Let x ~ N([mu], Z), and let S = ([Sigma]:[mu]). It is shown that if x'A1x is independent of x'Bx (x'A1x is distributed as a chi-square variable), then this property is inherited by every x'A2x for which S'A2S precedes S'A1S with respect to the range preordering (with respect to the rank...

Two restricted linear models M0 and M are considered. The models can differ in model matrices, dispersion matrices, and possibly in parameter restrictions. The validity under M of estimators that are admissible and linearly sufficient under M0 is studied. Criteria are derived for strong and week...

The general ridge estimators are characterized as linearly sufficient and admissible among the set of all linear estimators under a nonsingular linear model. Their counterparts, in some more general models, are presented.

The problem of characterizing dependence structures that preserve optimality is studied. The result is illustrated in the settings of block designs and repeated measurements designs. Some generalizations of earlier characterizations are established.