## The Annals of Mathematical Statistics

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

Hira Lal Koul

#### 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].

#### Article information

Source
Ann. Math. Statist., Volume 42, Number 2 (1971), 466-476.

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177693398

Digital Object Identifier
doi:10.1214/aoms/1177693398

Mathematical Reviews number (MathSciNet)
MR288896

Zentralblatt MATH identifier
0215.54204

JSTOR