Hostname: page-component-745bb68f8f-s22k5 Total loading time: 0 Render date: 2025-02-11T18:42:28.611Z Has data issue: false hasContentIssue false
  • English
  • Français

The convex hull of a spherically symmetric sample

Published online by Cambridge University Press:  01 July 2016

William F. Eddy  and
James D. Gale
William F. Eddy*
Affiliation:
Carnegie-Mellon University
James D. Gale*
Affiliation:
Carnegie-Mellon University
*
Postal address for both authors: Department of Statistics, Carnegie-Mellon University, Schenley Park, Pittsburgh, PA 15213, U.S.A.
Postal address for both authors: Department of Statistics, Carnegie-Mellon University, Schenley Park, Pittsburgh, PA 15213, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Abstract

Using the isomorphism between convex subsets of Euclidean space and continuous functions on the unit sphere we describe the probability measure of the convex hull of a random sample. When the sample is spherically symmetric the asymptotic behavior of this measure is determined. There are three distinct limit measures, each corresponding to one of the classical extreme-value distributions. Several properties of each limit are determined.

Keywords

EXTREME-VALUE PROCESSPOISSON RANDOM MEASURERANDOM SETSUPPORT FUNCTIONDIRECTIONWISE MAXIMUM
Type
Research Article
Information
Advances in Applied Probability , Volume 13 , Issue 4 , December 1981 , pp. 751 - 763
Copyright
Copyright © Applied Probability Trust 1981 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

∗

Research supported in part by NSF Grants MCS 78-02422 and MCS-80-05115 to Carnegie-Mellon University.

∗∗

Based in part on a portion of this author's Ph.D. thesis submitted to Carnegie-Mellon University.

References

Barnett, V. (1976) The ordering of multivariate data. J. R. Statist. Soc. A 139, 318–354.Google Scholar
Bebbington, A. C. (1978) A method of bivariate trimming for robust estimation of the correlation coefficient. Appl. Statist. 27, 221–271.Google Scholar
Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
Carnal, H. (1970) Die konvexe Hülle von n rotationssymmetrischverteilen Punkten. Z. Wahrscheinlichkeitsth. 15, 168–176.CrossRefGoogle Scholar
DeHaan, L. (1970) On Regular Variation and its Application to the Weak Convergence of Sample Extremes. Tract 32, Mathematisch Centrum, Amsterdam.Google Scholar
Eddy, W. F. (1980) The distribution of the convex hull of a Gaussian sample. J. Appl. Prob. 17, 686–695.CrossRefGoogle Scholar
Eddy, W. F. and Hartigan, J. A. (1977) Uniform convergence of the empirical distribution function over convex sets. Ann. Statist. 5, 370–374.CrossRefGoogle Scholar
Efron, B. (1965) The convex hull of a random set of points. Biometrika 52, 331–343.CrossRefGoogle Scholar
Eggleston, H. G. (1958) Convexity. Cambridge University Press.CrossRefGoogle Scholar
Fisher, L. (1969) Limiting sets and convex hulls of samples from product measures. Ann. Math. Statist. 40, 1824–1832.Google Scholar
Gale, J. D. (1980) The Asymptotic Distribution of the Convex Hull of a Random Sample. , Carnegie-Mellon University.Google Scholar
Geffroy, J. (1959) Contribution à la theorie des valeurs extrêmes. Publ. Inst. Statist. Univ. Paris VIII, 123–185.Google Scholar
Geffroy, J. (1961) Localisation asymptotique du polyèdre d'appui d'un échantillon laplacien à k dimensions. Publ. Inst. Statist. Univ. Paris X, 212–228.Google Scholar
Gnedenko, B. V. (1943) Sur la distribution limite du terme maximum d'un série aléatoire. Ann. Math. 44, 423–453.Google Scholar
Huber, P. J. (1972) Robust statistics: A review. Ann. Math. Statist. 43, 1041–1067.CrossRefGoogle Scholar
Jagers, P. (1974) Aspects of random measure and point processes. In Advances in Probability 3, ed. Ney, P. and Port, S. C. Dekker, New York.Google Scholar
Kendall, D. G. (1974) Foundations of a theory of random sets. In Stochastic Geometry, ed. Harding, E. F. and Kendall, D. G. Wiley, New York.Google Scholar
Mase, S. (1979) Random compact convex sets which are infinitely divisible with respect to Minkowski addition. Adv. Appl. Prob. 11, 834–850.CrossRefGoogle Scholar
Matheron, G. (1975) Random Sets and Integral Geometry. Wiley, New York.Google Scholar
Pickands, J. (1971) The two-dimensional Poisson process and extremal processes. J. Appl. Prob. 8, 745–756.CrossRefGoogle Scholar
Raynaud, H. (1965) Sur le comportement asymptotique de l'enveloppe convexe d'un nuage de points tirés au hasard dans ℝ n . C. R. Acad. Sci. Paris 261, 627–629.Google Scholar
Raynaud, H. (1970) Sur l'enveloppe convexe des nuages de points aléatoires dans ℝ n . I. J. Appl. Prob. 7, 35–48.Google Scholar
Rényi, A. and Sulanke, R. (1963), (1964) Über die konvexe Hülle von n zufällig gewählten Punkten, I and II. Z. Wahrscheinlichkeitsth. 2, 75–84; 3, 138–147.CrossRefGoogle Scholar
Resnick, S. I. (1975) Weak convergence to extremal processes. Ann. Prob. 3, 951–960.Google Scholar
Ripley, B. D. and Rasson, J.-P. (1977) Finding the edge of a Poisson forest. J. Appl. Prob. 14, 483–491.Google Scholar
Rogers, L. C. G. (1978) The probability that two samples in the plane have disjoint convex hulls. J. Appl. Prob. 15, 790–802.Google Scholar
Ruben, H. and Miles, R. E. (1980). A canonical decomposition of the probability measure of sets of isotropic random points in ℝn . J. Multivariate Anal. 10, 1–18.Google Scholar
Sager, T. W. (1979) An iterative method for estimating a multivariate mode and isopleth. J. Amer. Statist. Assoc. 74, 329–339.Google Scholar
Valentine, F. A. (1964) Convex Sets. McGraw-Hill, New York.Google Scholar
Yang, S. S. (1977) General distribution theory of the concomitants of order statistics. Ann. Statist. 5, 996–1002.CrossRefGoogle Scholar