R. Kala

The matrix equation AXB+CYD=E

Abstract A necessary and sufficient condition for the matrix equation AXB + CYD = E to be consistent is established, and if it is, a representation of the general solution is given, thus generalizing earlier results concerning the equation AX + YB = C .

The Matrix Equation AX - YB = C

Abstract A necessary and sufficient condition is established for solvability of the matrix equation AX − YB = C . The condition differs from that given by W.E. Roth. The general solution of the equation is also found.

Linear sufficiency with respect to a given vector of parametric functions

Solutions are given to the problems concerned with characterizing transformations of a general Gauss-Markov model [Y, Xβ, V] into [FY, FXβ, FVF′] such that the corresponding loss of information, if any, is irrelevant from the standpoint of determining the minimum dispersion linear unbiased estimator of a given vector of estimable parametric functions.

Range invariance of certain matrix products

A necessarv and sufficient condition is established for the product AB C to have its range.A (AB C), invariant with respect to the choice of a generalized inverse B .This result is then used to derive criteria for the invariance of the subspaces A(AB ).A(B C)A(B) and A(BB C) and also to deduce that the simultaneous invariance of the range of AB− C and the range of its conjugate transpose entails the invariance of the product AB−C itself.

On equalities between BLUES, WLSEs, and SLSEs†

Necessary and sufficient conditions for equalities between the best linear unbiased estimator, the weighted least-squares estimator, and the simple least-squares estimator of the expectation vector in a general Gauss-Markoff model are given in some alternative formulations. The main result states, somewhat surprisingly, that the weighted least-squares estimator cannot be identical with the simple least-squares estimator unless they both coincide with the best linear unbiased estimator.

Symmetrizers of matrices

Abstract A symmetrizer of a given pair of matrices, A and B , is defined as a matrix X for which the product AXB is symmetric. Right and left symmetrizers of a given matrix A are defined accordingly. The main results of the paper are general representations of all three types of symmetrizers. The problem considered arose in connection with certain questions pertaining to admissible linear estimation in a Gauss-Markoff model.

An extension of a rank criterion for the least squares estimator to be the best linear unbiased estimator

Abstract Among criteria for the least squares estimator in a linear model (y, Xβ, V) to be simultaneously the best linear unbiased estimator, one convenient for applications is that of Anderson (1971, 1972). His result, however, has been developed under assumptions of full column rank for X and nonsingularity for V. Subsequently, this result has been extended by Styan (1973) to the case when the restriction on X is removed. In this note, it is shown that also the restriction on V can be relaxed and, consequently, that Andersons criterion is applicable to the general linear model without any rank assumptions at all.

Simple least squares estimation versus best linear unbiased prediction

Abstract Necessary and sufficient conditions are developed for the simple least squares estimator to coincide with the best linear unbiased predictor. The conditions obtained are valid for a general linear model and are generalizations of the condition given by Watson (1972). Also, as a preliminary result, a new representation of the best linear unbiased predictor is established.

Covariance Adjustment When a Vector of Parameters is Restricted to a Given Subspace

The covariance adjustment of an estimator is considered in the case where a vector of unknown parameters is known a priori to be contained in a given subspace. The estimator obtained is shown to be at least as good as that produced by the customary covariance adjustment technique.

Some methods for constructing efficiency-balanced block designs

Abstract Three methods are presented for constructing connected efficiency-balanced block designs from other block designs with the same properties. The resulting designs differ from the original ones in the number of blocks and/or in the number of experimental units and their arrangement, while the number of treatments remains unaltered. Some remarks on the proposed methods of construction refer also to variance-balanced block designs.