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

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  • Bischoff, Wolfgang

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.

Suggested Citation

  • Bischoff, Wolfgang, 2000. "The Structure of a Linear Model: Sufficiency, Ancillarity, Invariance, Equivariance, and the Normal Distribution," Journal of Multivariate Analysis, Elsevier, vol. 73(2), pages 180-198, May.
    • Handle: RePEc:eee:jmvana:v:73:y:2000:i:2:p:180-198
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    References listed on IDEAS

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    1. W. Bischoff & H. Cremers & W. Fieger, 1987. "A characterization of the normal distribution by sufficiency of the least squares estimation," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 34(1), pages 259-273, December.
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