Geometrical Relations Among Variance Component Estimators

Abstract

In linear models with two variance components, mean squared errors (MSEs) of invariant quadratic estimators of linear combinations of the variance components are quadratics in the variance components. However, it is revealing to note that MSEs are linear in convex combinations of the squares and products of the variance components, so that surfaces of MSEs are subsets of planes. Lower bounds on MSEs form a concave surface. MSEs of admissible invariant quadratic estimators are tangent to this surface, while MSEs of inadmissible estimators are not. There are different bounds when attention is restricted to unbiased invariant quadratic estimators or the more general class obtained by dropping the restriction to unbiasedness. The purpose of this paper is to develop these simple relations and to illustrate them with graphs.