Empirical Bayes Estimation in Finite Population Sampling

Abstract

Abstract Empirical Bayes methods are becoming increasingly popular in statistics. Robbins (1955) introduced the method in the context of nonparametric estimation of a completely unspecified prior distribution. Subsequently, the method has been explored very successfully in a series of articles by Efron and Morris (1973, 1975, 1977) in a parametric framework. In the Efron—Morris setup, a family of parametric distributions is used as possible priors, but only when one or more of the parameters of the family of prior distributions is estimated from the data. Morris (1983) listed a number of areas where empirical Bayes methods are used. One of the main features of empirical Bayes analysis is to borrow strength from the ensemble—that is, use information from similar sources in constructing estimators and predictors in addition to the most directly available source of information. There are some situations in finite population sampling where such methods might be suitable. For instance, in many repetitive surveys such as household surveys, crop-cutting experiment, and so forth, we have at our disposal not only the current data, but also data from similar past experiments. This is particularly true when surveys are done routinely on, say, a monthly basis, and physical conditions of the sampling units do not change drastically over a period of time. In such cases, past information might be used profitably in arriving at suitable estimates of different characteristics of interest. For definiteness, consider a finite population with units labeled 1, 2, …, N. Let yi , denote the value of a single characteristic attached to the unit i. The vector y = (y 1 …, yN )' is the unknown state of nature, and it is assumed to belong to θ = RN . A subset s of {1, …, N} is called a sample. Let S denote the set of all possible samples. We consider only samples of size n. Consider the model yi = θ + εi(i = 1), …, N), where θ, ε1, …, εN are independently distributed with θ ∽ N(μ, σ2) and εi's iid N(0, τ2). Then the Bayes estimator of γ(y) = N-1 ∑i=1 N yi, under squared error loss is where , and M = τ2/σ2. In an empirical Bayes framework, it is assumed that we are at the mth stage of the sampling procedure and that sampling has been repeated (m − 1) times. The population size at the jth stage of the experiment is denoted by Nj, and at that stage we associate with the ith unit a certain characteristic, say, yi (j) (i = 1, …, Nj; j = 1, …, m). A fixed sample of size nj is taken at the jth stage, and a typical sample is denoted by sI(j = 1, …, m). Consider the model yi (j) = θ(j) + εi (j) (I = 1, …, Nj; j = 1, …, m), where θ(j)'s and εi (j)'s are all independently distributed with θ(j)'s iid N(μ σ2) and εi (j)'s iid N(0, τ2). Writing Bj = M/(M + nj) (j = 1, …, m), it follows from (1) that at the mth stage of the experiment, the Bayes estimator of where . In an empirical Bayes analysis, one or both of the parameters M and μ are unknown and need to be estimated from the data. This article proposes an estimator of M as a function of the usual F ratio of between and within mean squares and an estimator of μ based on the principle of maximum likelihood. A variety of properties of these empirical Bayes estimators are established both in the general case and in the special case when n1 = … = nm. These empirical Bayes estimators serve as a compromise between the classical and the Bayes estimators and perform quite satisfactorily in their risk performance.