×

On approximate likelihood inference in a Poisson mixed model. (English) Zbl 0902.62007

Summary: A two-step estimation approach is proposed for the fixed-effect parameters, random effects and their variance \(\sigma^2\) of a Poisson mixed model. In the first step, it is proposed to construct a small \(\sigma^2\)-based approximate likelihood function of the data and utilize this function to estimate the fixed-effect parameters and \(\sigma^2\). In the second step, the random effects are estimated by minimizing their posterior mean squared error. Methods of M. A. Waclawiw and K.-Y. Liang [J. Am. Stat. Assoc. 88, No. 421, 171-178 (1993)] based on so-called Stein-type estimating functions of N. E. Breslow and D. G. Clayton [ibid., 9-25 (1993)] based on penalized quasilikelihood are compared with the proposed likelihood method. The results of a simulation study on the performance of all three approaches are reported.

MSC:

62A01 Foundations and philosophical topics in statistics
62F10 Point estimation
62F12 Asymptotic properties of parametric estimators
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Billingsley, Probability and Measure. (1979)
[2] Breslow, Approximate inference in generalized linear mixed models, J. Amer. Statist. Assoc. 88 pp 9– (1993) · Zbl 0775.62195
[3] Breslow, Bias correction in generalized linear models with a single component of dispersion, Biometrika 82 pp 81– (1995) · Zbl 0823.62059
[4] Dempster, Maximum likelihood from incomplete observations, J. Roy. Statist. Soc. Ser. B 39 pp 1– (1977)
[5] Ferreira, Estimating equations in the présence of prior knowledge, Biometrika 69 pp 667– (1982)
[6] Godambe, An optimum property of regular maximum likelihood estimation, Ann. Math. Statist. 31 pp 1208– (1960) · Zbl 0118.34301
[7] Godambe, Estimating Functions pp 3– (1991)
[8] Green, Penalized likelihood for general semi-parametric regression models, Internat. Statist. Rev. 55 pp 245– (1987) · Zbl 0636.62068
[9] Lee, Hierarchical generalized linear models, J. Roy. Statist. Soc. Ser. B 58 pp 619– (1996) · Zbl 0880.62076
[10] Liang, Extension of the Stein procedure through the use of estimating functions, J. Amer. Statist. Assoc. 85 pp 435– (1990) · Zbl 0703.62030
[11] Liang, Longitudinal data analysis using generalized linear models, Biometrika 73 pp 13– (1986) · Zbl 0595.62110
[12] Morton, A generalized linear model with nested strata of extra-Poisson variation, Biometrika 74 pp 247– (1987) · Zbl 0621.62075
[13] Patterson, Recovery of interblock information when block sizes are unequal, Biometrika 58 pp 545– (1974)
[14] Puri, Nonparametric Methods in Multivariate Analysis. (1971) · Zbl 0237.62033
[15] Seber, Nonlinear Regression. (1989)
[16] Waclawiw, Prediction of random effects in the generalized linear model, J. Amer. Statist. Assoc. 88 pp 171– (1993) · Zbl 0775.62198
[17] Zeger, Generalized linear models with random effects: a gibbs sampling approach, J. Amer. Statist. Assoc. 86 pp 79– (1991)
[18] Zeger, Models for longitudinal data: A generalized estimating equation approach, Biometrics 44 pp 1049– (1988) · Zbl 0715.62136
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.