Estimating Equations for Estimation of Mcdonald Generalized Beta— Binomial Parameters
ABSTRACT
There has been a considerable recent attention in modeling over dispersed binomial data occurring in toxicology, biology, clinical medicine, epidemiology and other similar fields using a class of Binomial mixture distribution such as Beta Binomial distribution (BB) and Kumaraswamy-Binomial distribution (KB). A new three-parameter binomial mixture distribution namely, McDonald Generalized Beta Binomial (McGBB) distribution has been developed which is superior to KB and BB since studies have shown that it gives a better fit than the KB and BB distribution on both real life data set and on the extended simulation study in handling over dispersed binomial data. The dispersion parameter will be treated as nuisance in the analysis of proportions since our interest is in the parameters of McGBB distribution. In this paper, we consider estimation of parameters of this MCGBB model using Quasi-likelihood (QL) and Quadratic estimating functions (QEEs) with dispersion. By varying the coefficients of the QEE’s we obtain four sets of estimating equations which in turn yield four sets of estimates. We compare small sample relative efficiencies of the estimates based on QEEs and quasi-likelihood with the maximum likelihood estimates. The comparison is performed using real life data sets arising from alcohol consumption practices and simulated data. These comparisons show that estimates based on optimal QEEs and QL are highly efficient and are the best among all estimates investigated.
There has been a considerable recent attention in modeling over dispersed binomial data occurring in toxicology, biology, clinical medicine, epidemiology and other similar fields using a class of Binomial mixture distribution such as Beta Binomial distribution (BB) and Kumaraswamy-Binomial distribution (KB). A new three-parameter binomial mixture distribution namely, McDonald Generalized Beta Binomial (McGBB) distribution has been developed which is superior to KB and BB since studies have shown that it gives a better fit than the KB and BB distribution on both real life data set and on the extended simulation study in handling over dispersed binomial data. The dispersion parameter will be treated as nuisance in the analysis of proportions since our interest is in the parameters of McGBB distribution. In this paper, we consider estimation of parameters of this MCGBB model using Quasi-likelihood (QL) and Quadratic estimating functions (QEEs) with dispersion. By varying the coefficients of the QEE’s we obtain four sets of estimating equations which in turn yield four sets of estimates. We compare small sample relative efficiencies of the estimates based on QEEs and quasi-likelihood with the maximum likelihood estimates. The comparison is performed using real life data sets arising from alcohol consumption practices and simulated data. These comparisons show that estimates based on optimal QEEs and QL are highly efficient and are the best among all estimates investigated.
KEYWORDS
Maximum Likelihood, McDonald Generalized Beta Binomial, Simulation, Quadratic Estimating Equations, Quasi-Likelihood
Maximum Likelihood, McDonald Generalized Beta Binomial, Simulation, Quadratic Estimating Equations, Quasi-Likelihood
Cite this paper
Janiffer, N. , Islam, A. and Luke, O. (2014) Estimating Equations for Estimation of Mcdonald Generalized Beta— Binomial Parameters. Open Journal of Statistics, 4, 702-709. doi: 10.4236/ojs.2014.49065.
Janiffer, N. , Islam, A. and Luke, O. (2014) Estimating Equations for Estimation of Mcdonald Generalized Beta— Binomial Parameters. Open Journal of Statistics, 4, 702-709. doi: 10.4236/ojs.2014.49065.
References
[1] Breslow, N.E. (1990) Tests of Hypothesis in Over-Dispersed Poisson Regression and Other Quasi-Likelihood Models. Journal of the American Statistical Association, 85, 565-571.
http://dx.doi.org/10.1080/01621459.1990.10476236 [2] Moore, D.F. and Tsiatis, M. (1991) Robust Estimation of the Standard Error in Moment Methods for Extra Binomial/ Extra Poisson Variation. Biometrika, 47, 383-401.
http://dx.doi.org/10.2307/2532133 [3] NeIder, J.A. and Pregibon, D. (1987) An Extended Quasi-Likelihood Function. Biometrika, 74, 221-232.
http://dx.doi.org/10.1093/biomet/74.2.221 [4] Whittle, P. (1961) Gaussian Estimation in Stationary Time Series. Bulletin of the International Statistical Institute, 39, 1-26. [5] Crowder, M.J. (1985) Gaussian Estimation for Correlated Binomial Data. Journal of the Royal Statistical Society, Series B, 47, 229-237. [6] Davidian, M. and Carrol, R.J. (1987) Variance Function Estimation. Journal of the American Statistical Association, 82, 1079-1091.
http://dx.doi.org/10.1080/01621459.1987.10478543 [7] Crowder, M.J. (1987) On Linear Quadratic Estimating Functions. Biometrika, 74, 591-597.
http://dx.doi.org/10.1093/biomet/74.3.591 [8] Godambe, V.P. and Thompson, M.E. (1989) An Extension of Quasi-Likelihood Estimation. Journal of Statistical Planning and Inference, 22, 137-152.
http://dx.doi.org/10.1016/0378-3758(89)90106-7 [9] McDonald, J.B. (1984) Some Generalized Functions for the Size Distribution of Income. Econometrica: Journal of the Econometric Society, 52, 647-663.
http://dx.doi.org/10.2307/1913469 [10] McDonald, J.B. and Xu, Y.J. (1995) A Generalization of the Beta Distribution with Applications. Journal of Econometrics, 66, 133-152.
http://dx.doi.org/10.1016/0304-4076(94)01612-4 [11] Wedderburn, R.M. (1974) Quasi-Likelihood Functions, Generalized Linear Models and the Gauss Newton Method. Biometrics, 61, 439-447. [12] Alanko, T. and Lemmens, P.H. (1996) Response Effects in Consumption Surveys: An Application of the Betabinomial Model to Self-Reported Drinking Frequencies. Journal of Official Statistics, 12, 253-273. [13] Rodríguez-Avi, J., Conde-Sánchez, A., Sáez-Castillo, A.J. and Olmo-Jiménez, M.J. (2007) A Generalization of the Beta-Binomial Distribution. Journal of the Royal Statistical Society, Series C (Applied Statistics), 56, 51-61.
http://dx.doi.org/10.1111/j.1467-9876.2007.00564.x [14] Chandrabose, M., Pushpa, W. and Roshan, D. (2013) The McDonald Generalized Beta-Binomial Distribution: A New Binomial Mixture Distribution and Simulation Based Comparison with Its Nested Distributions in Handling Overdispersion. International Journal of Statistics and Probability, 2, 213-223.
[1] Breslow, N.E. (1990) Tests of Hypothesis in Over-Dispersed Poisson Regression and Other Quasi-Likelihood Models. Journal of the American Statistical Association, 85, 565-571.
http://dx.doi.org/10.1080/01621459.1990.10476236 [2] Moore, D.F. and Tsiatis, M. (1991) Robust Estimation of the Standard Error in Moment Methods for Extra Binomial/ Extra Poisson Variation. Biometrika, 47, 383-401.
http://dx.doi.org/10.2307/2532133 [3] NeIder, J.A. and Pregibon, D. (1987) An Extended Quasi-Likelihood Function. Biometrika, 74, 221-232.
http://dx.doi.org/10.1093/biomet/74.2.221 [4] Whittle, P. (1961) Gaussian Estimation in Stationary Time Series. Bulletin of the International Statistical Institute, 39, 1-26. [5] Crowder, M.J. (1985) Gaussian Estimation for Correlated Binomial Data. Journal of the Royal Statistical Society, Series B, 47, 229-237. [6] Davidian, M. and Carrol, R.J. (1987) Variance Function Estimation. Journal of the American Statistical Association, 82, 1079-1091.
http://dx.doi.org/10.1080/01621459.1987.10478543 [7] Crowder, M.J. (1987) On Linear Quadratic Estimating Functions. Biometrika, 74, 591-597.
http://dx.doi.org/10.1093/biomet/74.3.591 [8] Godambe, V.P. and Thompson, M.E. (1989) An Extension of Quasi-Likelihood Estimation. Journal of Statistical Planning and Inference, 22, 137-152.
http://dx.doi.org/10.1016/0378-3758(89)90106-7 [9] McDonald, J.B. (1984) Some Generalized Functions for the Size Distribution of Income. Econometrica: Journal of the Econometric Society, 52, 647-663.
http://dx.doi.org/10.2307/1913469 [10] McDonald, J.B. and Xu, Y.J. (1995) A Generalization of the Beta Distribution with Applications. Journal of Econometrics, 66, 133-152.
http://dx.doi.org/10.1016/0304-4076(94)01612-4 [11] Wedderburn, R.M. (1974) Quasi-Likelihood Functions, Generalized Linear Models and the Gauss Newton Method. Biometrics, 61, 439-447. [12] Alanko, T. and Lemmens, P.H. (1996) Response Effects in Consumption Surveys: An Application of the Betabinomial Model to Self-Reported Drinking Frequencies. Journal of Official Statistics, 12, 253-273. [13] Rodríguez-Avi, J., Conde-Sánchez, A., Sáez-Castillo, A.J. and Olmo-Jiménez, M.J. (2007) A Generalization of the Beta-Binomial Distribution. Journal of the Royal Statistical Society, Series C (Applied Statistics), 56, 51-61.
http://dx.doi.org/10.1111/j.1467-9876.2007.00564.x [14] Chandrabose, M., Pushpa, W. and Roshan, D. (2013) The McDonald Generalized Beta-Binomial Distribution: A New Binomial Mixture Distribution and Simulation Based Comparison with Its Nested Distributions in Handling Overdispersion. International Journal of Statistics and Probability, 2, 213-223.