Sandwich theorem in comparison of multivariate normal experiments

Abstract

For normal linear experiments E 1 = N(Aβ, ∑ σ iV i) and E 2 = N(Bβ, ∑ σ iW i) with unknown variance components σ i , i = 1,…, q, we consider some relations of type “ E 1 is at least as good as E 2”, among others with respect to (a) linear estimation, (b) quadratic estimation and (c) quadratic estimation of variance components. It is shown, in a simple way, that (b)⇒(c)⇒(a).