Asymptotic Behavior of a Class of Confidence Regions Based on Ranks in Regression

Abstract

Asymptotic behavior of a class of confidence regions, based on rank statistics, for the regression parameter vector is considered. These regions are shown to be asymptotically bounded and ellipsoidic in probability. Asymptotic normality of their center of gravities is also proved. It is noted that the asymptotic efficiencies of these regions when defined in terms of ratio of Lebesgue measures corresponds to that of corresponding test statistics that are used to define these regions. Similar conclusion holds for their center of gravities, where now asymptotic efficiency is defined as inverse ratio of their generalized limiting variances. Also a class of consistent estimators is given for some functionals of the underlying distributions. Finally simultaneous confidence intervals, based on the above center of gravity, for linear parametric functions are shown to have asymptotic coverage probability $1 - \alpha$. Basic to this work are two papers, one by the author [4] and one by Jureckova [3].