Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-27T11:01:08.367Z Has data issue: false hasContentIssue false
  • English
  • Français

A Stochastic Comparison for Arrangement Increasing Functions

Published online by Cambridge University Press:  12 September 2008

Abba M. Krieger  and
Paul R. Rosenbaum
Abba M. Krieger
Affiliation:
Department of Statistics, University of Pennsylvania, Philadelphia PA 19104–6302, USA
Paul R. Rosenbaum
Affiliation:
Department of Statistics, University of Pennsylvania, Philadelphia PA 19104–6302, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

Let h(·) be an arrangement increasing function, let X have an arrangement increasing density, and let XE be a random permutation of the coordinates of X. We prove E{h(XE)} ≤ E{h(X)}. This comparison is delicate in that similar results are sometimes true and sometimes false. In a finite distributive lattice, a similar comparison follows from Holley's inequality, but the set of permutations with the arrangement order is not a lattice. On the other hand, the set of permutations is a lattice, though not a distributive lattice, if it is endowed with a different partial order, but in this case the comparison does not hold.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 3 , Issue 3 , September 1994 , pp. 345 - 348
Copyright
Copyright © Cambridge University Press 1994

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]Berge, C. (1971) Principles of Combinatorics, Academic Press.Google Scholar
[2]Bollobás, B. (1986) Combinatorics, Cambridge University Press.Google Scholar
[3]D'Abadie, C. and Proschan, F. (1984) Stochastic versions of rearrangement inequalities. In: Tong, Y. (ed.) Inequalities in Statistics and Probability, Hayward, CA: IMS4–12.Google Scholar
[4]Eaton, M. (1982). A review of selected topics in probability inequalities. Annals of Statistics 10 11–43.CrossRefGoogle Scholar
[5]Hajek, J. (1970) Miscellaneous problems of rank test theory. In: Puri, M. L. (ed.) Nonparametric Techniques in Statistical Inference, Cambridge University Press, 3–20.Google Scholar
[6]Hollander, M., Proschan, F. and Sethuraman, J. (1977). Functions decreasing in transposition and their applications in ranking problems. Annals of Statistics 5 722–733.CrossRefGoogle Scholar
[7]Holley, R. (1974) Remarks on the FKG inequalities. Communications in Mathematical Physics 36 227–231.CrossRefGoogle Scholar
[8]Marshall, A. and Olkin, I. (1979) Inequalities: Theory of Majorization and Its Applications, Academic Press.Google Scholar
[9]Rosenbaum, P. (1989) On permutation tests for hidden biases in observational studies: An application of Holley's inequality to the Savage lattice. Annals of Statistics 17 643–653.CrossRefGoogle Scholar
[10]Savage, I. R. (1957) Contributions to the theory of rank order statistics: The trend case. Annals of Mathematical Statistics 28 968–977.CrossRefGoogle Scholar
[11]Savage, I. R. (1964) Contributions to the theory of rank order statistics: Applications of lattice theory. Review of the International Statistical Institute 32 52–63.CrossRefGoogle Scholar