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A generalization of the Binomial distribution based on the dependence ratio

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  • Gianfranco Lovison

Abstract

type="main" xml:id="stan12053-abs-0001"> We propose a generalization of the Binomial distribution, called DR-Binomial, which accommodates dependence among units through a model based on the dependence ratio (Ekholm et al., Biometrika, 82, 1995, 847). Properties of the DR-Binomial are discussed, and the constraints on its parameter space are studied in detail. Likelihood-based inference is presented, using both the joint and profile likelihoods; the usefulness of the DR-Binomial in applications is illustrated on a real dataset displaying negative unit-dependence, and hence under-dispersion compared with the Binomial. Although the DR-Binomial turns out to be a reparameterization of Altham's Additive-Binomial and Kupper–Haseman's Correlated-Binomial distribution, we believe its introduction is useful, both in terms of interpretability and mathematical tractability and in terms of generalizability to the Multinomial case.

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  • Gianfranco Lovison, 2015. "A generalization of the Binomial distribution based on the dependence ratio," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 69(2), pages 126-149, May.
    • Handle: RePEc:bla:stanee:v:69:y:2015:i:2:p:126-149
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    References listed on IDEAS

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    1. Patricia M. E. Altham, 1978. "Two Generalizations of the Binomial Distribution," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 27(2), pages 162-167, June.
    2. Lovison, G., 1998. "An alternative representation of Altham's multiplicative-binomial distribution," Statistics & Probability Letters, Elsevier, vol. 36(4), pages 415-420, January.
    3. Yu, Chang & Zelterman, Daniel, 2002. "Sums of dependent Bernoulli random variables and disease clustering," Statistics & Probability Letters, Elsevier, vol. 57(4), pages 363-373, May.
    4. Anders Ekholm & John W. McDonald & Peter W. F. Smith, 2000. "Association Models for a Multivariate Binary Response," Biometrics, The International Biometric Society, vol. 56(3), pages 712-718, September.
    5. Rand R. Wilcox, 1981. "A Review of the Beta-Binomial Model and its Extensions," Journal of Educational and Behavioral Statistics, , vol. 6(1), pages 3-32, March.
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