Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-22T18:45:27.820Z Has data issue: false hasContentIssue false
  • English
  • Français

DISPERSIVE ORDERING FOR THE MULTIVARIATE NORMAL DISTRIBUTION

Published online by Cambridge University Press:  05 January 2016

Xuan Leng ,
Jinsen Zhuang  and
Taizhong Hu
Xuan Leng
Affiliation:
Department of Statistics and Finance, School of Management, University of Science and Technology of China, Hefei, Anhui 230026, China E-mail: [email protected]
Jinsen Zhuang
Affiliation:
Department of Statistics and Finance, School of Management, University of Science and Technology of China, Hefei, Anhui 230026, China E-mail: [email protected]
Taizhong Hu
Affiliation:
Department of Statistics and Finance, School of Management, University of Science and Technology of China, Hefei, Anhui 230026, China E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Let (X1, …, Xn) be a multivariate normal random vector with any mean vector, variances equal to 1 and covariances equal and positive. Turner and Whitehead [9] established that the largest order statistic max{X1, …, Xn} is less than the standard normal random variable in the dispersive order. In this paper, we give a new and straightforward proof for this result. Several possible extensions of this result are also discussed.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 30 , Issue 2 , April 2016 , pp. 141 - 152
Copyright
Copyright © Cambridge University Press 2016 

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.)

References

1.Bickel & Lehmann (1976). Descriptive statistics for non-parametric models. III. Dispersion. Annals of Statistics 4: 1139–1158.Google Scholar
2.Jeon, J., Kochar, S., & Park, C.G. (2006). Dispersive ordering – some applications and examples. Statistical Papers 47: 227–247.CrossRefGoogle Scholar
3.Khaledi, B.-E. & Kochar, S. (2000). On dispersive ordering between order statistics in one-sample and two-sample problems. Statistics and Probability Letters 46: 257–261.Google Scholar
4.Lewis, T. & Thompson, J.W. (1981). Dispersive distributions and the connection between dispersivity and strong unimodality. Journal of Applied Probability 18: 76–90.CrossRefGoogle Scholar
5.Sampford, M.R. (1953). Some inequalities on Mill's ratio and related functions. Annals of Mathematical Statistics 24: 130–132.Google Scholar
6.Shaked, M. (1982). Dispersive ordering of distributions. Journal of Applied Probability 19: 310–320.Google Scholar
7.Shaked, M. & Shanthikumar, J.G. (2007). Stochastic Orders. New York: Springer.Google Scholar
8.Tong, Y.L. (1990). The Multivariate Normal Distribution. New York: Springer-Verlag.Google Scholar
9.Turner, A. & Whitehead, J. (2015). Partial stochastic dominance for the multivariate Gaussian distribution. Statistics and Probability Letters 103: 80–85.Google Scholar
10.Whitehead, J., Cleary, F., & Turner, A. (2015). Bayesian sample sizes for exploratory clinical trials comparing multiple experimental treatments with a control. Statistics in Medicine DOI: 10.1002/sim.6469.Google Scholar