Hostname: page-component-77c89778f8-gq7q9 Total loading time: 0 Render date: 2024-07-16T19:40:24.303Z Has data issue: false hasContentIssue false
  • English
  • Français

Moment inequalities of the Liapunov type

Part of: Distribution theory - Probability Distribution theory

Published online by Cambridge University Press:  09 April 2009

H. L. MacGillivray
H. L. MacGillivray
Affiliation:
Department of Mathematics, University of Queensland, St. Lucia, Queensland 4067, Australia
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.

Characterisations of the distribution of a non-negative random variable are sought for which the Liapunov moment inequality is extended to give inequalities between inverse powers of moment ratios, which are known as mean sizes in considerations of particle size distributions. A solution is found for continuous distributions, and the conditions applied to a number of well-known distributions. A further class of distributions is considered for which the new inequalities hold but the inequality direction is reversed for some orders of the moments. The study involves examination of the signs of the third central moments of a family of distributions, obtained by a log transformation, from the weighted, or moment, distributions induced by the non-negative random variable.

MSC classification

Secondary: 60E15: Inequalities; stochastic orderings 62E10: Characterization and structure theory
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 31 , Issue 2 , August 1981 , pp. 236 - 251
Copyright
Copyright © Australian Mathematical Society 1982

References

Allen, T. (1975), Particle size measurement 2nd ed. (Chapman and Hall, London).Google Scholar
Barlow, R. E., Marshall, A. W. and Proschan, F. (1963), ‘Properties of probability distributions with monotone hazard rate’, Ann. Math. Statist. 34, 375–389.CrossRefGoogle Scholar
Belz, M. H. (1947), ‘Note on the Liapounoff inequality for absolute moments’, Ann. Math. Statist. 18, 604–605.Google Scholar
Karlin, S., Proschan, F. and Barlow, R. E. (1961), ‘Moment inequalities of Pólya frequency functions’, Pacific J. Math. 11, 1023–1033.CrossRefGoogle Scholar
Karlin, S. (1968), Total positivity Vol. I (Stanford University Press, Stanford).Google Scholar
Lukacs, E. (1960), Characteristic functions (Charles Griffen and Co. Ltd., London).Google Scholar
Moran, P. A. P. (1969), An introduction to probability theory (Clarendon Press, Oxford).Google Scholar
Marsaglia, G., Marshall, A. W. and Proschan, F. (1965), ‘Moment crossings as related to density crossings’, J. Roy. Statist. Soc. Ser. B 27, 91–93.Google Scholar
Randolph, A. D. and Larson, M. A.(1971), Theory of particulate processes: analysis and techniques of continuous crystallization (Academic Press, New York).Google Scholar
White, E. T. (1971), Industrial crystallization (Department of Chemical Engineering, University of Queensland, Brisbane).Google Scholar