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Characterization of the logistic and loglogistic distributions by extreme value related stability with random sample size

Published online by Cambridge University Press:  14 July 2016

W. J. Voorn
W. J. Voorn*
Affiliation:
Universiteit van Amsterdam
*
∗Postal address: Universiteit van Amsterdam, Vakgroep Medische Fysica, Meibergdreef 15, 1105 AZ Amsterdam, The Netherlands.
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Abstract

Maximum stability of a distribution with respect to a positive integer random variable N is defined by the property that the type of distribution is not changed when considering the maximum value of N independent observations. The logistic distribution is proved to be the only symmetric distribution which is maximum stable with respect to each member of a sequence of positive integer random variables assuming value 1 with probability tending to 1. If a distribution is maximum stable with respect to such a sequence and minimum stable with respect to another, then it must be logistic, loglogistic or ‘backward' loglogistic. The only possible sample size distributions in these cases are geometric.

Keywords

EXTENDED LOGISTIC DISTRIBUTIONEXTENDED LOGLOGISTIC DISTRIBUTIONGEOMETRIC DISTRIBUTIONEXTENDED GEOMETRIC DISTRIBUTIONSYMMETRIC DISTRIBUTIONSINCREASING FAILURE RATE
Type
Research Papers
Information
Journal of Applied Probability , Volume 24 , Issue 4 , December 1987 , pp. 838 - 851
Copyright
Copyright © Applied Probability Trust 1987 

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References

Baringhaus, L. (1980) Eine simultane Charakterisierung der geometrischen Verteilung und der logistischen Verteilung. Metrika 27, 243–253.CrossRefGoogle Scholar
Barndorff-Nielsen, O. (1964) On the limit distribution of the maximum of a random number of independent random variables. Acta Math. Acad. Sci. Hungar. 15, 399–403.CrossRefGoogle Scholar
Berkson, J. (1951) Why I prefer logits to probits. Biometrics 7, 327–339.CrossRefGoogle Scholar
Berman, S. M. (1964) Limit theorems for the maximum term in stationary sequences. Ann. Math. Statist. 35, 502–516.Google Scholar
Feller, W. (1940) On the logistic law of growth and its empirical verification in biology. Acta Biotheoretica 5, 51–66.Google Scholar
Feller, W. (1971) An Introduction to Probability Theory and its Applications. Vol. 2, 2nd edn. Wiley, New York.Google Scholar
Galambos, J. (1978) The Asymptotic Theory of Extreme Order Statistics. Wiley, New York.Google Scholar
Gnedenko, B. V. (1982a) On some stability theorems. In Stability Problems for Stochastic Models. Proc. 6th Seminar, Moscow, ed. Kalashnikov, V. V. and Zolotarev, V. M., Lecture Notes in Mathematics 982, Springer-Verlag, Berlin, 24–31.Google Scholar
Gnedenko, B. V. (1982b) On limit theorems for a random number of random variables. In: Probability Theory and Mathematical Statistics. Proc. 4th USSR–Japan Symp., Tbilisi, USSR, ed. Itô, K. and Prokhorov, J. V., Lecture Notes in Mathematics, Springer-Verlag, Berlin, 167–176.Google Scholar
Kakosyan, A. V., Klebanov, L. B. and Melamed, J. A. (1984) Characterization of Distributions by the Method of Monotone Operators. Lecture Notes in Mathematics 1088, Springer-Verlag, Berlin.Google Scholar
Mogyoródi, J. (1967) On the limit distribution of the largest term in the order statistics of a sample of random size, Magyar Tud. Akad. Mat. Fiz. Oszt. Kozl. 17, 75–83 (in Hungarian).Google Scholar
Shah, B. K. and Dave, P. H. (1964) A note on log-logistic distribution. J. M. S. Univ. Baroda 12, 15–20.Google Scholar