J. Van Ryzin

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.

A histogram method of density estimation

Let Y1,Y2,…,Yn,(Y1≦ Y2≦ … ≦Yn) be the order statistics of a random sample from a distribution F with density f on the real line. A class of density estimates of the histogram type based on differences of the form Yj+k−Yj,k≧l,j=l,…n−k are proposed and studied. The estimates are shown to be both weakly and strongly consistent at ail points x=C(f), the continuity set of f, under suitable conditions.

Uniform consistency of a histogram density estimator and modal estimation

Let be an ordered sample of n independent observations, X1,X2,…,Xn of a random variable X with distribution function F(x) and density f(x) continuous on its support set . As a nonparametric histogram estimator of the density function f(x), consider an estimator fn (x) of the form: where {An (x)} is a suitably chosen sequence of non-negative integer-valued indexing random variables; and {kn} is also an appropriately defined sequence of positive integers which depends only on the sample size n . J. Van Ryzin (1973) has given conditions under which the above estimators are pointwise consistent. In this paper we establish conditions under which such a histogram density estimator is uniformly consistent almost surely. When the density has a unique mode, the results are used to obtain a strongly consistent estimator of the mode similar to that of Venter (1967).

Small sample relative performance of the spline smooth survival estimator

The spline smooth survival estimator of Klotz is compared by simulation with the Kaplan-Meier product limit estimator and the empirical Bayes estimator of Van Ryzin, Susarla, and Rai. Using integrated mean square error, the spline smooth estimator is seen to perform better than the Kaplan-Meier estimator for a variety of distributions and censoring percentages. The comparison with the estimator of Van Ryzin Susarla and Rai is less clear and is sometimes better and sometimes worse.

The methods for Smooth estimation of discrete distributions

Let X1,…,Xn be independent and identically distributed random variables with discrete density . This paper gives two methods for smooth estimation of pi. for each i. One method may be viewed as a generalization to the case of an infinite number of nonzero cell probabilities of the Fienberg and Holland (1970) estimator for the multinomial case. Some Monte Carlo simula-tions show that for a small sample, the proposed estimators yield smaller risks under squared error loss than the usual maximum likelihood estimator (m.l.e.). However, under mild conditions the proposed estimators are shown to have the same large sample properties (strong consistency and asymptotic normality) as the m.l.e.

EMPIRICAL BAYES SLIPPAGE TESTS

Introductionand Summary The empirical Bayes approach to statistical decision theory is applicable when one is confronted repeatedly and independently with the same decision problem. In such instances it is reasonable to formulate the component problems in the sequence as Bayes decision problems with respect to a completely unknown prior distribution on the parameter space and then use the accumulated observations to improve the decision rule at each stage. This approach is due to H. Robbins [4] and is best presented in his paper [5]. We assume familiarity with [5].