Texas Project NExT Journal
Vol. 2 (2004), pp. 11-20.

Empirical Bayes and Bayes Prediction of Finite Population Total Using Auxiliary Information

Mark S. Hamnera, John W. Seamanb, Jr., Dean M. Youngb

Key words: Superpopulation model; empirical Bayes estimator; general regression estimator; partitioned matrices

Abstract: The focus of this paper is on the prediction of a finite population total T by taking a sample of size n from a population of size N units. Classical theory models the data collection procedure with a sampling design, a probability function defined on the sample space, S, of all possible samples of size n. The sampling design along with unbiasedness requirements yields a classical approach to relating observed with unobserved population units. To assist in the population prediction we assume that a p-dimensional vector is known for each of the N population units. Known as auxiliary information, these p-dimensional vectors are used to obtain a very well-known classical estimator for finite population prediction known as the general regression estimator suggested by [1] and [5]. The general regression estimator is the popular design-based Horvitz-Thompson estimator plus an adjustment term. Sarndal [6] employ classical sampling design theory, using inclusion probabilities, and a regression model y = Xb + e, where X is the auxiliary information matrix, b is the unknown coefficient vector, and e N(0,V). The regression model, however, is used only as a means to obtain an estimate of b. Hence, unbiasedness and variance expressions for the general regression estimator are derived under the classical sampling design approach. In contrast, a superpopulation model provides the stochastic structure for Bayesian inferential purposes. The superpopulation model establishes the main relations between the observed and unobserved units of y. Using a specified superpopulation model, we derive an existing Bayesian estimator in matrix form and show the matrix mathematics involved in order to attain a Bayesian analog to a certain classical estimator.

a Department of Mathematics and Computer Science, Texas Woman's University, P.O. Box 425886, Denton, TX 76204-5886, USA.
Email address: mahmner@twu.edu

b Department of Statistical Sciences, Baylor University, P.O. Box 7140, Waco, TX, 76798-7140. USA

Received by Editor: October 18, 2004 and, in revised form November 22, 2004.
Posted: December 22, 2004.

Copyright 2004