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The Integrated Cauchy Functional Equation: Some Comments on Recent Papers

Part of: Distribution theory

Published online by Cambridge University Press:  01 July 2016

C. R. Rao ,
T. Sapatinas  and
D. N. Shanbhag
C. R. Rao*
Affiliation:
Pennsylvania State University
T. Sapatinas
Affiliation:
University of Sheffield
D. N. Shanbhag*
Affiliation:
University of Sheffield
*
* Postal address: Department of Statistics, Pennsylvania State University, University Park, PA 16802, USA.
*** Postal address: School of Mathematics and Statistics, The University, Sheffield S10 2TN, UK.
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Abstract

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We make some comments on recent papers involving the integrated Cauchy functional equation or specialized versions of it, and reveal in particular that these papers give an inaccurate picture of the current state of the literature on the topic.

Keywords

LAU–RAO THEOREMDENY'S THEOREMEXPOTENTIAL AND GEOMETRIC DISTRIBUTIONSPOISSON PROCESS

MSC classification

Primary: 62E10: Characterization and structure theory

Information

Type
Letters to the Editor
Information
Advances in Applied Probability , Volume 26 , Issue 3 , September 1994 , pp. 825 - 829
Copyright
Copyright © Applied Probability Trust 1994 

Footnotes

**

Present address: Department of Mathematical Statistics and Operational Research, Laver Building, Exeter University, Exeter, EX44QE, UK.

Research supported by US Army Research Grant DAA H04-93-G-0030.

References

Alzaid, A. A., Rao, C. R. and Shanbhag, D. N. (1987) Solution of the integrated Cauchy functional equation on the half line using exchangeability. Sankhya A 49, 189–194.Google Scholar
Chukova, S. and Dimitrov, B. (1992) On distributions having the almost-lack-of-memory property. J. Appl. Prob. 29, 691–698.Google Scholar
Davies, P. L. (1993) Review of Ramachandran and Lau (1991). Math. Rev. (93c:60016), 1518–1519.Google Scholar
Davies, P. L. and Shanbhag, D. N. (1987) A generalization of a theorem of Deny with applications in characterization theory. Quart. J. Math. Oxford (2) 38, 13–34.CrossRefGoogle Scholar
Dimitrov, B. and Khalil, Z. (1990) On a new characterization of the exponential distribution related to a queueing system with an unreliable server. J. Appl. Prob. 27, 221–226.Google Scholar
Fosam, E. B., Rao, C. R. and Shanbhag, D. N. (1993) Comments on some results involving the integrated Cauchy functional equation. Statist. Prob. Lett. 17, 299–302.CrossRefGoogle Scholar
Fosam, E. B. and Shanbhag, O. N. (1994) Certain characterizations of exponential and geometric distributions. J. R. Statist. Soc. B 56, 157–160.Google Scholar
Gupta, P. L. and Gupta, R. C. (1986) A characterization of the Poisson process. J. Appl. Prob. 23, 233–235.CrossRefGoogle Scholar
Van Harn, K. and Steutel, F. W. (1991) On a characterization of the exponential distribution. J. Appl. Prob. 28, 947–949.Google Scholar
Huang, W. J., Li, S. H. and Su, J. C. (1993) Some characterizations of the Poisson process and geometric renewal process. J. Appl. Prob. 30, 121–130.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
Kakosyan, A. V., Klebanov, L. B. and Melamed, J. A. (1984) Characterization of Distributions by Method of Intensively Monotonic Operators. Lecture Notes in Mathematics 1088, Springer-Verlag, New York.Google Scholar
Lau, K. S. and Prakasa Rao, B. L. S. (1990) Characterization of the exponential distribution by the relevation transform. J. Appl. Prob. 27, 726–729.CrossRefGoogle Scholar
Lau, K. S. and Prakasa Rao, B. L. S. (1992) Letter to the editor. J. Appl. Prob. 29, 1003–1004.Google Scholar
Lau, K. S. and Rao, C. R. (1982) Integrated Cauchy functional equation and characterizations of the exponential law. Sankhya A 44, 72–90.Google Scholar
Liang, T. C. and Balakrishnan, N. (1992) A characterization of the exponential distribution through conditional independence. J. R. Statist. Soc. B 54, 269–271.Google Scholar
Marsaglia, G. and Tubilla, A. (1975) A note on the lack of memory property of the exponential distribution. Ann. Prob. 3, 352–354.CrossRefGoogle Scholar
Pakes, A. G. and Khattree, R. (1992) Length-biasing, characterizations of laws and the moment problem. Austral. J. Statist. 34, 307–322.CrossRefGoogle Scholar
Puri, P. S. and Rubin, H. (1974) On a characterization of the family of distributions with constant multivariate failure rates. Ann. Prob. 2, 738–740.CrossRefGoogle Scholar
Rao, C. R. (1983) An extension of Deny's theorem and its applications to characterization of probability distributions. In A Festchrift for Eric Lehmann, ed. Bickel, P. J. et al., pp. 348–366, Wadsworth, Monterey, CA.Google Scholar
Rao, C. R. and Shanbhag, D. N. (1986) Recent results on characterization of probability distributions: a unified approach through extensions of Deny's theorem. Adv. Appl. Prob. 18, 660–678.Google Scholar
Rao, C. R. and Shanbhag, D. N. (1989) Recent advances on the integrated Cauchy functional equation and related results in applied probability. In Papers in Probability, Statistics and Mathematics in Honor of S. Karlin, ed. Anderson, T. W. et al., pp. 239–253, Academic Press, New York.CrossRefGoogle Scholar
Sapatinas, T. (1993) Characterization and Identifiability Results in some Stochastic Models. Unpublished Ph.D. Thesis, Sheffield University.Google Scholar
Shanbhag, D. N. (1977) An extension of the Rao-Rubin characterization of the Poisson distribution. J. Appl. Prob. 14, 640–646.Google Scholar
Shanbhag, D. N. (1991) Review of Dimitrov and Khalil (1990) Math. Rev. (91j:60147), 5590.Google Scholar
Shimizu, R. (1978) Solution to a functional equation and its applications 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. 31, 309–313.Google Scholar
Witte, H.-J. (1993) Some characterizations of exponential or geometric distributions in a nonstationary record value model. J. Appl. Prob. 30, 373–381.CrossRefGoogle Scholar