Nonnegative Estimation of Variance Components for a Nested Three-Way Random Model

Abstract

A nonnegative variance estimation procedure is suggested for an unbalanced data where two factors are nested in another. Since the involved factors are all random, the approach is based on a nested three-way random model. The proposed method for the estimation of variance components is compared with Henderson’s Method I and III in view of the same estimation procedure based on the method of moments. Although both the Henderson’s Method I and III are known to be useful for the estimation of variance components for balanced or unbalanced data, they often yield negative values as variance estimates whereas the estimates by the suggested method are never be negative. Hence, it points out what makes this happen and discusses how to fix the problem. The proposed method shows how to define sums of squares and the orthogonal coefficient matrices that are necessary for the evaluation of expectations. All the matrices of the quadratic forms for computing sums of squares are symmetric and idempotent. It also reveals the individual coefficient of each variance component does not change from equation to equation.