Weak generalized inverses and minimum variance linear unbiased estimation

Abstract

This paper presents a unified account of the theory of least squares and its adaptations to statis· tic al models more complicated than the classical one. Firs t comes a developme nt of the properties of weak general ized matrix inverses, a useful variant of the more familiar pseudo·inverse. These properties are e mployed in a proof of the usual Gauss theorem, and in analyzin g the case in which known linear res traints are obeyed by the para me ters. Anothe r s itu ation treated is that of a s ingular variance-co variance matrix for the observations . Applications include the case of equi-correlated variables (i ncluding es timation despite ignorance of the corre lation), linear " res tra ints" subject to random error, and step wise linear es timation.