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

In the general Gauss-Markoff model (Y, Xβ, σ²V), 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.