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The Scientific World Journal
Volume 2013 (2013), Article ID 468418, 7 pages
http://dx.doi.org/10.1155/2013/468418
Research Article

A Characterization of the Compound Multiparameter Hermite Gamma Distribution via Gauss’s Principle

Wolters Kluwer Financial Services, Seefeldstrasse 69, 8008 Zürich, Switzerland

Received 13 August 2013; Accepted 22 September 2013

Academic Editors: N. Marwan, J. Pacheco, and S. Umarov

Copyright © 2013 Werner Hürlimann. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. C. F. Gauss, Theoria Motus Corporum Coelestium in Sectionibus Conicis Solem Ambientum, Perthes and Besser, Hamburg, Germany, 1809, English Translation by C. H. Davis, Theory of the Motions of the Heavenly Bodies Moving about the Sun in Conic Sections, Dover, New York, NY, USA, 1857, Reprint 1963.
  2. S. M. Stigler, “Gauss and the invention of least squares,” The Annals of Statistics, vol. 9, no. 3, pp. 465–474, 1981. View at Publisher · View at Google Scholar
  3. G. W. Stewart, Theory of the Combination of Observations Least Subject to Errors: Part One, Part Two, Supplement, vol. 11 of Classics in Applied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia, Pa, USA, 1995, English Translation of C. F. Gauss, Theoria Combinationis Observationum Erroribus Minimis Obnoxiae, 1821–1826.
  4. H. Poincaré, Calcul des Probabilities, Gauthier-Villars, Paris, France, 2nd edition, 1912.
  5. H. Teicher, “Maximum likelihood characterizations of distributions,” The Annals of Mathematical Statistics, vol. 32, no. 4, pp. 1214–1222, 1961. View at Publisher · View at Google Scholar
  6. T. S. Ferguson, “Location and scale parameters in exponential families of distributions,” The Annals of Mathematical Statistics, vol. 33, no. 3, pp. 986–1001, 1962. View at Publisher · View at Google Scholar
  7. T. S. Ferguson, “Correction to “location and scale parameters in exponential families of distributions”,” The Annals of Mathematical Statistics, vol. 34, no. 4, p. 1603, 1963. View at Publisher · View at Google Scholar
  8. A. W. Marshall and I. Olkin, “Maximum likelihood characterizations of distributions,” Statistica Sinica, vol. 3, pp. 157–171, 1993. View at Google Scholar
  9. L. Bondesson, “A generalization of Poincaré's characterization of exponential families,” Journal of Statistical Planning and Inference, vol. 63, no. 2, pp. 147–155, 1997. View at Google Scholar · View at Scopus
  10. A. Azzalini and M. G. Genton, “On Gauss's characterization of the normal distribution,” Bernoulli, vol. 13, no. 1, pp. 169–174, 2007. View at Publisher · View at Google Scholar · View at Scopus
  11. L. L. Campbell, “Equivalence of Gauss's principle and minimum discrimination information estimation of probabilities,” The Annals of Mathematical Statistics, vol. 41, no. 3, pp. 1011–1015, 1970. View at Publisher · View at Google Scholar
  12. P. Puig, “A note on the harmonic law: a two-parameter family of distributions for ratios,” Statistics and Probability Letters, vol. 78, no. 3, pp. 320–326, 2008. View at Publisher · View at Google Scholar · View at Scopus
  13. M. Duerinckx, C. Ley, and Y. Swan, “Maximum likelihood characterization of distributions,” preprint, http://arxiv-web3.library.cornell.edu/abs/1207.2805.
  14. S.-I. Amari, Differential-Geometrical Methods in Statistics, vol. 28 of Lecture Notes in Statistics, Springer, Heidelberg, Germany, 1985.
  15. D. A. Sprott, “Estimating the parameters of a convolution by maximum likelihood estimation,” Journal of the American Statistical Association, vol. 78, no. 382, pp. 457–460, 1983. View at Publisher · View at Google Scholar
  16. W. Hürlimann, “On parameter orthogonality to the mean,” Statistical Papers, vol. 33, no. 1, pp. 69–74, 1992. View at Publisher · View at Google Scholar
  17. S. Hudson, “New results for a class of univariate distributions,” Communications in Statistics, vol. 31, no. 2, pp. 239–247, 2002. View at Google Scholar · View at Scopus
  18. W. Hürlimann, “On mean scaled insurance risk models,” Blätter der DGVFM, vol. 23, no. 3, pp. 251–265, 1998. View at Publisher · View at Google Scholar
  19. W. Hürlimann, “Measuring operational risk using a mean scaled individual risk model,” Applied Mathematics and Computation, vol. 152, no. 2, pp. 425–447, 2004. View at Publisher · View at Google Scholar · View at Scopus
  20. P. Puig and J. Valero, “Characterization of count data distributions involving additivity and binomial subsampling,” Bernoulli, vol. 13, no. 2, pp. 544–555, 2007. View at Publisher · View at Google Scholar · View at Scopus
  21. G. E. Willmot and D. A. Sprott, “A note on parameters orthogonal to the mean,” Biometrika, vol. 81, no. 2, pp. 409–412, 1994. View at Publisher · View at Google Scholar · View at Scopus
  22. W. Hürlimann, “On maximum likelihood estimation for count data models,” Insurance Mathematics and Economics, vol. 9, pp. 39–49, 1990. View at Google Scholar · View at Scopus
  23. W. Hürlimann, “Correction note: on maximum likelihood estimation for count data models,” Insurance Mathematics and Economics, vol. 10, no. 1, p. 81, 1991. View at Google Scholar · View at Scopus
  24. W. Hürlimann, “Pseudo compound Poisson distributions in risk theory,” ASTIN Bulletin, vol. 20, pp. 57–79, 1990. View at Google Scholar
  25. P. Puig, “Characterizing additively closed discrete models by a property of their maximum likelihood estimators, with an application to generalized Hermite distributions,” Journal of the American Statistical Association, vol. 98, no. 463, pp. 687–692, 2003. View at Publisher · View at Google Scholar · View at Scopus
  26. P. Puig and J. Valero, “Count data distributions: some characterizations with applications,” Journal of the American Statistical Association, vol. 101, no. 473, pp. 332–340, 2006. View at Publisher · View at Google Scholar · View at Scopus
  27. C. D. Kemp and A. W. Kemp, “Some properties of the Hermite distribution,” Biometrika, vol. 52, no. 3, pp. 381–394, 1965. View at Google Scholar · View at Scopus
  28. R. K. Milne and M. Westcott, “Generalized multivariate Hermite distributions and related point processes,” Annals of the Institute of Statistical Mathematics, vol. 45, no. 2, pp. 367–381, 1993. View at Publisher · View at Google Scholar · View at Scopus
  29. A. W. Kemp and C. D. Kemp, “An alternative derivation of the Hermite distribution,” Biometrika, vol. 53, pp. 627–628, 1966. View at Google Scholar
  30. R. P. Gupta and G. C. Jain, “A generalized Hermite distribution and its properties,” SIAM Journal on Applied Mathematics, vol. 27, no. 2, pp. 359–363, 1974. View at Google Scholar · View at Scopus
  31. C. S. Kumar, “Extended generalized hypergeometric probability distributions,” Statistics and Probability Letters, vol. 59, no. 2, pp. 1–7, 2002. View at Google Scholar · View at Scopus
  32. P. Lévy, “Sur les exponentielles de polynômes et sur l'arithmétique des produits de lois de Poisson,” Annales Scientifiques de l'Ecole Normale Supérieure, vol. 54, pp. 231–292, 1937. View at Google Scholar
  33. E. Lukacs, Characteristic Functions, Griffin, London, UK, 2nd edition, 1970.
  34. L. N. Johnson, S. Kotz, and A. W. Kemp, Univariate Discrete Distributions, John Wiley & Sons, New York, NY, USA, 2nd edition, 1992.
  35. W. Feller, An Introduction to Probability Theory and Its Applications, vol. 1, John Wiley & Sons, 3rd edition, 1968.
  36. W. Hürlimann, “Robust confidence bounds for the mean of some count data models,” Blätter der DGVFM, vol. 25, no. 4, pp. 795–811, 2002. View at Publisher · View at Google Scholar