1 Estimation of variance components

Abstract

Publisher Summary This chapter discusses the usual mixed linear model on variance components. The unknown parameters of this model are called “variance components.” The ANOVA technique provides good estimators in balanced designs, but such estimators may be inefficient in more general linear models. A completely different approach is the ML (maximum likelihood) method. The likelihood of the unknown parameters is based on observed Y and the likelihood equations are obtained by computing the derivatives of likelihood with respect to the parameters. The marginal likelihood based on the maximal invariant of Y and obtained is called “marginal maximum likelihood (MML) equations.” The general large sample properties associated with ML estimators are misleading in the absence of studies on the orders of sample sizes for which these properties hold in particular cases. The bias in MML estimators may not be large even in small samples. The chapter also discusses a general method called “minimum norm quadratic estimation” (MINQE). The method is applicable in situations where ML and MML fail. It offers a wide scope in the choice of the norm depending on the nature of the model and prior information available and there is an automatic provision for incorporating available prior information on the unknown parameters. The MINQE equation provides a natural numerical algorithm for computing the ML or MML estimator. For a suitable choice of the norm, the MINQ estimators provide minimum variance estimators of θ when Y is normally distributed.