Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-25T05:08:21.965Z Has data issue: false hasContentIssue false
  • English
  • Français

Poisson mixtures and quasi-infinite divisibility of distributions

Published online by Cambridge University Press:  14 July 2016

Prem S. Puri  and
Charles M. Goldie
Prem S. Puri*
Affiliation:
Purdue University
Charles M. Goldie*
Affiliation:
University of Sussex
*
∗Postal address: Department of Statistics, Purdue University, Mathematical Sciences Building, West Lafayette, IN 47907, U.S.A.
∗∗Postal address: Mathematics Division, School of Mathematical and Physical Sciences, University of Sussex, Falmer, Brighton BN1 9QH, U.K.
Get access Rights & Permissions [Opens in a new window]

Abstract

Any probability distribution on [0,∞) can function as the mixing distribution for a Poisson mixture, i.e. a mixture of Poisson distributions. The mixing distribution is called quasi-infinitely divisible (q.i.d.) if it renders the Poisson mixture infinitely divisible, or λ-q.i.d. if it does so after scaling by a factor λ> 0, or ∗-q.i.d. if it is λ-q.i.d. for some λ. These classes of distributions include the infinitely divisible distributions, and each exhibits many of the properties of the latter class but in weakened form. The paper presents the main properties of the classes and the class of Poisson mixtures, including characterisations of membership, relation with cumulants, and closure properties. Examples are given that establish among other things strict inclusions between the classes of mixing distributions.

Keywords

CUMULANTSINFINITE DIVISIBILITYMIXTURE OF DISTRIBUTIONSPOISSON MIXTUREQUASI-INFINITE DIVISIBILITY
Type
Research Papers
Information
Journal of Applied Probability , Volume 16 , Issue 1 , March 1979 , pp. 138 - 153
Copyright
Copyright © Applied Probability Trust 1979 

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

These investigations were started while this author was working on the project at the Statistics Laboratory, University of California, Berkeley, supported by the U.S. Energy Research and Development Agency and completed at Purdue University with the support of U.S. National Science Foundation Grant No. 0199–50–13995.

References

Bartholomew, D. J. (1969) Conditions for mixtures of exponential densities to be probability densities. Ann. Math. Statist. 40, 2183–2188.Google Scholar
Bates, G. E. and Neyman, J. (1952a) Contribution to the theory of accident proneness. I. An optimistic model of the correlation between light and severe accidents. Univ. Calif. Pub. Statist. 1, 215–254.Google Scholar
Bates, G. E. and Neyman, J. (1952b) Contribution to the theory of accident proneness. II. True or false contagion. Univ. Calif. Pub. Statist. 1, 255–276.Google Scholar
Feller, W. (1943) On a general class of ‘contagious’ distributions. Ann. Math. Statist. 14, 389–400.Google Scholar
Feller, W. (1968) An Introduction to Probability Theory and its Applications, Vol. 1, 3rd edn. Wiley, New York.Google Scholar
Feller, W. (1971) An Introduction to Probability Theory and its Applications, Vol. 2, 2nd edn. Wiley, New York.Google Scholar
Goldie, C. M. (1967) A class of infinitely divisible distributions. Proc. Camb. Phil. Soc. 63, 1141–1143.Google Scholar
Haight, F. A. (1967) Handbook of the Poisson Distribution. Wiley, New York.Google Scholar
Keilson, J. and Steutel, F. W. (1972) Families of infinitely divisible distributions closed under mixing and convolution. Ann. Math. Statist. 43, 242–250.CrossRefGoogle Scholar
Keilson, J. and Steutel, F. W. (1974) Mixtures of distributions, moment inequalities, and measures of exponentiality and normality. Ann. Prob. 2, 112–130.Google Scholar
Lukacs, E. (1960) Characteristic Functions. Griffin, London.Google Scholar
Maceda, E. C. (1948) On the compound and generalized Poisson distributions. Ann. Math. Statist. 19, 414–416.Google Scholar
Neyman, J. (1939) On a new class of contagious distributions, applicable in entomology and bacteriology. Ann. Math. Statist. 10, 35–37.Google Scholar
Puri, P. S. (1976) Poisson mixtures and quasi-infinite divisibility of distributions. I.M.S. Bull. 5, abstract 76t-127, p. 204.Google Scholar
Steutel, F. W. (1970) Preservation of Infinite Divisibility Under Mixing and Related Topics. Mathematical Centre Tracts 33, Amsterdam.Google Scholar
Teicher, H. (1961) Identifiability of mixtures. Ann. Math. Statist. 32, 244–248.Google Scholar
Tortrat, A. (1969) Sur les mélanges de lois indéfinement divisibles. C. R. Acad. Sci. Paris A 269, A784–A786.Google Scholar