On Covariance in Finite Population Sampling

Abstract

A number of covariance results in finite population sampling are brought together in this article. Instructive methods are given to derive these results. Whilst sampling variance provides a measure of the precision of estimators in finite population surveys, the role of sampling covariance is however different but construc- tive, as it generally works to enhance the precision of estimators. An orthodox example of sampling covariance at work can be found in simple random samples drawn without replacement, in which variance of the sample mean estimator is only (I -f)N/(N- 1) times the variance of the same estimator derived from a simple random sample drawn with replacement, wheref= n/N is the sampling fraction. The factor is less than unity for n > 1. Despite this, one is rather surprised to find that little attention is given in sampling texts on how sampling covariances are at work; in some instances, sampling covariances are simply ignored as pointed out by Scott & Seber (1983). Sampling covariances seem to be least understood by students. One purpose of this article is to bring together a number of results on sampling covariances under a simple random sampling without replacement design. Though most of these results are not new, they are not readily available in the literature. Furthermore the methods used to derive these results are instructive and hopefully will give some insight to the student on how sampling covariances are at work.