Variance Component Estimation by the Method of Least-Squares

Abstract

Motivated by the fact that the method of least-squares is one of the leading principles in parameter estimation, we introduce and develop the method of least-squares variance component estimation (LS-VCE). The results are presented both for the model of observation equations and for the model of condition equations. LS-VCE has many attractive features. It provides a unified least-squares framework for estimating the unknown parameters of both the functional and stochastic model. Also, our existing body of knowledge of least-squares theory is directly applicable to LS-VCE. LS-VCE has a similar insightful geometric interpretation as standard least-squares. Properties of the normal equations, estimability, orthogonal projectors, precision of estimators, nonlinearity, and prior information on VCE can be easily established. Also measures of inconsistency, such as the quadratic form of residuals and the w-test statistic can directly be given. This will lead us to apply hypotheses testing to the stochastic model.