Jordan algebras and Bayesian quadratic estimation of variance components
In a variance-component model there are in general no uniformly best (quadratic) unbiased estimators for linear functions of variance components. \textit{J. Seely} [Ann. Math. Statistics 42, 710-721 (1971; Zbl 0249.62067)] has found a necessary and sufficient condition for their existence. He found that a certain class of square matrices should form a Jordan algebra, or, as he called it, a quadratic subspace. This condition is in particular met if the variance-component model under investigation is a balanced mixed linear model. It is the purpose of this paper to give some properties of Jordan algebras: simple formulae for \(A\circ B\), \(A^ +\), \(A^ -\); solutions of matrix equations \(AXB=C\), and so on. Applications are the determination of locally optimal estimators and Bayesian quadratic estimators of variance components.
uniformly best quadratic unbiased estimators
Bayesian quadratic estimators of variance components
scientific article; zbMATH DE number 57572