Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-25T18:05:47.494Z Has data issue: false hasContentIssue false
  • English
  • Français

Some Characterizations of the One-Parameter Family of Probability Distributions

Published online by Cambridge University Press:  20 November 2018

A. M. Mathai
A. M. Mathai*
Affiliation:
McGill University
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider one-parameter families of distributions which are of the general exponential type, the general series distribution type, or which can be transformed into the exponential type by a one-to-one transformation. In this paper we establish theorems to the effect that such distributions may be characterized by a simple differential equation involving the mean function. It is illustrated that almost all the classical one-parameter families of distributions are characterized by these theorems. Multivariate generalizations are given, and it is also noticed that the functional form of the normalizing factor determines the specific distribution in each general family.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 9 , Issue 1 , April 1966 , pp. 95 - 102
Copyright
Copyright © Canadian Mathematical Society 1966

References

1. Khatri, C.G., On certain properties of the power series distribution. Biometrika, vol. 46, (1959) pp. 486-490.Google Scholar
2. Kosambi, D.D., Characterization properties of series distributions. Proc. Indian Academy of Sciences, vol. 15 (1949) pp.109-113.Google Scholar
3. Noack, A., A class of random variables with discrete distributions. Ann. Math. Statist, vol. 21,(1950) pp. 127-132.Google Scholar
4. Patil, G.P., A characterization of the exponential type distribution. Biometrika, vol.50, (1961) pp. 205-207.Google Scholar
5. Patil, G.P. and Shorrock, R., On certain properties of the exponential type families. J. Roy Statist. Soc. Series B, vol. 27, (1965) part 1.Google Scholar