric Bayesian survival analysis: a simulation study

This paper discusses shrinkage estimation in nonparametric Bayesian survival analysis using censored data. The shrinkage estimators proposed are based on estimating the parameter measure of a prior Dirichlet process in a nonparametric Bayesian survival curve estimator which is the posterior mean of this process. The shrinkage is toward a prior family of exponential survival curves. The estimators are then compared by simulation with the wholly nonparametric estimator of Kaplan-Meier and the maximum likelihood estimator for the exponential family. These comparisons are done in cases where the exponential assumption is both

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

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

Nonparametric Bayesian Estimation of Survival Curves from Incomplete Observations

Abstract This article presents a nonparametric Bayesian estimator of a survival curve based on incomplete or arbitrarily right-censored data. This estimator, a Bayes estimator under a squared-error loss function assuming a Dirichlet process prior, is shown to be a Bayesian extension of the usual product limit (Kaplan-Meier) nonparametric estimator.

Nonparametric Bayesian Estimation of Survival Curves from Incomplete Observations

Abstract This article presents a nonparametric Bayesian estimator of a survival curve based on incomplete or arbitrarily right-censored data. This estimator, a Bayes estimator under a squared-error loss function assuming a Dirichlet process prior, is shown to be a Bayesian extension of the usual product limit (Kaplan-Meier) nonparametric estimator.