Completely robust statistics for dependence in the general linear model (case of full rank) (Q1063358)

The authors study the general linear model \(Y=X\beta +\epsilon\) where X is an \(n\times p\) matrix of constants, \(\beta\) is a p-vector of unknown parameters, \(\epsilon\) is a normal random n-vector with the errors \(\epsilon_ i\) dependent, i.e. \(\epsilon\) \(\sim N(0,\Sigma)\), with \(\Sigma\) positive definite unknown covariance matrix of \(\epsilon\). In particular is investigated if there exist matrices \(\Sigma \neq \sigma^ 2I\), such that the F-statistic, normally used to test hypotheses on the parameters of the linear model, still has Snedecor's F- distribution. The authors present the definition of completely robust statistics for dependent data and recall previous results they obtained for ANOVA, ANCOVA and BIB designs. Further they show that for the general linear model \(\sigma^ 2\) I is the only matrix which leads to a statistic that, for the hypothesis \(\beta =\beta_ 0\), has an F- distribution.