The structure of a linear model : sufficiency, ancillarity, invariance, equivariance and the normal distribution

Kurzfassung/Abstract

Consider a general linear model $Y=X\beta+Z$, where $\text{Cov}\,Z$ 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 $\beta$ if $Z$ is normally distributed. Further we can solve the converse problem as well.