Browse > Article
http://dx.doi.org/10.29220/CSAM.2018.25.1.079

Independence and maximal volume of d-dimensional random convex hull  

Son, Won (Department of Statistics, Seoul National University)
Park, Seongoh (Department of Statistics, Seoul National University)
Lim, Johan (Department of Statistics, Seoul National University)
Publication Information
Communications for Statistical Applications and Methods / v.25, no.1, 2018 , pp. 79-89 More about this Journal
Abstract
In this paper, we study the maximal property of the volume of the convex hull of d-dimensional independent random vectors. We show that the volume of the random convex hull from a multivariate location-scale family indexed by ${\Sigma}$ is stochastically maximized in simple stochastic order when ${\Sigma}$ is diagonal. The claim can be applied to a broad class of multivariate distributions that include skewed/unskewed multivariate t-distributions. We numerically investigate the proven stochastic relationship between the dependent and independent random convex hulls with the Gaussian random convex hull. The numerical results confirm our theoretical findings and the maximal property of the volume of the independent random convex hull.
Keywords
convex hull; independence; multivariate location-scale family; simple stochastic order; stochastic geometry;
Citations & Related Records
Times Cited By KSCI : 2  (Citation Analysis)
연도 인용수 순위
1 Affentranger F (1991). The convex hull of random points with spherically symmetric distributions, Rendiconti del Seminario Matematico Universita e Politecnico di Torino, 49, 359-383.
2 Asimit AV, Furman E, and Vernic R (2010). On a multivariate Pareto distribution, Insurance: Mathematics and Economics, 46, 308-316.   DOI
3 Azzalini A and Capitanio A (2003). Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t-distribution, Journal of the Royal Statistical Society. Series B (Statistical Methodology), 65, 367-389.   DOI
4 Barany I and Vu V (2007). Central limit theorems for Gaussian polytopes, The Annals of Probability, 35, 1593-1621.   DOI
5 Barber CB, Dobkin DP, and Huhdanpaa H (1996). The Quickhull algorithm for convex hulls, ACM Transactions on Mathematical Software (TOMS), 22, 469-483.   DOI
6 Barnett V (1976). The ordering of multivariate data, Journal of the Royal Statistical Society. Series A (General), 139, 318-355.
7 Fang KT, Kotz S, and Ng KW (1990). Symmetric Multivariate and Related Distributions, MA Springer US, Boston.
8 Carpenter M and Diawara N (2007). A multivariate gamma distribution and its characterizations, American Journal of Mathematical and Management Sciences, 27, 499-507.
9 Cook RD (1979). Influential observations in linear regression, Journal of the American Statistical Association, 74, 169-174.   DOI
10 Cover T and Gamal AE (1983). An information - theoretic proof of Hadamard's inequality (Corresp.), IEEE Transactions on Information Theory, 29, 930-931.   DOI
11 Fawcett T and Niculescu-Mizil A (2007). PAV and the ROC convex hull, Machine Learning, 68, 97-106.   DOI
12 Hug D (2013). Random polytopes. In Spodarev E (eds) Stochastic Geometry, Spatial Statistics and Random Fields, Lecture Notes in Mathematics, (Vol. 2068, pp. 205-238). Springer-Verlag, Berlin-Heidelberg.
13 Hug D and Reitzner M (2005). Gaussian polytopes: variances and limit theorems, Advances in Applied Probability, 37, 297-320.   DOI
14 Lim J and Won JH (2012). ROC convex hull and nonparametric maximum likelihood estimation, Machine Learning, 88, 433-444.   DOI
15 Ng CT, Lim J, Lee KE, Yu D, and Choi S (2014). A fast algorithm to sample the number of vertexes and the area of the random convex hull on the unit square, Computational Statistics, 29, 1187-1205.
16 Ollila E, Oja H, and Croux C (2003). The affine equivariant sign covariance matrix: asymptotic behavior and efficiencies, Journal of Multivariate Analysis, 87, 328-355.   DOI
17 Zhao J and Kim HM (2016). Power t distribution, Communications for Statistical Applications and Methods, 23, 321-334.   DOI
18 Renyi A and Sulanke R (1963). Uber die konvexe Hulle von n zufallig gewahlten Punkten, Zeitschrift Fur Wahrscheinlichkeitstheorie und Verwandte Gebiete, 2, 75-84.   DOI
19 Shao J (2003). Mathematical Statistics (2nd ed), Springer, New York.
20 Son W, Ng CT, and Lim J (2015). A new integral representation of the coverage probability of a random convex hull, Communications of Statistical Applications and Methods, 22, 69-80.   DOI
21 Renyi A and Sulanke R (1964). Uber die konvexe Hulle von n zufallig gewahlten Punkten, Zeitschrift Fur Wahrscheinlichkeitstheorie Und Verwandte Gebiete, 3, 138-147.