The Best Strategy for Estimating the Mean of a Finite Population

Abstract

If the finite population is homogeneous and the statistician's resources permit him to take a sample size $m$ at the most, it is shown here that his best strategy for estimating the population mean is to draw a sample of size $m$ by simple random sampling without replacement and to take the sample mean as the estimate. The strategy is the best in the sense that in the entire class of unbiased strategies subject to the restriction that the size of any observed sample does not exceed $m$, this strategy minimizes for any convex loss function both the maximum and the average risks over the set of parameter points arising from permutations of the labels of the population units. Similar decision-theoretic justification is also derived for the customary strategy for a two-stage cluster-sampling design.