The estimation of variance components in unbalanced models

Abstract

The estimation of variance components in random and mixed factorial analysis of variance models has been the topic of much research. When all the cells of the model have the same amount of data, uniformly, minimum variance, unbiased estimates are known. In this situation closed formed expressions are available for the estimates of the variance components. When different amounts of data are available in the cells no estimation procedure is agreed upon and uniformly, minimum variance, unbiased estimates are not known. In addition, none of the current estimation procedures have a known closed form expression for the estimates. The procedures are either iterative or a solution to a system of linear equations. This manuscript considers an estimation technique that averages the estimates obtained from subsets of the data that are balanced and contain m pieces of data per cell. Here m is less than or equal to the minimum cell size and greater than 1. A closed form expression will be derived for the estimates, small sample variances determined, similarities to the form of the estimates in the balanced case examined, and an examination of the efficiencies of the estimates will be made. The averaging idea will also be considered for the estimation of the fixed effects.