Variance of a Weighted Mean

Abstract

In many areas of statistical practice the problem arises of combining several estimates of an unknown quantity to obtain an estimate of improved precision. For example, suppose two or more technicians have performed assays on several samples from a homogeneous material. It is desired to average their results in the best manner possible, allowing for the fact that technicians differ in precision. In general the relative precisions are not known exactly, but estimates are available from current or previous experimental data. A similar problem arises in the analysis of incomplete block experiments. The intra-block and estimates of varietal means have different variances, and the recovery of inter-block information is an attempt to combine these estimates in the most efficient manner. Although similar methods are applicable, the experimental design problem differs from the simple case of weighted means in several important respects. An investigation of this problem in the case of simple lattice designs will be presented in a subsequent paper. The model for the first problem may be described as follows. Let U1, UkfU?. be k estimates of a parameter ,u, being independently and normally distributed with mean ,u and variances o, . . . Also let s2, . , s. 2 be independent unbiased estimates of the _2 having