Restricted Liu estimator in generalized linear models: Monte Carlo simulation studies on gamma and Poisson distributed responses

Year 2019, Volume: 48 Issue: 4, 1250 - 1276, 08.08.2019

Fikriye Kurtoğlu , M. Revan Özkale

Abstract

In this study, we introduce iterative restricted Liu estimator to combat multicollinearity in generalized linear models. We also obtain necessary and sufficient conditions for the superiority of the first-order approximated restricted Liu estimator over the first-order approximated maximum likelihood and Liu estimators by the approximated mean squared error criterion. The results are illustrated by conducting simulation studies and numerical examples.

Keywords

Restricted Liu estimation, Multicollinearity, Generalized linear models, Mean squared error, Gamma distribution, Poisson distribution

References

• [1] D. A. Belsley, E. Kuh and R. E. Welsch. Regression diagnostics:Identifying influential data and sources of collinearity, John Wiley and Sons, 1980.
• [2] A. Ben-Israel and T. N. E. Greville. Generalized inverses: Theory and applications, John Wiley and Sons, 1974.
• [3] S. Chatterjee and A. S. Hadi. Sensitivity analysis in linear regression, John Wiley and Sons, 1988.
• [4] F. A. Graybill. Theory and application of the linear model, Duxbury Press, 1976.
• [5] J. W. Hardin and J. M. Hilbe. Generalized linear models and extensions, Stata Press, 2012.
• [6] D. A. Harville. Matrix algebra from a statistician’s perspective, Springer, 2008.
• [7] S. Kaçıranlar, S. Sakallıoğlu, F. Akdeniz, G. P. Styan and H. J. Werner. A new biased estimator in linear regression and a detailed analysis of the widely-analysed dataset on Portland cement, Sankhya: The Indian J Stat Ser B. 61, 443-459, 1999.
• [8] F. Kurtoğlu and M. R. Özkale. Liu estimation in generalized linear models: application on gamma distributed response variable, Statistical Papers 57(4), 911-928, 2016.
• [9] K. Liu. A new class of biased estimate in linear regression, Comm Statist Theory Methods 22(2), 393-402, 1993.
• [10] M. J. Mackinnon and M. L. Puterman. Collinearity in generalized linear models, Comm Statist Theory Methods 18(9), 3463-3472, 1989.
• [11] B. D. Marx. A continuum of principal component generalized linear regressions, Comput Statist Data Analysis 13, 385-393, 1992.
• [12] B. D. Marx and E. P. Smith. Principal component estimation for generalized linear regression, Biometrika 77(1), 23-31, 1990.
• [13] G. C. McDonald and D. I. Galarneau. A monte carlo evaluation of ridge-type estimators, J Amer Statist Assoc. 70, 407-416, 1975.
• [14] R. H. Myers. Classical and modern regression with applications, Duxbury Press, 1990.
• [15] H. Nyquist. Restricted estimation of generalized linear models, J Roy Statist Soc Ser C. 40(1), 133-141, 1991.
• [16] M. R. Özkale. The relative efficiency of the restricted estimators in linear regression models, J App Statist. 41(5), 998-1027, 2014.
• [17] B. Segerstedt. On ordinary ridge regression in generalized linear models, Commun Statist Theory Methods 21(8), 2227-2246, 1992.
• [18] G. Trenkler and H. Toutenburg. Mean squared error matrix comparisons between biased estimators an overview of recent results, Statistical Papers 31(1), 165-179, 1990.