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Characterizations of the geometric distribution by distributional properties

Published online by Cambridge University Press:  14 July 2016

Mynt Zijlstra
Mynt Zijlstra*
Affiliation:
Nederlandse Philips Bedrijven BV
*
∗ Postal address: Nederlandse Philips Bedrijven BV, ISA-CQM, Building VN-706, P.O. Box 218, 5600 MD Eindhoven, The Netherlands.
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Abstract

Some new characterizations of the geometric distribution are studied. A generalization of the characterization by the well-known ‘lack-of-memory' property is given together with some closely related characterizations. Furthermore the modified geometric distribution is characterized by a distributional property of the difference of two successive order statistics. The latter result extends work of Puri and Rubin (1970). Finally the geometric distribution is characterized by a conditional distribution property of the difference of two arbitrary order statistics, which generalizes a result by Arnold (1980). Some of the results given answer open questions put in earlier papers.

Keywords

LACK OF MEMORYORDER STATISTICS

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 20 , Issue 4 , December 1983 , pp. 843 - 850
Copyright
Copyright © Applied Probability Trust 1983 

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References

Arnold, B. C. (1980) Two characterizations of the geometric distribution. J. Appl. Prob. 17, 570–573.CrossRefGoogle Scholar
Arnold, B. C. and Ghosh, M. (1976) A characterization of geometric distributions by distributional properties of order statistics. Scand. Actuarial J., 232–234.CrossRefGoogle Scholar
El-Neweihi, E. and Govindarajulu, Z. (1979) Characterizations of geometric distribution and discrete IFR(DFR) distributions using order statistics. J. Statist. Planning Inf. 3, 85–90.Google Scholar
Engel, J. and Zijlstra, M. (1980) A characterization of the gamma distribution by the negative binomial distribution. J. Appl. Prob. 17, 1138–1144.Google Scholar
Galambos, J. (1975) Characterizations of probability distribution by properties of order statistics II. In Statistical Distributions in Scientific Work, ed. Patil, G. P. et al., Reidel, Dordrecht.Google Scholar
Galambos, J. and Kotz, S. (1978) Characterizations of Probability Distributions. Lecture Notes in Mathematics 675, Springer-Verlag, Berlin.CrossRefGoogle Scholar
Govindarajulu, Z. (1980) Characterization of the geometric distribution using properties of order statistics. J. Statist. Planning Inf. 4, 237–247.CrossRefGoogle Scholar
Johnson, N. L. and Kotz, S. (1969) Distributions in Statistics, Vol. 1. Wiley, New York.Google Scholar
Krishnaji, N. (1971) Note on a characterizing property of the exponential distribution. Ann. Math. Statist. 42, 361–362.Google Scholar
Puri, P. S. and Rubin, H. (1970) A characterization based on the absolute difference of two i.i.d. random variables. Ann. Math. Statist. 41, 2113–2122.Google Scholar
Rossberg, H. J. (1972) Characterization of the exponential and the Pareto distribution by means of some properties of the distributions which the differences and quotients of order statistics are subject to. Math. Operationsforsch. Statist. 3, 207–216.Google Scholar
Shanbhag, D. N. (1970) The characterizations for exponential and geometric distributions. J. Amer. Statist. Assoc. 65, 1256–1259.CrossRefGoogle Scholar
Shanbhag, D. N. (1977) An extension of the Rao–Rubin characterization of the Poisson distribution. J. Appl. Prob. 14, 640–646.Google Scholar
Shimizu, R. (1978) Solution to a functional equation and its application to some characterization problems. Sankhya A 40, 319–332.Google Scholar
Shimizu, R. (1979) On a lack of memory property of the exponential distribution. Ann. Inst. Statist. Math. A 31, 309–313.Google Scholar
Srivastava, R. C. (1979) Two characterizations of the geometric distribution by record values. Sankhya B 40, 276–278.Google Scholar