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Sufficiency and completeness in the linear model. (English) Zbl 0614.62080
The author considers first the linear model $$Ey=X\beta$$, Cov y$$=V$$ and defines, following J. K. Baksalary and R. Kala [Ann. Stat. 9, 913-916 (1981; Zbl 0471.62067)] and the reviewer [Sankhyā, Ser. A 45, 88-98 (1983; Zbl 0535.62007)], when a linear statistic $$z=Ly$$ is linearly sufficient or linearly complete. New characterizations for there concepts are given (Proposition 3.1). Next, the model $$Ey=X\beta$$, Cov y$$=\sigma^ 2V$$ is dealt with. It is defined and characterizations are given when (Ly,y’Ty) is quadratically sufficient.
As an application the author discusses stepwise regression and gives a necessary and sufficient condition when the first step estimators of expectation and residual variance, respectively, form, together with additional observations, a quadratically sufficient statistic.
Reviewer: H.Drygas

##### MSC:
 62J05 Linear regression; mixed models 62B05 Sufficient statistics and fields
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##### References:
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