Bias reduction in exponential family nonlinear models
Kosmidis, Ioannis and Firth, D. (David). (2009) Bias reduction in exponential family nonlinear models. Biometrika, Vol.96 (No.4). pp. 793-804. ISSN 0006-3444Full text not available from this repository.
Official URL: http://dx.doi.org/10.1093/biomet/asp055
In Firth (1993, Biometrika) it was shown how the leading term in the asymptotic bias of the maximum likelihood estimator is removed by adjusting the score vector, and that in canonical-link generalized linear models the method is equivalent to maximizing a penalized likelihood that is easily implemented via iterative adjustment of the data. Here a more general family of bias-reducing adjustments is developed for a broad class of univariate and multivariate generalized nonlinear models. The resulting formulae for the adjusted score vector are computationally convenient, and in univariate models they directly suggest implementation through an iterative scheme of data adjustment. For generalized linear models a necessary and sufficient condition is given for the existence of a penalized likelihood interpretation of the method. An illustrative application to the Goodman row-column association model shows how the computational simplicity and statistical benefits of bias reduction extend beyond generalized linear models.
|Item Type:||Journal Article|
|Subjects:||Q Science > QH Natural history > QH301 Biology
Q Science > QA Mathematics
|Divisions:||Faculty of Science > Statistics|
|Library of Congress Subject Headings (LCSH):||Asymptotic expansions, Nonlinear theories, Multivariate analysis, Linear models (Statistics), Estimation theory|
|Journal or Publication Title:||Biometrika|
|Number of Pages:||12|
|Page Range:||pp. 793-804|
|Funder:||Engineering and Physical Sciences Research Council (EPSRC), Economic and Social Research Council (Great Britain) (ESRC)|
|References:||BULL, S. B., LEWINGER, J. B. & LEE, S. S. F. (2007). Confidence intervals for multinomial logistic regression in sparse data. Statist. Med. 26, 903–18. BULL, S. B.,MAK, C.&GREENWOOD, C. (2002).Amodified score function estimator for multinomial logistic regression in small samples. Comp. Statist. Data Anal. 39, 57–74. COOK, R. D., TSAI, C.-L. & WEI, B. C. (1986). Bias in nonlinear regression. Biometrika 73, 615–23. CORDEIRO, G. M. & MCCULLAGH, P. (1991). Bias correction in generalized linear models. J. R. Statist. Soc. B 53, 629–43. CRAM´ER, H. (1946). Mathematical Methods of Statistics. Princeton, NJ: Princeton University Press. FAHRMEIR, L. & TUTZ, G. (2001). Multivariate Statistical Modelling Based on Generalized Linear Models. NewYork: Springer. FIRTH, D. (1992a). Bias reduction, the Jeffreys prior and GLIM. In Advances in GLIM and Statistical Modelling: Proc. GLIM 92 Conf., Ed. L. Fahrmeir, B. Francis, R. Gilchrist & G. Tutz, pp. 91–100. New York: Springer. FIRTH, D. (1992b). Generalized linear models and Jeffreys priors: an iterative generalized least-squares approach. In Computational Statistics I, Ed. Y. Dodge and J. Whittaker, pp. 553–7. Heidelberg: Physica. FIRTH, D. (1993). Bias reduction of maximum likelihood estimates. Biometrika 80, 27–38. GOODMAN, L. A. (1979). Simple models for the analysis of association in cross-classifications having ordered categories. J. Am. Statist. Assoc. 74, 537–52. GOODMAN, L. A. (1981). Association models and canonical correlation in the analysis of cross-classifications having ordered categories. J. Am. Statist. Assoc. 76, 320–34. GOODMAN, L. A. (1985). The analysis of cross-classified data having ordered and/or unordered categories: association models, correlation models, and asymmetry models for contingency tables with or without missing entries. Ann. Statist. 13, 10–69. HEINZE, G. & SCHEMPER, M. (2002). A solution to the problem of separation in logistic regression. Statist. Med. 21, 2409–19. HEINZE, G. & SCHEMPER, M. (2004). A solution to the problem of monotone likelihood in Cox regression. Biometrics 57, 114–9. JEFFREYS, H. (1946). An invariant form for the prior probability in estimation problems. Proc. R. Soc. Lond. 186, 453–61. MAGNUS, J. R.&NEUDECKER,H. (1999). Matrix Differential Calculus with Applications in Statistics and Econometrics. Chichester: Wiley. MCCULLAGH, P. & NELDER, J. (1989). Generalized Linear Models, 2nd ed. London: Chapman and Hall. MEHRABI, Y. & MATTHEWS, J. N. S. (1995). Likelihood-based methods for bias reduction in limiting dilution assays. Biometrics 51, 1543–9. PETTITT, A. N., KELLY, J. M. & GAO, J. T. (1998). Bias correction for censored data with exponential lifetimes. Statist. Sinica 8, 941–64. SARTORI, N. (2006). Bias prevention of maximum likelihood estimates for scalar skew normal and skew t distributions. J. Statist. Plan. Infer. 136, 4259–75. WEI, B. (1997). Exponential Family Nonlinear Models. New York: Springer. ZORN, C. (2005). A solution to separation in binary response models. Polit. Anal. 13, 157–70.|
Actions (login required)