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A Correction for an Underdispersed Event Count Probability Distribution

  • Rainer Winkelmann, Curtis S. Signorino and Gary King


We demonstrate that the expected value and variance commonly given for a well-known probability distribution are incorrect. We also provide corrected versions and report changes in a computer program to account for the known practical uses of this distribution.

The probability distribution in question, named the continuous parameter binomial (CPB) by King (1989a), has been known for at least six decades. The publications that reported the moments incorrectly, or were at least unclear about them, include a dated article (Guldberg 1931), a dissertation and several resulting published articles (Katz 1945, 1965), a popular reference book (Johnson and Kotz 1969),' work in political science on event count regression models (King 1989a), and extensions of these event count models in econometrics (Winkelmann and Zimmermann 1991). Event count regression models have become increasingly common in empirical political science research; some recent examples include Wang et al. (1993) and Krause (1994).



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Abramowitz, M., and Stegun, I., eds. 1965. Handbook of Mathematical Functions. New York: Dover.
Guldberg, Alf. 1931. “On Discontinuous Frequency-Functions and Statistical Series.” Skandinavisk Aktvarietionskrift 14: 167–97.
Graham, Ronald L., Knuth, Donald E., and Patashnik, Oren. 1989. Concrete Mathematics: A Foundation for Computer Science. Reading: Addison-Wesley.
Johnson, Norman L., and Kotz, Samuel. 1969. Distributions in Statistics: Discrete Distributions. New York: Wiley.
Katz, Leo. 1965. “Unified Treatment of a Broad Class of Discrete Probability Distributions.” In Patil, Ganapati P., ed., Classical and Contagious Discrete Distributions. Calcutta: Statistical Publishing Society.
Katz, Leo. 1945. “Characteristics of Frequency Functions Defined by First Order Difference Equations.” Ph.D. diss. University of Michigan.
King, Gary. 1989a. “Variance Specification in Event Count Models: From Restrictive Assumptions to a Generalized Estimator.” American Journal of Political Science 33: 762784.
King, Gary. 1989b. Unifying Political Methodology: The Likelihood Theory of Statistical Inference. New York: Cambridge University Press.
Krause, George A. 1994. “Federal Reserve Policy Decision Making: Political and Bureaucratic Influences.” American Journal of Political Science 38: 124–44.
Wang, T. Y., Dixon, William J., Muller, Edward N., and Seligson, Mitchell A. 1993. “Inequality and Political Violence Revisited.” American Political Science Review 87: 979–93.
Winkelmann, Rainer, and Zimmermann, Klaus F. 1991. “A New Approach for Modeling Economic Count Data.” Economics Letters 37: 139–43.
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Political Analysis
  • ISSN: 1047-1987
  • EISSN: 1476-4989
  • URL: /core/journals/political-analysis
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