International Journal of Computational Geometry & Applications c  World Scientiﬁc Publishing Company Quantile Approximation for Robust Statistical Estimation and k-Enclosing Problems ∗ DAVID M. MOUNT † Department of Computer Science, University of Maryland, College Park, Maryland 20742 E-mail: mount@cs.umd.edu NATHAN S. NETANYAHU ‡ Dept. of Mathematics and Computer Science, Bar-Ilan University, Ramat-Gan 52900, Israel and Ctr. for Automation Research, University of Maryland, College Park, Maryland 20742 E-mail: nathan@macs.biu.ac.il CHRISTINE D. PIATKO The Johns Hopkins University Applied Physics Laboratory, Laurel, Maryland, 20723 E-mail: christine.piatko@jhuapl.edu RUTH SILVERMAN Center for Automation Research, University of Maryland, College Park, Maryland, 20742 E-mail: ruth@cfar.umd.edu ANGELA Y. WU Department of Computer Science and Information Systems, American University, Washington, DC 20016 E-mail: awu@american.edu Received received date Revised revised date Communicated by Editor’s name ABSTRACT Given a set P of n points in R d , a fundamental problem in computational geometry is concerned with ﬁnding the smallest shape of some type that encloses all the points of P . Well-known instances of this problem include ﬁnding the smallest enclosing box, minimum volume ball, and minimum volume annulus. In this paper we consider the following variant: Given a set of n points in R d , ﬁnd the smallest shape in question that contains at least k points or a certain quantile of the data. This type of problem is known as a k-enclosing problem. We present a simple algorithmic framework for computing quantile approximations for the minimum strip, ellipsoid, and annulus containing a given quantile of the points. The algorithms run in O(n log n) time. Keywords: Robust estimation, LMS regression, minimum enclosing disk, minimum vol- ume ball/ellipsoid/annulus estimator. * A preliminary version of this paper appeared in Proceedings of the Tenth Canadian Conference on Computational Geometry, Montr´ eal, Qu´ ebec, Canada, August 9–12, 1998, pp. 18–19. † This work was supported in part by National Science Foundation grant CCR–9712379. ‡ This research was carried out in part while the author was also aﬃliated with the Center of Excellence in Space Data and Information Sciences (CESDIS), Code 930.5, NASA/GSFC. 1