A New Estimator for the Gaussian Linear Regression Model with Multicollinearity


  • Issam Dawoud Department of Mathematics, Al-Aqsa University, Gaza, Palestine
  • B. M. Golam Kibria Department of Mathematics and Statistics, Florida International University, Miami, FL, USA
  • Adewale F. Lukman Department of Physical Sciences, Landmark University, Omu-Aran, Nigeria
  • Onifade C. Olufemi Department of Statistics, University of Abuja, Abuja, Nigeria


Liu estimator, mean squared error, proposed two-parameter estimator, real-life dataset application, ridge estimator, simulation


In this study, a new two-parameter biased estimator for estimating the parameter of a multicollinearity linear regression model is developed. We compared the performance of the proposed estimator with some existing estimators in terms of the mean squared error matrix. The theoretical comparisons and simulation results revealed that the proposed two-parameter estimator dominate other estimators under certain conditions using the mean squared error. The simulation result shows that the ordinary least squares estimator has the lowest performance. Among the one-parameter estimator, the ridge regression estimator dominates the Liu estimator. However, the proposed method dominates both the one-parameter and two-parameter estimators. We also discuss the estimations of the biasing parameters. A famous real-life dataset is analyzed to support both theoretical and simulation results. 


Akdeniz F, Kaçiranlar S. On the almost unbiased generalized Liu estimator and unbiased estimation of the bias and MSE. Commun Stat - Theory Methods. 1995; 24(7): 1789-1797.

Aladeitan BB, Adebimpe O, Lukman AF, Oludoun O, Abiodun OE. Modified Kibria-Lukman (MKL) estimator for the Poisson regression model: application and simulation. F1000Research. 2021; 10, 548. https://doi.org/10.12688/f1000research.53987.1.

Amin M, Akram MN, Majid A. On the estimation of Bell regression model using ridge estimator. Commun Stat - Simul Comput. 2023; 52(3): 854-867.

Arashi M, Roozbeh M, Hamzah NA, Gasparini M. Ridge regression and its applications in genetic studies. Plos One. 2021; 16(4): e0245376.

Dawoud I, Kibria BMG. A new biased estimator to combat the multicollinearity of the Gaussian linear regression model. Stats. 2020a; 3(4): 526-541.

Dawoud I, Kibria BMG. A new two parameter biased estimator for the unrestricted linear regression model: Theory, Simulation and Application. Int J Clin Biostat Biom. 2020b; 6(2): 1-10.

Eledum H, Alkhalifa AA. Generalized two stages ridge regression estimator for multicollinearity and autocorrelated errors. Canadian Journal on Science and Engineering Mathematics. 2012; 3(3): 79-85.

Eledum H, Zahri M. Relaxation method for two stages ridge regression estimator. Int J Pure Appl Math. 2013; 85(4): 653-667.

Farebrother RW. Further results on the mean square error of Ridge regression. J R Stat Soc. 1976; 38: 248-250.

Gibbons DG. A simulation study of some ridge estimators. J Am Stat Assoc. 1981; 76(373): 131–139.

Hoerl AE, Kennard RW. Ridge regression: biased estimation for nonorthogonal problems. Technometrics. 1970;12(1): 55-67.

Hoerl AE, Kannard RW, Baldwin KF. Ridge regression: some simulations. Commun Stat. 1975; 4(2): 105-123.

Khalaf G, Shukur G. Choosing ridge parameter for regression problems. Commun Stat - Theory Methods. 2005; 34(5): 1177-1182.

Kibria BMG. Performance of some new ridge regression estimators. Commun Stat Simul Comput. 2003; 32(2): 419-435.

Kibria BMG, Banik S. Some ridge regression estimators and their performances. J Mod Appl Stat Methods. 2016; 15(1): 206-238.

Kibria BMG, Lukman AF. A new Ridge-Type estimator for the linear regression model: simulations and applications. Scientifica. 2020; Article ID 9758378, 16 pages.

Liu K. A new class of biased estimate in linear regression. Commun Stat - Theory Methods. 1993; 22: 393-402.

Liu K. Using Liu-type estimator to combat collinearity. Commun Stat - Theory Methods. 2003; 32(5): 1009-1020.

Li Y, Yang H. Anew Liu-type estimator in linear regression model. Stat Pap. 2012; 53(2): 427-437.

Lukman AF, Osowole OI, Ayinde K. Two stage robust Ridge method in a linear regression model. J Mod Appl Stat Methods. 2015; 14(2): 53-67.

Lukman AF, Ayinde K. Review and classifications of the ridge parameter estimation techniques. Hacet J Math Stat. 2017; 46(5): 953-967.

Lukman AF, Ayinde K, Ajiboye SA. Monte-Carlo study of some classification-based Ridge parameter estimators. J Mod Appl Stat Methods. 2017; 16(1): 428-451.

Lukman AF, Olatunji A. Newly proposed biased Ridge estimator: an application to the Nigerian Economy. Pakistan J Stat. 2018; 34(2), 91-98.

Lukman AF, Ayinde K, Binuomote S, Clement OA. Modified ridge-type estimator to combat multicollinearity: application to chemical data. J Chemom. 2019a; 33(5): e3125.

Lukman AF, Ayinde K, Kun SS, Adewuyi E. A modified new two-parameter estimator in a linear regression model. Model Simul Eng. 2019b, https://doi.org/10.1155/2019/6342702.

Lukman AF, Kibria BMG, Ayinde K, Jegede SL. Modified one-parameter Liu estimator for the linear regression model. Model Simul Eng. 2020, https://doi.org/10.1155/2020/9574304.

Månsson K, Kibria BMG, Shukur G. Performance of some weighted Liu estimators for logit regression model: an application to Swedish accident data. Commun Stat - Theory Methods. 2015;44(2): 363-375.

Mayer LS, Willke TA. On biased estimation in linear models. Technometrics. 1973; 15(3): 497-508.

Newhouse JP, Oman SD. An evaluation of ridge estimators. A report prepared for United States air force project RAND. 1971.

Ozkale MR, Kaçiranlar S. The restricted and unrestricted two-parameter estimators. Commun Stat - Theory Methods. 2007; 36(15): 2707-2725.

Qasim M, Amin M, Omer T. Performance of some new Liu parameters for the linear regression model. Commun Stat - Theory Methods. 2020; 49(17): 4178-4196.

Qasim M, Månsson K, Sjölander P, Kibria BMG. A new class of efficient and debiased two-step shrinkage estimators: method and application. J Appl Stat. 2022; 49(16): 4181-4205.

Roozbeh M. Optimal QR-based estimation in partially linear regression models with correlated errors using GCV criterion. Comput Stat Data Anal. 2018; 117: 45-61.

Sakallıoglu S, Kaçıranlar S. A new biased estimator based on ridge estimation. Stat Pap. 2008; 49(4): 669-689.

Suhail M, Chand S, Kibria BMG. Quantile-based robust ridge m-estimator for linear regression model in presence of multicollinearity and outliers. Commun Stat Simul Comput. 2021; 50(11): 3194-3206.

Stein C. Inadmissibility of the usual estimator for mean of multivariate normal distribution. In Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1956; 19541955, Vol. I, 197-206. Berkeley: University of California Press.

Swindel BF. Good ridge estimators based on prior information. Commun Stat - Theory Methods. 1976; 5(11):1065-1075.

Trenkler G, Toutenburg H. Mean squared error matrix comparisons between biased estimators-an overview of recent results. Stat Pap. 1990; 31(1): 165-179.

Ugwuowo FI, Oranye EH, Arum KC. On the Jackknife Kibria-Lukman estimator for the linear regression model. Commun Stat Simul Comput. 2021; https://doi.org/10.1080/03610918.2021.

Yang H, Chang X. A new two-parameter estimator in linear regression. Commun Stat - Theory Methods. 2010; 39(6): 923-934.




How to Cite

Dawoud, I. ., M. Golam Kibria, B. ., F. Lukman, A. ., & C. Olufemi, O. . (2023). A New Estimator for the Gaussian Linear Regression Model with Multicollinearity. Thailand Statistician, 21(4), 910–925. Retrieved from https://ph02.tci-thaijo.org/index.php/thaistat/article/view/251068