Some Simplifying Results on BLUEs

Abstract

When the error covariance matrix, V, for the model Y = X@ + el is nonsingular it is known that the Best Linear Unbissed Estimator (BLUE) of A'@ is l'(X'V-lX)+XY-lY, in which M + represents the Moore-Penrose generalized inverse. When V is singular this does not generalize to 1'(X'V+X)+X'V+Y. This paper discusses approaches to finding BLUES. Simple proofs and new results are presented, primarily baaed on a study of the equation BLUE (1'0) = w'Y +a, where WY is a proper random variable and a is a scalar constant,

Matrix rank and inertia formulas in the analysis of general linear models

“…In Section 5, we first give a group of results on the OLSP of (2) and its statistical properties. We then establish a group of formulas for calculating the rank and inertia in (10) and use the formulas to characterize the equality and inequalities in (12). The connections between the OLSP and BLUP of (2), as well as the equality and inequalities between the dispersion matrices of the OLSP and BLUP of (2) are investigated in Section 6.…”

“…Even so, people are still able to obtain more and more new and valuable results on statistical inference of GLMs. Estimation ofˇas well as prediction of " " " in (1) are major concerns in the statistical inference of (1), and it is always desirable, as claimed in [10,11], to simultaneously identify estimators and predictors of all unknown parameters in GLMs. As formulated in [10,11], a general vector of linear parametric functions involving the two unknown parameter vectorsˇand " " " in (1) is given by…”