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A crossing theorem for distribution functions and their moments

Published online by Cambridge University Press:  17 April 2009

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

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It is proved here that for two distribution functions with equal moments up to order n, the number of crossings is at least n, and if exactly n, the remaining odd or even moments (for n even or odd respectively) do not cross again. This both generalises and extends a number of previous results.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 31 , Issue 3 , June 1985 , pp. 413 - 419
Copyright
Copyright © Australian Mathematical Society 1985

References

[1]Ali, M.M., “Stochastic orderings and kurtosis measure”, J. Amer. Statist. Assoc. 69 (1974), 543–545.CrossRefGoogle Scholar
[2]Dyson, F.J., “A note on kurtosis”, J. Roy. Statist. Soc. Ser. B 5 (1943), 360–361.CrossRefGoogle Scholar
[3]Karlin, S., Proschan, F. and Barlow, R.E., “Moment inequalities of Pólya frequency functions”, Pacific J. Math. 11 (1961), 1023–1033.CrossRefGoogle Scholar
[4]Karlin, S., Total positivity, Vol. I (Standford University Press, Stanford, 1968).Google Scholar
[5]MacGillivray, H.L., “The mean, median, mode inequality and skewness for a class of densities”, Austral. J. Statist. 23 (1981), 247–250.CrossRefGoogle Scholar
[6]Marsaglia, G., Marshall, A.W. and Proschan, F., “Moment crossings as related to density crossings”, J. Roy. Statist. Soc. Ser. B 27 (1965), 91–93.Google Scholar
[7]Oja, H., “On location, scale, kewness and kurtosis of univariate distributions”, Scand. J. Statist. 8 (1981), 154–168.Google Scholar
[8]Runnenburg, J.Th., “Mean, median, mode”, Statist. Neerlandica 32 (1978), 73–79.CrossRefGoogle Scholar
[9]Stoyan, D., Comparison methods for queues and other stochastic models (John Wiley & Sons, London, New York; Akademie-Verlag, Berlin, 1983).Google Scholar
[10]van Zwet, W.R., Convex transformations of random variables (Mathematische Centrum, Amsterdam, 1964).Google Scholar