Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-13T00:41:34.895Z Has data issue: false hasContentIssue false
  • English
  • Français

A Note on Conditioning and Stochastic Domination for Order Statistics

Part of: Distribution theory - Probability Nonparametric inference

Published online by Cambridge University Press:  14 July 2016

Devdatt Dubhashi  and
Olle Häggström
Devdatt Dubhashi*
Affiliation:
Chalmers University of Technology
Olle Häggström*
Affiliation:
Chalmers University of Technology
*
∗Postal address: Department of Computing Science, Chalmers University of Technology, S-412 96 Göteborg, Sweden.
∗∗Postal address: Department of Mathematicical Statistics, Chalmers University of Technology, S-412 96 Göteborg, Sweden. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

For an order statistic (X1:n,…,Xn:n) of a collection of independent but not necessarily identically distributed random variables, and any i ∈ {1,…,n}, the conditional distribution of (Xi+1:n,…,Xn:n) given Xi:n > s is shown to be stochastically increasing in s. This answers a question by Hu and Xie (2006).

Keywords

Order statisticscouplingstochastic domination

MSC classification

Secondary: 62G30: Order statistics; empirical distribution functions 60E15: Inequalities; stochastic orderings
Type
Research Article
Information
Journal of Applied Probability , Volume 45 , Issue 2 , June 2008 , pp. 575 - 579
Copyright
Copyright © Applied Probability Trust 2008 

Footnotes

Supported by the Swedish Research Council and by the G-ran Gustafsson Foundation for Research in the Natural Sciences and Medicine.

References

[1] Boland, P. J., Hollander, M., Joag-Dev, K. and Kochar, S. (1996). Bivariate dependence properties of order statistics. J. Multivariate Anal. 56, 75–89.Google Scholar
[2] Efron, B. (1965). Increasing properties of Pólya frequency functions. Ann. Math. Statist. 36, 272–279.Google Scholar
[3] Häggström, O. (2007). Problem solving is often a matter of cooking up an appropriate Markov chain. Scand. J. Statist. 34, 768–780.Google Scholar
[4] Harris, T. E. (1960). Lower bound for the critical probability in a certain percolation process. Proc. Camb. Philos. Soc. 56, 13–20.Google Scholar
[5] Hu, T. and Xie, C. (2006). Negative dependence in the balls and bins experiment with applications to order statistics. J. Multivariate Anal. 97, 1342–1354.Google Scholar
[6] Lindvall, T. (1992). Lectures on the Coupling Method. John Wiley, New York.Google Scholar
[7] Newton, I. (1707). Arithmetica Universalis: Sive de Compositione et Resolutione Arithmetica Liber.Google Scholar
[8] Strassen, V. (1965). The existence of probability measures with given marginals. Ann. Math. Statist. 36, 423–439.Google Scholar