Communications for Statistical Applications and Methods

The Korean Statistical Society (한국통계학회)
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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)
  • Received : 2017.10.16
  • Accepted : 2017.12.20
  • Published : 2018.01.31
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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.

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References

  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. https://doi.org/10.1016/j.insmatheco.2009.11.004
  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. https://doi.org/10.1111/1467-9868.00391
  4. Barany I and Vu V (2007). Central limit theorems for Gaussian polytopes, The Annals of Probability, 35, 1593-1621. https://doi.org/10.1214/009117906000000791
  5. Barber CB, Dobkin DP, and Huhdanpaa H (1996). The Quickhull algorithm for convex hulls, ACM Transactions on Mathematical Software (TOMS), 22, 469-483. https://doi.org/10.1145/235815.235821
  6. Barnett V (1976). The ordering of multivariate data, Journal of the Royal Statistical Society. Series A (General), 139, 318-355.
  7. Carpenter M and Diawara N (2007). A multivariate gamma distribution and its characterizations, American Journal of Mathematical and Management Sciences, 27, 499-507.
  8. Cook RD (1979). Influential observations in linear regression, Journal of the American Statistical Association, 74, 169-174. https://doi.org/10.1080/01621459.1979.10481634
  9. Cover T and Gamal AE (1983). An information - theoretic proof of Hadamard's inequality (Corresp.), IEEE Transactions on Information Theory, 29, 930-931. https://doi.org/10.1109/TIT.1983.1056751
  10. Fang KT, Kotz S, and Ng KW (1990). Symmetric Multivariate and Related Distributions, MA Springer US, Boston.
  11. Fawcett T and Niculescu-Mizil A (2007). PAV and the ROC convex hull, Machine Learning, 68, 97-106. https://doi.org/10.1007/s10994-007-5011-0
  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. https://doi.org/10.1017/S0001867800000197
  14. Lim J and Won JH (2012). ROC convex hull and nonparametric maximum likelihood estimation, Machine Learning, 88, 433-444. https://doi.org/10.1007/s10994-012-5290-y
  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. https://doi.org/10.1016/S0047-259X(03)00045-9
  17. Renyi A and Sulanke R (1963). Uber die konvexe Hulle von n zufallig gewahlten Punkten, Zeitschrift Fur Wahrscheinlichkeitstheorie und Verwandte Gebiete, 2, 75-84. https://doi.org/10.1007/BF00535300
  18. Renyi A and Sulanke R (1964). Uber die konvexe Hulle von n zufallig gewahlten Punkten, Zeitschrift Fur Wahrscheinlichkeitstheorie Und Verwandte Gebiete, 3, 138-147.
  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. https://doi.org/10.5351/CSAM.2015.22.1.069
  21. Zhao J and Kim HM (2016). Power t distribution, Communications for Statistical Applications and Methods, 23, 321-334. https://doi.org/10.5351/CSAM.2016.23.4.321