Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-02T21:34:43.006Z Has data issue: false hasContentIssue false
  • English
  • Français

Permutation probabilities for gamma random variables

Published online by Cambridge University Press:  14 July 2016

Robert J. Henery
Robert J. Henery*
Affiliation:
University of Strathclyde
*
∗ Postal address: Department of Mathematics, University of Strathclyde, Livingstone Tower, 26 Richmond St, Glasgow G1 1XH, U.K.
Get access Rights & Permissions [Opens in a new window]

Abstract

The order statistics of a set of independent gamma variables, in general not identically distributed, may serve as a basis for ordering players in a hypothetical game. An alternative formulation in terms of negative binomial variables leads to an expression for the probability that the random gammas are in a given order. This expression may contain rather many terms and some approximations are discussed — firstly as the gamma parameters αi tend to equality with all ni the same, and secondly when the probability of an inversion is small. In another interpretation the probabilities discussed arise in the statement of confidence limits for the ratios of population variances, and here the inversion probability is small enough usually that lower and upper bounds may be given for the probability that the sample variances occur in their expected order. These bounds are calculated from the probability that two variables are in expected order, and for gamma variables this probability is obtained from the F-distribution.

Keywords

GAMMA DISTRIBUTIONNEGATIVE BINOMIALAPPROXIMATIONBOUNDSTREND TESTS
Type
Research Papers
Information
Journal of Applied Probability , Volume 20 , Issue 4 , December 1983 , pp. 822 - 834
Copyright
Copyright © Applied Probability Trust 1983 

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

Bechhofer, R. E. and Sobel, M. (1954) A single-sample multiple decision procedure for ranking variances of normal populations. Ann. Math. Statist. 25, 273–289.CrossRefGoogle Scholar
Engel, J. and Zijlstra, M. (1980) A characterisation of the gamma distribution by the negative binomial distribution. J. Appl. Prob. 17, 1138–1144.Google Scholar
Gibbons, J. D., Olkin, I. and Sobel, M. (1977) Selecting and Ordering Populations: A New Statistical Methodology. Wiley, New York.Google Scholar
Henery, R. J. (1981a) Permutation probabilities as models for horse races. J. R. Statist. Soc. B 43, 86–91.Google Scholar
Henery, R. J. (1981b) Place probabilities in normal order statistics models for horse races. J. Appl. Prob. 18, 839–852.Google Scholar
Johnson, N. L. and Kotz, S. (1969) Distributions in Statistics. Wiley, New York.Google Scholar
Marshall, A. W. and Olkin, I. (1979) Inequalities: Theory of Majorization and its Applications. Academic Press, New York.Google Scholar
Plackett, R. L. (1954) A reduction formula for normal multivariate integrals. Biometrika 41, 351–360.Google Scholar
Savage, I. R. (1957) Contributions to the theory of rank order statistics — the ‘trend’ case. Ann. Math. Statist. 28, 968–977.Google Scholar
Schafer, R. E. and Rutemiller, H. C. (1975) Some characteristics of a ranking procedure for population parameters based on chi-square statistics. Technometrics 17, 327–331.CrossRefGoogle Scholar