Bayes invariant quadratic estimation in general linear regression models (Q1193988)

The authors consider variance-component models which by a maximal invariant are transformed to an observation vector \(t\) with \(Et=0\), \(Ett'=\sum^ k_{i=1}\sigma_ i^ 2W_ i,W_ k\) a projection-matrix and \(\text{im}(W_ i)\subseteq \text{im}(W_ k),\) \(i=1,\dots ,k-1\). At first, the representation of Bayes invariant quadratic unbiased estimators is found. Secondly, in the two variance components model an explicit formula for the Bayes estimator is given. In general, a basis for a Jordan algebra containing \(\text{span}(W_ 1,\dots,W_ k)\) has to be found. This can be done in the case of a three components model under the additional assumption of orthogonality and in models of the kind \(Ey=\mu 1_ n\), \(\text{Cov }y=\sigma_ 1^ 2D'D+\sigma_ 2^ 2\Delta'\Delta +\sigma_ 3^ 2I_ n\) if \(D'D\) and \(\Delta'\Delta\) are incidence matrices and the orthogonality condition \(D(I-n^{-1}1_ n1'_ n)D'=0\) is met. Also, partially balanced incomplete block designs (PBIBD) are considered.