Linear Spaces and Minimum Variance Unbiased Estimation

Abstract

Consideration is given to minimum variance unbiased estimation when the choice of estimators is restricted to a finite-dimensional linear space. The discussion gives generalizations and minor extensions of known results in linear model theory utilizing both the coordinate-free approach of Kruskal and the usual parametric representations. Included are (i) a restatement of a theorem on minimum variance unbiased estimation by Lehmann and Scheffe; (ii) a minor extension of a theorem by Zyskind on best linear unbiased estimation; (iii) a generalization of the covariance adjustment procedure described by Rao; (iv) a generalization of the normal equations; and (v) criteria for existence of minimum variance unbiased estimators by means of invariant subspaces. Illustrative examples are included.