Beta-Binomial-Poisson Mixture Model and Its Parameter Estimation: An Application to Number of Female Child Births
DOI:
https://doi.org/10.1285/i20705948v19n1p62-78Keywords:
EM algorithm, Method of Moments, Mixture model, Son preferenceAbstract
This study investigates parameter estimation for the beta-binomial-Poisson mixture model, which accounts for overdispersion and heterogeneity in count data, making it relevant for demographic stuties such as number of female
births. The model was applied to NFHS-V data from five Indian states, where son preference influences reproductive behavior. Traditional methods, including Maximum Likelihood Estimation (MLE) and the Method of Moments (MoM), face challenges due to latent variables and model complexity.
We explore the Expectation-Maximization (EM) algorithm as a robust alternative. Results show that EM yields more stable and realistic parameter estimates, while MoM often produces poor fits or unrealistic values. Chi-square tests indicate that EM achieves a substantially better fit than MoM, although discrepancies remain, suggesting the need for more advanced modelling.
References
Casella, G. and Berger, R. L. (2002). Statistical Inference. Thomson Learning, Australia; Pacific Grove, CA, 2nd edition.
Dandekar, V. . M. (1955). Certain modified forms of binomial and poisson distributions. Sankhya: The Indian Journal of Statistics (1933-1960), 15(3):237–250.
Dempster, A. P., Laird, N. M., and Rubin, D. B. (1977). Maximum likelihood from incomplete data via the em algorithm. Journal of the Royal Statistical Society: Series B (Methodological), 39(1):1–22.
Fisher, R. A. (1922). On the mathematical foundations of theoretical statistics. Philosophical transactions of the Royal Society of London. Series A, containing papers of a mathematical or physical character, 222(594-604):309–368.
Kumar, A. (2020). A probability model for the number of female child births. Journal of Statistics Applications & Probability, 9(3):525–534.
Lin, M., Lucas Jr, H. C., and Shmueli, G. (2013). Research commentary—too big to fail: large samples and the p-value problem. Information Systems Research, 24(4):906–917.
McLachlan, G. J. and Peel, D. (2000). Finite Mixture Models. Wiley-Interscience.
McLachlan, G. J. and Peel, D. (2008). Finite Mixture Models. John Wiley & Sons.
Pathak, K. (1966). A probability distribution for the number of conceptions. Sankhya: The Indian Journal of Statistics, Series B, 28:213–218.
Pearson, K. (1900). On the criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can be reasonably supposed to have arisen from random sampling. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 50(302):157–175.
Rahim, S. A., Manson, G., and Aziz, M. (2021). Data clustering based on gaussian mixture model and expectation-maximization algorithm for data-driven structural health monitoring system. International Journal of Integrated Engineering, 13(7):167–175.
Rai, P. K., Pareek, S., and Joshi, H. (2014). On the estimation of probability model for the number of female child births among females. Journal of Data Science, 12(3):137–156.
Redner, R. A. and Walker, H. F. (1984). Mixture densities, maximum likelihood and the em algorithm. SIAM Review, 26(2):195–239.
Ross, S. M. (2014). Introduction to probability models. Academic Press.
Roy, S., Sharma, P., Singh, K., and Srivastava, R. (2023). On a statistical model useful for demographics: Estimating the mean number of children ever born through the distribution of male births with an application to data from India. Journal of Reliability
and Statistical Studies, 16(1):57–80.
Sammaknejad, N., Zhao, Y., and Huang, B. (2019). A review of the expectation maximization algorithm in data-driven process identification. Journal of Process Control, 73:123–136.
Singh, B. P., Maheshwari, S., and Gupta, P. K. (2015). A probability model for sex composition of children in the presence of son preference. Demography India, 44(1 & 2):50–57.
Singh, K., Singh, B. P., and Singh, N. (2012). A probabilistic study of variation in number of child deaths. Journal of Rajasthan Statistical Association, 1(1):54–67.
Yadava, R. C. (2016). Stochastic models for human fertility. Demography India, 45(1 & 2):1–16.
Yadava, R. C., Kumar, A., and Srivastava, U. (2013). Sex ratio at birth: A model based approach. Mathematical Social Sciences, 65(1):36–39.
Zhu, J., Eickhoff, J. C., and Kaiser, M. S. (2003). Modeling the dependence between number of trials and success probability in beta-binomial-poisson mixture distributions. Biometrics, 59(4):955–961.
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