Representations of best linear unbiased estimators in the Gauss-Markoff model with a singular dispersion matrix

Abstract

In the general Gauss-Markoff model ( Y, Xβ, σ 2 V), when V is singular, there exist linear functions of Y which vanish with probability 1 imposing some restrictions on Y as well as on the unknown β. In all earlier work on linear estimation, representations of best-linear unbiased estimators (BLUE's) are obtained under the assumption: “ L′Y is unbiased for Xβ ⇒ L′X = X.” Such a condition is not, however, necessary. The present paper provides all possible representations of the BLUE's some of which violate the condition L′X = X. Representations of X for given classes of BLUE's are also obtained.