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

Wolfgang Bischoff

Journal of Multivariate Analysis, 2000, vol. 73, issue 2, 180-198

Abstract: Consider a general linear model Y=X[beta]+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 [beta] if Z is normally distributed. Further we can solve the converse problem as well.

Keywords: linear model; sufficiency; specific sufficiency; ancillarity; invariance; equivariance; normal distribution; partially known covariance matrices (search for similar items in EconPapers)
Date: 2000
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