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Two Generalizations of the Binomial Distribution

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  • Patricia M. E. Altham

Abstract

The sum of k independent and identically distributed (0, 1) variables has a binomial distribution. If the variables are identically distributed but not independent, this may be generalized to a two‐parameter distribution where the k variables are assumed to have a symmetric joint distribution with no second‐ or higher‐order “interactions”. Two distinct generalizations are obtained, depending on whether the “multiplicative” or “additive” definition of “interaction” for discrete variables is used. The multiplicative generalization gives rise to a two‐parameter exponential family, which naturally includes the binomial as a special case. Whereas with a beta‐binomially distributed variable the variance always exceeds the corresponding binomial variance, the “additive” or “multiplicative” generalizations allow the variance to be greater or less than the corresponding binomial quantity. The properties of these two distributions are discussed, and both distributions are fitted, successfully, to data given by Skellam (1948) on the secondary association of chromosomes in Brassica.

Suggested Citation

  • 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.
    • Handle: RePEc:bla:jorssc:v:27:y:1978:i:2:p:162-167
    DOI: 10.2307/2346943
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    1. Zhen Pang & Anthony Y. C. Kuk, 2005. "A Shared Response Model for Clustered Binary Data in Developmental Toxicity Studies," Biometrics, The International Biometric Society, vol. 61(4), pages 1076-1084, December.
    2. 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.
    3. Pires, Rubiane M. & Diniz, Carlos A.R., 2012. "Correlated binomial regression models," Computational Statistics & Data Analysis, Elsevier, vol. 56(8), pages 2513-2525.
    4. Chang Yu & Daniel Zelterman, 2002. "Statistical Inference for Familial Disease Clusters," Biometrics, The International Biometric Society, vol. 58(3), pages 481-491, September.
    5. Iain L. MacDonald, 2021. "Is EM really necessary here? Examples where it seems simpler not to use EM," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 105(4), pages 629-647, December.
    6. Dittrich, R. & Hatzinger, R. & Katzenbeisser, W., 2002. "Modelling dependencies in paired comparison data: A log-linear approach," Computational Statistics & Data Analysis, Elsevier, vol. 40(1), pages 39-57, July.
    7. Yu, Chang & Zelterman, Daniel, 2008. "Sums of exchangeable Bernoulli random variables for family and litter frequency data," Computational Statistics & Data Analysis, Elsevier, vol. 52(3), pages 1636-1649, January.
    8. C. D. Lai & K. Govindaraju & M. Xie, 1998. "Effects of correlation on fraction non-conforming statistical process control procedures," Journal of Applied Statistics, Taylor & Francis Journals, vol. 25(4), pages 535-543.
    9. Kolossiatis, M. & Griffin, J.E. & Steel, M.F.J., 2011. "Modeling overdispersion with the normalized tempered stable distribution," Computational Statistics & Data Analysis, Elsevier, vol. 55(7), pages 2288-2301, July.
    10. Zhen Pang & Anthony Y. C. Kuk, 2007. "Test of Marginal Compatibility and Smoothing Methods for Exchangeable Binary Data with Unequal Cluster Sizes," Biometrics, The International Biometric Society, vol. 63(1), pages 218-227, March.
    11. Timothy I. Cannings & Richard J. Samworth, 2017. "Random-projection ensemble classification," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 79(4), pages 959-1035, September.
    12. Catalina Stefanescu & Bruce W. Turnbull, 2003. "Likelihood Inference for Exchangeable Binary Data with Varying Cluster Sizes," Biometrics, The International Biometric Society, vol. 59(1), pages 18-24, March.
    13. Ardo Hout & Graciela Muniz-Terrera, 2019. "Hidden three-state survival model for bivariate longitudinal count data," Lifetime Data Analysis: An International Journal Devoted to Statistical Methods and Applications for Time-to-Event Data, Springer, vol. 25(3), pages 529-545, July.
    14. 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.
    15. Maria A. Spassova, 2019. "Statistical Approach to Identify Threshold and Point of Departure in Dose–Response Data," Risk Analysis, John Wiley & Sons, vol. 39(4), pages 940-956, April.
    16. Lovison, G., 1998. "An alternative representation of Altham's multiplicative-binomial distribution," Statistics & Probability Letters, Elsevier, vol. 36(4), pages 415-420, January.
    17. Dankmar Böhning, 2015. "Power series mixtures and the ratio plot with applications to zero-truncated count distribution modelling," METRON, Springer;Sapienza Università di Roma, vol. 73(2), pages 201-216, August.
    18. Borges, Patrick & Rodrigues, Josemar & Balakrishnan, Narayanaswamy & Bazán, Jorge, 2014. "A COM–Poisson type generalization of the binomial distribution and its properties and applications," Statistics & Probability Letters, Elsevier, vol. 87(C), pages 158-166.
    19. Sueli Mingoti, 2003. "A note on the sample size required in sequential tests for the generalized binomial distribution," Journal of Applied Statistics, Taylor & Francis Journals, vol. 30(8), pages 873-879.
    20. Molenberghs, Geert & Declerck, Lieven & Aerts, Marc, 1998. "Misspecifying the likelihood for clustered binary data," Computational Statistics & Data Analysis, Elsevier, vol. 26(3), pages 327-349, January.

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