Minqe theory and its relation to ML and MML estimation of variance components

Abstract

Starting with the general linear model Y=Xβ+ε where E(εε')=θ₁V₁+ ... +θ_pV_p, the theory of minimum norm quadratic estimation (MINQE) of the parameter θ=(θ₁... θ_p)' is developed. The method depends on the choice of a natural quadratic estimator of θ in terms of the unobservable variable ε and comparing it with a quadratic estimator Y'AY in terms of the observable variable Y. The matrix A is determined by minimizing the difference between two quadratic forms. By placing restrictions on Y'AY such as unbiasedness(U), invariance(I) under translation of Y by Xβ, different kinds of MINQU's such as MINQE(I), MINQE(U), MINQE(U, I), etc. are generated. A class of iterated MINQE's(IMINQE's) is developed to obtain estimators free from apriori information used in the construction of MINQE's. This class is shown to include maximum likelihood (ML) and marginal ML (MML) estimators. Thus the MINQE principle provides a unified theory of estimation of variance components.