Abstract
A new generalization of the discrete Poisson distribution was introduced by Consul and Jain (1970, 1973) and studied by Charalambides (1974), Kumar and Consul (1979), Janardan and Schaeffer (1977), Janardan el al. (1979), Consul and Shoukri (1984, 1985), and many others. This model contains two parameters and possesses the over-dispersion and the underdispersion properties which make it a good descriptive model for aggregation patterns in biology, ecology and many other disciplines. Although the negative binomial distribution possesses the over-dispersion property, it is found that the parametric structure of the generalized Poisson distribution (GPD) permits a relatively simpler inferential procedure.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Charalambides, C. A. (1974), “The distribution of sufficient statistics of truncated generalized logarithmic series, Poisson, and negative binomial distributions.” Canadiern Journal of Statistics 2, 261–275.
Consul, P. C. (1981), “Relation of modified power series distributions to Lagrangian probability distributions.” Communications in Statistics A, Theory and Methods 10, 2039–2046.
Consul, P. C., and H. C. Gupta (1980), “The generalized negative binomial distribution and its characterization by zero regression.” Siam Journal on Applied Mathematics 39, 231–237.
Consul, P. C, and G. C. Jain (1970), “A generalization of the Poisson distribution.” Technical report #114. Department of Mathematics, Statistics, and Computing Science, University of Calgary, Calgary, Alberta, Canada.
Consul, P. C., and G. C. Jain (1973), “A generalization of the Poisson distribution.” Technometrics 15, 791–799.
Consul, P. C., and L. R. Shenton (1972), “Use of the Lagrange expansion for generating discrete generalized probability distributions.” Siam Journal on Applied Mathematics 23, 239–248.
Consul, P. C, and L. R. Shenton (1973), “Some interesting properties of Lagrange distributions.” Communications in Statistics 2, 263–272.
Consul, P. C., and M. M. Shoukri (1984), “Maximum likelihood estimation for the generalized Poisson distribution.” Communications in Statistics A, Theory and Methods 13, 1533–1547.
Consul, P. C., and M. M. Shoukri (1985), “The generalized Poisson distribution when the sample mean is larger than the sample variance.” Communications in Statistics B, Simulation and Computation 14, 667–681.
Fisher, R. A. (1930), The Genetical Theory of Natural Selection. London and New York: Oxford University Press.
Hill, J. M., and C. A. Gulati (1981), “The random walk associated with the game of roulette.” Journal of Applied Probability 18, 931–936.
Jain, G. C., and P. C. Consul (1971), “A generalization of the negative binomial distribution.” Siam Journal on Applied Mathematics 21, 501–513.
Janardan, K. G., W. Kerster, and D. J. Schaeffer (1979), “Biological applications of the Lagrangian Poisson distribution.” BioSciences 29, 599–602.
Janardan, K. G., and D. J. Schaeffer (1977), “Models for the analysis of chromosomal aberrations in human leukocytes.” Biometrical Journal 19, 599–612.
Kumar, A., and P. C. Consul (1979), “Minimum variance unbiased estimator for modified power series distribution.” Communications in Statistics A, Theory and Methods 9, 1261–1275.
Mohanty, S. G. (1966), “On a generalized two-coin tossing problem.” Biometrische Zeitschrift 8, 266–272.
Neyman, J. (1963), “Certain chance mechanisms involving discrete distributions.” Classical and Contagious Discrete Distributions, Calcutta: Statistical Publishing Society.
Takács, L. (1962), “A generalization of the ballot problem and its applications in the theory of queues.” Journal of the American Statistical Association 57, 327–337.
Whittaker, E. T., and G. N. Watson (1963), A Course of Modern Analysis. Cambridge, MA: Cambridge University Press.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1987 D. Reidel Publishing Company, Dordrecht, Holland
About this chapter
Cite this chapter
Shoukri, M.M., Consul, P.C. (1987). Some Chance Mechanisms Generating the Generalized Poisson Probability Models. In: MacNeill, I.B., Umphrey, G.J., Donner, A., Jandhyala, V.K. (eds) Biostatistics. The University of Western Ontario Series in Philosophy of Science, vol 38. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-4794-8_15
Download citation
DOI: https://doi.org/10.1007/978-94-009-4794-8_15
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-8626-4
Online ISBN: 978-94-009-4794-8
eBook Packages: Springer Book Archive