A study of the influence of the 'natural restrictions' on estimation problems in the singular Gauss-Markov model

Abstract

It is known that if the Gauss-Markov model M = {Y,Xβ, σ²V} has the column space of the model matrix X not contained in the column space of the dispersion matrix σ²V, then the vector of parameters β has to satisfy certain linear equations. However, these equations become restrictions on β in the usual sense only when the random vector Y involved in them is replaced by an observed outcome y. In this paper, explicit solutions to several statistical problems are derived in two situations: when β is unconstrained and when β is constrained by the 'natural restrictions' mentioned above. The problems considered are: linear unbiased estimation and best linear unbiased estimation of an identifiable vector of parametric functions, comparison of estimators of any vector of parametric functions with respect to the matrix risk, and admissibility among the class of all linear estimators with respect to the matrix risk and with respect to the mean square error. The solutions corresponding to the unconstrained and constrained cases are compared to show in what sense β may be considered to be free to vary without loss of generality.