Texas Project NExT Journal

Vol. 2 (2004), pp. 11-20.

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

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**Mark S. Hamner**^{a}, John W. Seaman^{b}, Jr., Dean M.
Young^{b}

**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