Non-Negative Maximum Likelihood Estimators of Variance Components in a Simple Linear Model

Abstract

The balanced two-stage nested random effects model is treated in most courses covering the general linear model where the estimation of the variances of the random effects, or the so-called variance components, is of particular interest. The minimum variance unbiased estimators of the variance components are well known (see Graybill [1, p. 342], Searle [3, p. 405]). The maximum likelihood (ml) estimators are easily derived in an unconstrained parameter space, but the estimates may assume negative values (e.g. Searle [3, p. 418]). The derivation of the ml estimators in a restricted parameter space which guarantees non-negative estimates has been given by Herbach [2, pp. 949951]. Thompson and Moore [4] give a simple pool the minimum violator algorithm, developed from the result of maximizing the likelihood function of mean squares subject to a set of constraints which always yields non-negative estimates. The estimators thus obtained differ only slightly from the ml estimators obtained in [2]. In this note we present a simpler analytic method of obtaining the ml estimators in a restricted parameter space which can be used in an introductory linear models course. The model can be written as