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.
Highlights
Estimating functions have for sometimes been a key concept and subject of inquiry in research and it is known to be the most general method of estimation
The usual procedure is to take a parametric model, such as, the McDonald Generalized beta-binomial model to allow over as well as under dispersion and obtain maximum likelihood estimates of the parameters McDonald Generalized Beta Binomial (McGBB) distribution is a three-parameter distribution which is superior to Kumaraswamy-Binomial distribution (KB) in handling over dispersed binomial data
This procedure may produce inefficient or biased estimates when the parametric model does not fit the data well. More robust estimates, such as moment estimates, quasi-likelihood estimates (Breslow, 1990 [1]; Moore and Tsiatis, 1991 [2]), extended quasi-likelihood estimates (Nelder and Pregibon, 1987 [3]), the Gaussian likelihood estimates (Whittle, 1961 [4]; Crowder, 1985 [5]), estimates based on the pseudo-likelihood estimating equations of Davidian and Carrol (1987) [6] and estimates based on quadratic estimating functions of Crowder (1987) [7] and Godambe and Thompson (1989) [8] can be considered
Summary
Estimating functions have for sometimes been a key concept and subject of inquiry in research and it is known to be the most general method of estimation The basis of this method is a set of simultaneous equations involving both the data and the unknown model parameters. The usual procedure is to take a parametric model, such as, the McDonald Generalized beta-binomial model to allow over as well as under dispersion and obtain maximum likelihood estimates of the parameters McDonald Generalized Beta Binomial (McGBB) distribution is a three-parameter distribution which is superior to KB in handling over dispersed binomial data. In this paper we consider estimating the parameters of McDonald Generalized Beta Binomial by the quadratic estimating equations (QEE’s) of Crowder (1987) [7] and Godambe and Thompson (1989) [8] and compared the small sample efficiency and bias properties of these estimates with the maximum likelihood estimates.
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