Best quadratic unbiased estimation in variance-covariance component models²

Abstract

This paper deals with the existence of best quadratic unbiased estimators in variance covariance component models. It extends and unifies results previously obtained by Seely, Zyskind, Klonecki, Zmyślony, Gnot, Kleffe and Pincus. The author considers a quasinormally distributed random vector y such that Ey = Xβ, Cov y∈L, where L is a linear space of symmetric square matrices. Conditions for the existence of a BLUE of Ey and a BQUE of Cov y (Eyy′) are investigated. A BLUE exists iff symmetry conditions for certain matrices are met while a BQUE exists iff some modified quadratic subspace conditions are met. At the end of the paper three examples are studied in which all these conditions are met: The Random Coefficient Regression Model, the multivariate linear model and the Behrens-Fisher model. The proofs of the theorems are obtained by considering linear model in y and yy′, respectively.