Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-27T12:28:40.089Z Has data issue: false hasContentIssue false
  • English
  • Français

Recent results on characterization of probability distributions: a unified approach through extensions of Deny&s theorem

Published online by Cambridge University Press:  01 July 2016

C. Radhakrishna Rao  and
D. N. Shanbhag
C. Radhakrishna Rao*
Affiliation:
University of Pittsburgh
D. N. Shanbhag*
Affiliation:
University of Sheffield
*
∗Postal address: Center for Multivariate Analysis, Fifth Floor, Thackeray Hall, University of Pittsburgh, Pittsburgh, PA 15260, USA.
∗∗Postal address: Department of Probability and Statistics, University of Sheffield, Sheffield S3 7RH, UK.
Get access Rights & Permissions [Opens in a new window]

Abstract

The problem of identifying solutions of general convolution equations relative to a group has been studied in two classical papers by Choquet and Deny (1960) and Deny (1961). Recently, Lau and Rao (1982) have considered the analogous problem relative to a certain semigroup of the real line, which extends the results of Marsaglia and Tubilla (1975) and a lemma of Shanbhag (1977). The extended versions of Deny&s theorem contained in the papers by Lau and Rao, and Shanbhag (which we refer to as LRS theorems) yield as special cases improved versions of several characterizations of exponential, Weibull, stable, Pareto, geometric, Poisson and negative binomial distributions obtained by various authors during the last few years. In this paper we review some of the recent contributions to characterization of probability distributions (whose authors do not seem to be aware of LRS theorems or special cases existing earlier) and show how improved versions of these results follow as immediate corollaries to LRS theorems. We also give a short proof of Lau–Rao theorem based on Deny&s theorem and thus establish a direct link between the results of Deny (1961) and those of Lau and Rao (1982). A variant of Lau–Rao theorem is proved and applied to some characterization problems.

Keywords

EXCHANGEABLE RANDOM VARIABLESEXPONENTIAL, GEOMETRIC, PARETO AND STABLE DISTRIBUTIONSINTEGRATED CAUCHY FUNCTIONAL EQUATIONSLAU–RAO THEOREMSHANBHAG&S LEMMA
Type
Research Article
Information
Advances in Applied Probability , Volume 18 , Issue 3 , September 1986 , pp. 660 - 678
Copyright
Copyright © Applied Probability Trust 1986 

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.)

Footnotes

This work was supported by the Air Force Office of Scientific Research under Contract F49620-82-K-0001. Reproduction in whole or in part is permitted for any purpose of the United States Government.

References

Alzaid, A. A. (1983) Some Contributions to Characterization Theory. Ph.D. thesis, University of Sheffield.Google Scholar
Alzaid, A. A., Rao, C. R. and Shanbhag, D. N. (1983) Solutions of certain functional equations and related results on probability distributions. (Submitted for publication.) Google Scholar
Bellman, R. and Cooke, K. L. (1963) Differential-Difference Equations. Academic Press, New York.Google Scholar
Choquet, G. and Deny, J. (1960) Sur l’équation de convolution µ = µ ∗ s. C. R. Acad. Sci. Paris 250, 799–801.Google Scholar
Dallas, A. C. (1981) Record values and exponential distribution. J. Appl. Prob. 18, 949–951.CrossRefGoogle Scholar
Davies, P. L. and Shanbhag, D. N. (1984) A generalization of a theorem of Deny with applications in characterization theory. (Submitted for publication.) Google Scholar
Deheuvels, P. (1984) The characterizations of distributions by order statistics and record values–a unified approach. J. Appl. Prob. 21, 326–334.CrossRefGoogle Scholar
Deny, J. (1961) Sur l’équation de convolution µ∗ s. Semin. Theor. Potent. M. Brelot, Fac. Sci. Paris , 1959–60, 4 e ann.Google Scholar
Feller, W. (1966) An Introduction to Probability Theory and its Applications , Vol. 2. Wiley, New York.Google Scholar
Grosswald, E., Kotz, S. and Johnson, N. L. (1980) Characterizations of the exponential distributions by relevation-type equations. J. Appl. Prob. 17, 874–877.CrossRefGoogle Scholar
Gupta, R. C. (1984) Relationships between order statistics and record values and some characterization results. J. Appl. Prob. 21, 433–438.CrossRefGoogle Scholar
Isham, V., Shanbhag, D. N. and Westcott, M. (1975). A characterization of the Poisson process using forward recurrence times. Math. Proc. Camb. Phil. Soc. 78, 513–515.CrossRefGoogle Scholar
Klebanov, L. B. (1980) Several results connected with characterization of the exponential distribution. Teor. Verojatnost i Primenen. 25, 628–633.Google Scholar
Kotz, S. and Shanbhag, D. N. (1980) Some new approaches to probability distributions. Adv. Appl. Prob. 12, 903–921.CrossRefGoogle Scholar
Lau, Ka-Sing and Rao, , Radhakrishna, C. (1982) Integrated Cauchy functional equation and characterizations of the exponential law. Sankhya A 44, 72–90.Google Scholar
Lau, Ka-Sing and Rao, , Radhakrishna, C. (1984a) Integrated Cauchy functional equation on the whole line. Sankhya A 46, 311–319.Google Scholar
Lau, Ka-Sing and Rao, , Radhakrishna, C. (1984b) On the integral equation Technical Report #84-09, University of Pittsburgh.Google Scholar
Marsaglia, G. and Tubilla, A. (1975) A note on the lack of memory property of the exponential distributions. Ann. Prob. 3, 352–354.CrossRefGoogle Scholar
Rao, , Radhakrishna, C. (1983) An extension of Deny&s theorem and its application to characterization of probability distributions. In A Festschrift for Eric Lehmann (Eds. Bickel, P. J. et al.), Wadsworth, Monterey, CA, 348–365.Google Scholar
Rao, , Radhakrishna, C. and Rubin, H. (1964) On a characterization of the Poisson distribution. Sankhya A 26, 295–298.Google Scholar
Rao, M. B. and Shanbhag, D. N. (1982) Damage models. In Encyclopedia of Statistical Sciences , Vol. 2, Wiley, New York, 262–265.Google Scholar
Ramachandran, B. (1979) On the strong memoryless property of the exponential and geometric laws. Sankhya A 41, 244–251.Google Scholar
Ramachandran, B. (1982) On the equation Sankhya A 44, 364–371.Google Scholar
Ressel, P. (1984) De Finetti-type theorems: An analytical approach. Technical Report.CrossRefGoogle Scholar
Richards, D. St. P. (1981) Exponential distributions on partially ordered semiabelian groups. Technical Report, University of North Carolina.Google Scholar
Rossberg, J. J. (1972) Characterization of the exponential and the Pareto distributions 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.CrossRefGoogle Scholar
Shanbhag, D. N. (1977) An extension of the Rao-Rubin characterization. J. Appl. Prob. 14, 640–646.CrossRefGoogle Scholar
Shanbhag, D. N. (1983) Review No. 62038 on C. R. Rao and Srivastava. Math. Rev. Google Scholar
Shimuzu, R. (1978) Solution to a functional equation and its application to some characterization problems. Sankhya A 40, 319–332.Google Scholar
Srivastava, R. C. and Singh, J. (1975) On characterizations of the binomial and Poisson distributions based on a damage model. In Statistical Distributions in Scientific Work. 3, Ed. Patil, G. P. et al. Reidel, Dordrecht, 271–278.Google Scholar
Titchmarsh, E. C. (1949) The Theory of Functions. Clarendon Press, Oxford.Google Scholar
Westcott, M. (1981) Letter to the editor. J. Appl. Prob. 18, 568.CrossRefGoogle Scholar
Wolinska-Welcz, A. and Szynal, D. (1984) On a solution of Dugué&s problem for a class of couples of lattice distributions. Unpublished.Google Scholar