Asymptotic Behavior of Wilcoxon Type Confidence Regions in Multiple Linear Regression

Abstract

In the multiple linear regression model a class of confidence regions based on rank statistics is constructed. The asymptotic behavior of the center of gravity of a region corresponding to the Wilcoxon type rank statistic is considered. Section 1 consists of assumptions, introduction and notation. Section 2 consists of a monotonicity lemma. A uniform continuity theorem for Wilcoxon type linear rank statistics is proved in Section 3. This and the monotonicity lemma are used to show that the regions are bounded for large sample size, with large probability. Section 4 gives the asymptotic normality of the above mentioned quantity and defines a consistent estimator of $(\int f^2(x))$. Finally, the Appendix contains some results on relative compactness of some stochastic processes which are used in Section 3.