×

Negative binomial and mixed Poisson regression. (English) Zbl 0632.62060

A number of methods have been proposed for dealing with extra-Poisson variation when doing regression analysis of count data. This paper studies negative-binomial regression models and examines efficiency and robustness properties of inference procedures based on them. The methods are compared with quasilikelihood methods.

MSC:

62J02 General nonlinear regression
62F12 Asymptotic properties of parametric estimators
62F35 Robustness and adaptive procedures (parametric inference)
62E20 Asymptotic distribution theory in statistics
Full Text: DOI

References:

[1] Anscombe, Sampling theory of the negative binomial and logarithmic series distributions, Biometrika 37 pp 358– (1950) · Zbl 0039.14202 · doi:10.1093/biomet/37.3-4.358
[2] Armitage, Studies in the variability of pock counts, J. Hygiene Camb. 55 pp 564– (1957)
[3] Breslow, Extra-Poisson variation in log-linear models, Appl. Statist. 33 pp 38– (1984)
[4] Burridge, J. (1986). Mean-variance relationships, generalized linear models and GLIM. Unpublished.
[5] Chernoff, On the distribution of the likelihood ratio, Ann. Math. Statist. 25 pp 573– (1954) · Zbl 0056.37102
[6] Collings, Testing goodness of fit for the Poisson assumptions when observations are not identically distributed, J. Amer. Statist. Assoc. 80 pp 411– (1985) · Zbl 0578.62024
[7] Cox, D.R. (1961). Tests of separate families of hypotheses. Proc. Fourth Berkeley Symp. Math. Statist. Probab., 105-123. · Zbl 0201.52102
[8] Cox, Some remarks on over-dispersion, Biometrika 70 pp 269– (1983)
[9] Crowder, Gaussian estimation for correlated binomial data, J. Roy. Statist. Soc. Ser. B 47 pp 229– (1985)
[10] Crowder, On linear and quadratic estimating functions, Biometrika 74 pp 591– (1987) · Zbl 0635.62077
[11] Dean, C., and Lawless, J.F. (1987). Testing for overdispersion in Poisson regression models. Unpublished.
[12] Efron, Double exponential families and their use in generalized linear regression, J. Amer. Statist. Assoc. 81 pp 709– (1986) · Zbl 0611.62072
[13] Engel, Models for response data showing extra-Poisson variation, Statist. Neerlandica 38 pp 159– (1984)
[14] Finney, Radioligand assay, Biometrics 32 pp 721–
[15] Firth, On the efficiency of quasi-likelihood estimation, Biometrika 74 pp 233– (1987) · Zbl 0622.62034
[16] Frome, The analysis of rates using Poisson regression models, Biometrics 39 pp 665– (1983) · Zbl 0534.62073
[17] Frome, Regression analysis of Poisson-distributed data, J. Amer. Statist. Assoc. 68 pp 935– (1973) · Zbl 0271.62086
[18] Gaver, Robust empirical Bayes analysis of event rates, Technometrics 29 pp 1– (1987) · Zbl 0611.62124
[19] Godambe, V.P., and Thompson, M.E. (1987). An extension of quasi-likelihood estimation. J. Statist. Plann. Inference, to appear. · Zbl 0681.62036
[20] Haberman, The Analysis of Frequency Data. (1974) · Zbl 0325.62017
[21] Hinde, Compound Poisson regression models pp 109– (1982) · Zbl 0505.62050
[22] Holford, The estimation of age, period and cohort effects for vital rates, Biometrics 39 pp 311– (1983)
[23] Inagaki, Asymptotic relations between the likelihood estimating function and the maximum likelihood estimator, Ann. Inst. Statist. Math. 265 pp 1– (1973) · Zbl 0332.62022
[24] Lawless, J.F. (1987). Regression methods for Poisson process data. J. Amer. Statist. Assoc., Sept. 1987. · Zbl 0657.62103
[25] Manton, A variance components approach to categorical data models with heterogeneous cell populations: Analysis of spatial gradients in lung cancer mortality rates in North Carolina counties, Biometrics 37 pp 259– (1981)
[26] Margolin, Statistical anaylsis of the Ames salmonella/microsome test, Proc. Nat. Acad. Sci. U.S.A. 76 pp 3779– (1981)
[27] McCullagh, Quasi-likelihood functions, Ann. Statist. 11 pp 59– (1983) · Zbl 0507.62025
[28] McCullagh, Generalized Linear Models. (1983) · doi:10.1007/978-1-4899-3244-0
[29] Moore, D.F. (1985). Ph.D. Thesis, Univ. of Washington, Seattle.
[30] Moore, Asymptotic properties of moment estimators for overdispersed counts and proportions, Biometrika 23 pp 583– (1986) · Zbl 0614.62031
[31] Moran, Maximum likelihood estimation in non-standard conditions, Proc. Cambridge Philos. Soc. 70 pp 441– (1971) · Zbl 0224.62013
[32] Nelder, J.A., and Pregibon, D. (1987). Quasi-likelihood and generalized linear models. Biometrika, to appear. · Zbl 0621.62078
[33] Olkin, A comparison of m estimators for the bionomial distribution, J. Amer. Statist. Assoc. 76 pp 637– (1981)
[34] Paul, Inference sensitivity for Poisson mixtures, Biometrika 65 pp 591– (1978) · Zbl 0394.62038
[35] Sichel, Asymptotic efficiencies of three methods of estimation for the inverse Gaussian-Poisson distribution, Biometrika 69 pp 467– (1982)
[36] Stirling, Iteratively reweighted least squares for models with a linear part, Appl. Statist. 33 pp 1– (1984) · Zbl 0555.62029
[37] White, Maximum likelihood estimation of misspecified models, Econometrica 50 pp 1– (1982) · Zbl 0478.62088
[38] Whittle, Gaussian estimation in stationary time series, Bull. Internat. Statist. Inst. 39 pp 1– (1961) · Zbl 0116.11403
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.