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LARGE SAMPLE THEORY FOR A BAYESIAN NONPARAMETRIC SURVIVAL CURVE ESTIMATOR BASED ON CENSORED SAMPLES

IP.com Disclosure Number: IPCOM000128709D
Original Publication Date: 1976-Dec-31
Included in the Prior Art Database: 2005-Sep-16
Document File: 20 page(s) / 39K

Publishing Venue

Software Patent Institute

Related People

V. Susarla: AUTHOR [+4]

Abstract

Recently, attention has been drawn to the consideration of obtaining nonparametric Bayes estimates of a distribution function assuming a manageable prior (resulting in a manageable posterior distribution) on the space of dis- tribution functions F on R Towards this goal, Ferguson [4] introduced a class of priors, known as Dirichlet process priors, on F which enjoy the property that the posterior distribution is again a Dirichlet process. Ferguson used this fact to obtain the Bayes estimator of the right sided cumulative distribution function F (F(x) denotes the probability in (x,-) and this useful convention is borrowed from Efron [3]) under a weighted squared error loss function. It can be readily seen that this Bayes estimator of F will have all the asymptotic properties enjoyed by the maximum likeli- hood estimator of F if there is no prior on F and the observations are i.i.d. with an unknown right c.d.f. F 01 An important problem in survival analysis is,that of estimating either parametrically or nonparametrically the survival curve [Equation ommitted] (See, for example, Gross and Clark [6]). While treating this problem of estimating survival curves based on incomplete data, the authors [11] obtained the Bayes estimator of F under a weighted squared error loss function when the independent observations from F are randomly censored on the right under Dirichlet process priors of Ferguson [4]. They demonstrated that this Bayes estimator is an extension of the above mentioned Bayes estimator of Ferguson [4] and in a certain sense, also of the well-known Kaplan-Meier (KM) estimator [7] which maximizes the likelihood of the observations. Efron [3] and in a more detailed manner, Breslow and Crowley [1] showed that the KM estimator is weakly consistent and asymptotically normal Sponsorect oy: 1) Tne United States Army under Contract No. DAAG29-75-C-0024; 2) National Science Foundation under Grant No. MSC-76-05952; 3) National Institute of General Medical SciencesY DHEW., under Grant No. I-RO-1-GM 23129; and under the assumption that all the censoring random variables are i.i.d. continuous random variables. The object of this paper is to show that our Bayes estimator has good limiting properties including mean-square consistency (m.s.c.), almost sure consistency (a.s.c.) and asymptotic normality assuming that the observations are i.i.d. with right c.d.f. F 0 and that the censoring random variables are i.i.d. with a continuous distribution function. Efron [3] and Breslow and Crowley [1] have neither rate of convergence results for their weak consistency nor do they have m.s.c., while we obtain rates for both m.s.c. and a.s.c. Our methods of proof, in contrast with those of Breslow and Crowley [1], involve the analysis of the expectation and the variance of the logarithm of W n (u) involved in the Bayes estimator given in (2.2).

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THIS DOCUMENT IS AN APPROXIMATE REPRESENTATION OF THE ORIGINAL.

LARGE SAMPLE THEORY FOR A BAYESIAN NONPARAMETRIC SURVIVAL CURVE ESTIMATOR BASED ON CENSORED SAMPLES

V. Susarla and J. Van Ryzin

1. INTRODUCTION AND SUMMARY

Recently, attention has been drawn to the consideration of obtaining nonparametric Bayes estimates of a distribution function assuming a manageable prior (resulting in a manageable posterior distribution) on the space of dis- tribution functions F on R Towards this goal, Ferguson [4] introduced a class of priors, known as Dirichlet process priors, on F which enjoy the property that the posterior distribution is again a Dirichlet process. Ferguson used this fact to obtain the Bayes estimator of the right sided cumulative distribution function F (F(x) denotes the probability in (x,-) and this useful convention is borrowed from Efron [3]) under a weighted squared error loss function. It can be readily seen that this Bayes estimator of F will have all the asymptotic properties enjoyed by the maximum likeli- hood estimator of F if there is no prior on F and the observations are i.i.d. with an unknown right c.d.f. F 01

An important problem in survival analysis is,that of estimating either parametrically or nonparametrically the survival curve

(Equation Omitted)

(See, for example, Gross and Clark [6]). While treating this problem of estimating survival curves based on incomplete data, the authors [11] obtained the Bayes estimator of F under a weighted squared error loss function when the independent observations from F are randomly censored on the right under Dirichlet process priors of Ferguson [4]. They demonstrated that this Bayes estimator is an extension of the above mentioned Bayes estimator of Ferguson [4] and in a certain sense, also of the well-known Kaplan-Meier (KM) estimator [7] which maximizes the likelihood of the observations. Efron [3] and in a more detailed manner, Breslow and Crowley [1] showed that the KM estimator is weakly consistent and asymptotically normal Sponsorect oy: 1) Tne United States Army under Contract No. DAAG29-75-C-0024; 2) National Science Foundation under Grant No. MSC-76-05952;

3) National Institute of General Medical SciencesY DHEW., under Grant No. I-RO-1-GM 23129; and under the assumption that all the censoring random variables are i.i.d. continuous random variables.

The object of this paper is to show that our Bayes estimator has good limiting properties including mean-square consistency (m.s.c.), almost sure consistency (a.s.c.) and asymptotic normality assuming that the observations are i.i.d. with right c.d.f. F 0 and that the censoring random variables are i.i.d. with a continuous distribution function. Efron [3] and Breslow and Crowley [1] have neither rate of convergence results for their weak consistency nor do they have m.s.c., while we obtain rates for both m.s.c. and a.s.c. Our methods of proof, in contrast with those of Breslow and Crowley [1], involve the analysis of...