Sufficiency and completeness in the linear model
The author considers first the linear model \(Ey=X\beta\), Cov y\(=V\) and defines, following \textit{J. K. Baksalary} and \textit{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.
scientific article; zbMATH DE number 3994798