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On Difference Equations, Probability Models and the “Generalized Event Count” Distribution

Published online by Cambridge University Press:  04 January 2017

Timothy W. Amato
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Abstract

In this article, the mathematical and probabilistic foundations of Gary King's “generalized event count” (GEC) model for dealing with unequally dispersed event count data are explored. It is shown that the GEC model is a probability model that joins together the binomial, negative binomial, and Poisson distributions. Some aspects of the GEC's reparameterization are described and extended and it is shown how different reparameterizations lead to different interpretations of the dispersion parameter. The common mathematical and statistical structure of “unequally dispersed” event count models as models that require estimation of the “number of trials” parameter along with the “probability” component is derived. Some questions pertaining to estimation of this class of models are raised for future discussion.

Type
Symposium on the Generalized Event Count Estimator
Information
Political Analysis , Volume 6 , 1996 , pp. 175 - 212
Copyright
Copyright © Society for Political Methodology 

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References

Achen, Christopher. 1993. MLE, Aggregate Data, and the Montana Test. MimeoGoogle Scholar
Amemiya, Takeshi. 1985. Advanced Econometrics. Cambridge: Harvard University Press.Google Scholar
Anscombe, F. J. 1950. Sampling Theory of the Negative Binomial and Logarithmic Series Distributions. Biometrika 37: 358–82.Google Scholar
Casella, George, and Berger, Roger. 1990. Statistical Inference. Pacific Grove: Brooks-Cole.Google Scholar
Clark, S. J., and Perry, J. N. 1989. Estimation of the Negative Binomial Parameter к by Maximum Likelihood. Biometrics 45: 309–16.Google Scholar
Crowder, M. J. 1978. Beta-Binomial ANOVA for Proportions. Applied Statistics 27: 34–7.Google Scholar
Davidson, R. and MacKinnon, J. G. 1993. Estimation and Inference in Econometrics. New York: Oxford University Press.Google Scholar
Feldman, D. and Fox, M. 1968. Estimation of the Parameter n in the Binomial Distribution. Journal of the American Statistical Association 63: 150–58.Google Scholar
Fisher, Ronald. 1941. The Negative Binomial Distribution. Annals of Eugenics 11: 182–87.Google Scholar
Gelman, Andrew and King, Gary. 1990. Estimating Incumbency Advantage Without Bias. American Journal of Political Science 34: 1142–65.Google Scholar
Gourieroux, C., Monfort, A., and Trognon, A. 1984a. Pseudo Maximum Likelihood: Theory. Econometrica 52: 681–700.Google Scholar
Gourieroux, C., Monfort, A., and Trognon, A. 1984b. Pseudo Maximum Likelihood Methods: Applications to Poisson Models. Econometrica 52: 701–20.Google Scholar
Greene, W. 1993. Econometric Analysis 2nd. ed. New York: Macmillan.Google Scholar
Greenwood, M. and Yule, G. U. 1920. An Enquiry Into the Nature of Frequency Distributions of Multiple Happenings, With Particular Reference to the Occurrence of Multiple Attacks of Disease or Repeated Accidents. Journal of the Royal Statistical Society, Series A, 83: 255–79.Google Scholar
Hausman, J., Hall, B., and Griliches, Z. 1984. Econometric Models for Count Data with an Application to the Patents-R&D Relationship. Econometrica 52: 909–38.Google Scholar
Hogg, Robert and Craig, Allen. 1978. Introduction to Mathematical Statistics 4th. ed. New York: Macmillan.Google Scholar
Johnson, N. and Kotz, S. 1969. Distributions in Statistics: Discrete Distributions. New York: Wiley.Google Scholar
Johnson, N., Kotz, S. and Kemp, A. 1992. Distributions in Statistics: Discrete Distributions. 2nd. ed. New York: Wiley.Google Scholar
Katz, Leo. 1965. Unified Treatment of a Broad Class of Discrete Probability Distributions. In Patil, G.P. ed. Classical and Contagious Discrete Distributions. Calcutta: Statistical Publishing Society.Google Scholar
Kendall, M. G. and Stuart, A. 1977. The Advanced Theory of Statistics, Vol. 1, 4th. ed. London: Charles Griffin.Google Scholar
King, Gary. 1989a. Unifying Political Methodology. New York: Cambridge University Press.Google Scholar
King, Gary. 1989b. Variance Specification in Event Count Models: From Restrictive Assumptions a Generalized Estimator. American Journal of Political Science 33: 762–84.Google Scholar
King, Gary, Alt, J., Burns, N., and Laver, M. 1990. A Unified Model of Cabinet Dissolution in Parliamentary Democracies. American Journal of Political Science 33: 846–71.Google Scholar
Lancaster, Tony. 1990. The Econometric Analysis of Transition Data. New York: Cambridge University Press.Google Scholar
Lawless, Jerrold. 1987. Negative Binomial and Mixed Poisson Regression. Canadian Journal of Statistics 15: 209–25.Google Scholar
LeCam, Lucien. 1990. Maximum Likelihood: An Introduction. International Statistical Review 58: 153–71.Google Scholar
Lee, Lung-Fei. 1986. Specification Tests for Poisson Regression Models. International Economic Review 27: 689–706.Google Scholar
Lindley, D., and Phillips, L. 1976. Inference for a Bernoulli Process (A Bayesian View). American Statistician 30: 112–19.Google Scholar
McCullagh, P. and Nelder, J. 1989. Generalized Linear Models, 2nd. ed. London: Chapman and Hall.Google Scholar
Nelder, J., and Lee, Y. 1992. Likelihood, Quasi-Likelihood, and Pseudolikelihood: Some Comparisons. Journal of the Royal Statistical Society, Series B, 54: 273–84.Google Scholar
Olkin, I., Petkau, A. J., and Zidek, J. D. 1981. A Comparison of n Estimators for the Binomial Distribution. Journal of the American Statistical Association 76: 637–42.Google Scholar
Pratt, John. 1981. Concavity of the Loglikelihood. Journal of the American Statistical Association 76: 103–6.Google Scholar
Ross, G., and Preece, D. 1985. The Negative Binomial Distribution. The Statistician 94: 323–36.Google Scholar
Theil, H. 1971. Principles of Econometrics. New York: Wiley.Google Scholar
Williams, D. A. 1982. Extra Binomial Variation in Logistic Linear Models. Applied Statistics 31: 144–48.Google Scholar
Winkelmann, R., Signorino, C. and King, G. 1995. A Correction for an Underdispersed Event Count Probability Distribution. In Political Analysis, 5: edited by Freeman, John R. Ann Arbor: University of Michigan Press: 215–28.Google Scholar
Wooldridge, Jeffrey. 1991. Specification Testing and Quasi Maximum Likelihood Estimation. Journal of Econometrics 48: 29–55.Google Scholar