Unified Approach to Estimation in Linear Models with Fixed and Mixed Effects

摘要

A unified approach is developed for the estimation of unknown fixed parameters and prediction of random effects in a mixed Gauss-Markoff linear model. It is shown that both the estimators and their mean square errors can be expressed in terms of the elements of a g-inverse of a partitioned matrix which can be set up in terms of the matrices used in expressing the model. No assumptions are made on the ranks of the matrices involved. The method is parallel to the one developed by the author in the case of the fixed effects Gauss-Markoff model using a g-inverse of a partitioned matrix. A new concept of generalized normal equations is introduced for the simultaneous estimation of fixed parameters, random effects and random error. All the results are deduced from a general lemma on an optimization problem. This paper is self contained as all the algebraic results used are stated and proved. The unified theory developed in an earlier paper (Rao, 1988) is somewhat simplified.