Efficiency properties of cell means variance component estimates (Q1330178)

The authors consider a variance-components model of the kind \[ Y = \sum_{r \in F} U_ r \theta_ r + \sum_{u \in R} U_ u \xi + \varepsilon, \] arising from a factorial experiment with \(k\) factors. \(F\) represents the set of fixed factors, \(R\) represents the set of random factors. By the use of some linear transformation (``cell mean transformation'') the model \(\overline {Y} = \overline {X}\theta + \overline {U}\xi + \overline {\varepsilon}\) is obtained. The image space is decomposed into certain pairwise orthogonal subspaces. If \(\overline {Y}_ r\) represents the projection of \(\overline {Y}\) on the \(r\)-th space, \(q_ r = \| \overline {Y}_ r \|^ 2\) can be computed. The expectation of \(q_ r\) is a linear function of the variances and these -- alike to Henderson's method -- unbiased estimators can easily be computed, since the system is triangular. The efficiency of these estimators compared with the MINQUE-estimators is studied. It would be interesting to investigate the used transformation from the sufficiency (linear, quadratic) point of view.