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An improved method for the computation of maximum likelihood estimates for multinomial overdispersion models. (English) Zbl 1430.62015
Summary: We consider the maximum likelihood estimation of two commonly used overdispersion models, namely, the Dirichlet-multinomial distribution (DM), due to [J. E. Mosimann, Biometrika 49, 65–82 (1962; Zbl 0105.12502)], and a finite mixture distribution (FM) proposed by the authors [Biometrika 80, No. 2, 363–371 (1993; Zbl 0778.62013); J. Am. Stat. Assoc. 93, No. 443, 1078–1087 (1998; Zbl 1064.62527)]. These models have been successfully used in the literature for modeling overdispersion in multinomial data. Maximum likelihood estimation of the parameters of these models using the classical Fisher scoring method poses certain computational challenges. In the case of DM, the challenges are overcome by noting that the Fisher information matrix can be computed using the beta-binomial distribution (BB), which is the univariate version of DM. On the other hand, in the case of FM, an approximation theorem can be used to obtain a two-stage procedure for computing the maximum likelihood estimates. Simulation results show that the two-stage procedure is faster without loosing any accuracy.

MSC:
62-08 Computational methods for problems pertaining to statistics
62F10 Point estimation
Software:
nlmdl
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References:
[1] Brier, S.S., 1979. Categorical data models for complex sampling schemes. Ph.D. Dissertation, School of Statistics, University of Minnesota, Minneapolis, MN (unpublished).
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[5] Morel, J.G.; Nagaraj, N.K., A finite mixture distribution for modelling multinomial extra variation, Biometrika, 80, 363-371, (1993) · Zbl 0778.62013
[6] Mosimann, J.E., On the compound multinomial distribution, the multivariate \(\beta\) distribution and correlation among proportions, Biometrika, 49, 65-82, (1962) · Zbl 0105.12502
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