The Structure of a Linear Model: Sufficiency, Ancillarity, Invariance, Equivariance, and the Normal Distribution

Consider a general linear model Y=Xβ+Z where Cov Z may be known only partially. We investigate carefully the notions of sufficiency, ancillarity, invariance, and equivariance and related notions for projectors in a general linear model. In this way we can prove a Basu type theorem. This result can be used to give the relation between the sufficiency of the generalized least-squares estimator and the assumption that Z is normally distributed. So we can generalize the well-known result that the generalized least-squares estimator is sufficient for β if Z is normally distributed. Further we can solve the converse problem as well.