Two parameter Ridge estimator in the inverse Gaussian regression model

Abstract

It is well known that multicollinearity, which occurs among the explanatory variables, has adverse effects on the maximum likelihood estimator in the inverse Gaussian regression model. Biased estimators are proposed to cope with the multicollinearity problem in the inverse Gaussian regression model. The main interest of this article is to introduce a new biased estimator. Also, we compare newly proposed estimator with the other estimators given in the literature. We conduct a Monte Carlo simulation and provide a real data example to illustrate the performance of the proposed estimator over the maximum likelihood and Ridge estimators. As a result of the simulation study and real data example, the newly proposed estimator is superior to the other estimators used in this study.

Keywords

• Inverse Gaussian Regression
• Biased estimator
• Two parameter Ridge estimator
• Multicollinearity

References

1. [1] M.N. Akram, M. Amin and M. Qasim, A new Liu-type estimator for the inverse Gaussian regression model, J. Stat. Comput. Simul. 90 (7), 1153–1172, 2020.
2. [2] Z.Y. Algamal, Performance of Ridge estimator in inverse Gaussian regression model, Comm. Statist. Theory Methods 48 (15), 3836–3849, 2019.
3. [3] M. Amin, M. Qasim, S. Afzal and K. Naveed, New Ridge estimators in the inverse Gaussian regression model: Monte Carlo simulation and application to chemical data, Comm. Statist. Simulation Comput.,doi: 10.1080/03610918.2020.1797794, 2020.
4. [4] M. Amin, M. Qasim and M. Amanullah, Performance of Asar and Genç and Huang and Yang’s two-parameter estimation methods for the gamma regression model, Iran. J. Sci. Technol. Trans. A Sci. 43, 2951–2963, 2019.
5. [5] Y. Asar, Liu Type Logistic Estimators, PhD thesis, Institute of Science, Selcuk University, Konya, Turkey, 2015.
6. [6] Y. Asar and A. Genç, Two-parameter Ridge estimator in the binary logistic regression, Comm. Statist. Simulation Comput. 46 (9), 7088–7099, 2017.
7. [7] K.A. Brownlee, Statistical Theory and Methodology in Science and Engineering, Wiley, New York, 1965.
8. [8] R.S. Chhikara and J.L. Folks, The Inverse Gaussian Distribution: Theory, Methodology and Applications, Marcel Dekker, New York, 1989.

9. [9] H. Ertaş, S. Toker and S. Kaçıranlar,Robust two parameter Ridge M-estimator for linear regression, J. Appl. Stat. 42 (7), 1490–1502, 2015.
10. [10] R.W. Farebrother, Further results on the mean square error of Ridge regression, J. R. Stat. Soc. Ser. B. Stat. Methodol. 38, 248–250, 1976.
11. [11] A.E. Hoerl and R.W. Kennard, Ridge regression: biased estimation for nonorthogonal problems, Technometrics 12 (1), 55–67, 1970.
12. [12] J. Huang and H. Yang, A two-parameter estimator in the negative binomial regression model, Comm. Statist. Simulation Comput. 84 (1), 124–134, 2014.
13. [13] S. Lipovetsky, Two parameter Ridge regression and its convergence o the eventual pairwise model, Math Comput Model 44, 304–318, 2006.
14. [14] S. Lipovetsky and W.M. Conklin, Ridge regression in two-parameter solution, Appl. Stoch. Models Bus. Ind. 21 (6), 525–540, 2005.
15. [15] G.C. McDonald and D.I. Galarneau, A Monte Carlo evaluation of some Ridge-type estimators, J. Amer. Statist. Assoc. 70 (350), 407–416, 1975.
16. [16] K. Naveed, M. Amin, S. Afzal and M. Qasim, New shrinkage parameters for the inverse Gaussian Liu regression, Comm. Statist. Theory Methods, doi: 10.1080/03610926.2020.1791339, 2020.
17. [17] M.R. Özkale and S. Kaçıranlar, The restricted and unrestricted two-parameter estimators, Comm. Statist. Theory Methods 36, 2707–2725, 2007.
18. [18] A. Punzo,A new look at the inverse Gaussian distribution with applications to insurance and economic data, J. Appl. Stat. 46 (7), 1260–1287, 2019.
19. [19] V. Seshadri, The Inverse Gaussian Distribution: Statistical Theory and Applications, Springer, New York, Volume 137 of Notes in Statistics, 2012.
20. [20] S. Toker, Investigating the two parameter analysis of Lipovetsky for simultaneous systems, Statist. Papers 61, 2059–2089, 2020.
21. [21] S. Toker and S. Kaçıranlar, On the performance of two parameter ridge estimator under the mean square error criterion, Appl. Math. Comput. 219, 4718–4728, 2013.
22. [22] S. Toker, G. Şiray and M.Qasim, Developing a first order two parameter estimator for the generalized linear models, 11th International Statistics Congress (ISC2019), Muğla, Turkey, 2019.
23. [23] G. Trenkler and H. Toutenburg, Mean squared error matrix comparisons between biased estimators-an overwiev of recent results, Statist. Papers 31, 165–179, 1990.
24. [24] M.C.K. Tweedie,Statistical properties of inverse Gaussian distributions, I, Ann. Math. Statist. 28 (2), 362–377, 1957