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

Author： Bischoff W.  

Publisher： Academic Press

ISSN： 0047-259X

Source： Journal of Multivariate Analysis, Vol.73, Iss.2, 2000-05, pp. : 180-198

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Abstract

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. Copyright 2000 Academic Press.

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