Robust Liu-Type Estimator for SUR Model

  • Tarek Omara Kafrelsheikh University
Keywords: Multicollinearity, outliers, Liu-type estimator, S estimator, MM estimator, SUR model.


The Liu-type estimator is one of the shrink estimators that is used to remedy for a problem of multicollinearityin SUR model, but it is sensitive to the outlier. In this paper, we introduce the S Liu-type (SLiu-type) and MM Liu-type estimator (MMLiu-type) for SUR model. These estimators merge Liu-type estimator with S-estimator and with MM-estimator which makes it have high robustness at the different level of efficiency and at the same time prevents the bad effects of multicollinearity. Moreover, to get more robust features, we have modified the Liu-type estimator by making it depend on MM estimator instead of GLS estimator. The asymptotical properties for the suggested estimator were discussed and we used the fast and robust bootstrap (FRB) to obtain the suggested robust estimators. Furthermore, we run the simulation study to show the extent of excellence for the suggested robust estimators relative to the other estimators by many factors.


M. Alkhamisi , G. Khalaf , and G. Shukur , Some Modifications for Choosing Ridge Parameters, Communications in Statistics-Theory and Methods, Vol.35, 2006.

O. Arslan and N. Billor , Robust Liu estimator for regression based on M-estimator, Journal of applied statistics, Vol.27, 2000.

K. Betl , A. Ozlem and Y. Berna , Robust ridge and robust Liu estimator for regression based on the LTS estimator, Journal of Applied Statistics, Vol.40, No.3,2013.

M. Bilodeau and P. Duchesne , Robust estimation of the SUR model , The Canadian Journal of Statistics, Vol. 28,2000.

A. El- Houssainy , M. Sayed, A. Alaa, A. Naglaa and M Tarek , Robust cross validation in SUR ridge estimators and SUR robust ridge estimators, Journal of statistic theory and application, Vol.10, No.1,2011.

L. Firinguetti , Ridge Regression in the Context of a System of Seemingly Unrelated Regression Equations, Journal of Statistics Computation and Simulation, Vol. 56, 1997.

J. Huber, Robust Statistics, Wiley, New York, 1991.

W. Jibo, Improved Liu-Type estimator of parameters in two seemingly unrelated regressions, International scholarly research Notices, Vol. 2014, Article ID 679835, 2014.

B. Kibria and M. Golam , Performance of Some New Ridge Regression Estimators, Communication in Statistics- Simulation and Computation, vol.32, No.2, 2003.

R.Koenker and S. Portnoy, M-estimation of multivariate regressions, Journal of the American Statistical Association, Vol. 85, 1990.

P. Krist and V. Stefan, Robust Inference for Seemingly Unrelated Regression Models, Journal of Multivariate Analysis, vol. 167, 2018.

Liu K. Using Liu-type estimator to combat collinearity , Communications in Statistics - Theory and Methods, vol.32, No.5., 2003.

H. Lopuhaa, On the relation between S-estimators and M-estimators of multivariate location and covariance, The Annals of Statistics, vol.17, 1989.

R. Maronna , Robust ridge regression for high-dimensional Data, Technometrics, Vo.53, 2011.

J. Rousseeuw and V. Yohai , Robust regression by means of S-estimators. In Robust and Nonlinear Time Series Analysis, Springer, New York, 1984.

J. Rousseeuw , Least median of squares regression, journal of the American statistical association, Vo.79, 1984.

M. Salibian-Barrera, S. Van and G. Willems, Fast and robust bootstrap, Statistical Methods and Applications, Vol. 17, 2008.

G. Shukur and Z. Zeebari , Median regression for SUR models with the same explanatory variables in each equation, Journal of Applied Statistics, Vol. 39, No.8, 2012.

V. Srivastava and D. Giles , Seemingly Unrelated Regression Equations Models: Estimation and Inference Dekker,1987.

O. Tarek , MM and MM Ridge Estimators for SUR Model, International Journal of Statistics and Applications, Vol.7, No.1, 2017.

O. Tarek , Stochastic Restricted Liu Type estimator for SUR model, Pakistan Journal of Statistics and Operation Research, vol. 14, no. 4, 2018.

O. Tarek, Weight LAD and Weight LAD Ridge Estimator for Seemingly unrelated regression Models, Advances and Applications in Statistics, Vo52, No.6., 2018.

A. Yasin A. and G. Asir , New Shrinkage Parameters for the Liu-type Logistic Estimators, Journal Communications in Statistics -Simulation and Computation, vol. 45, no.3, 2016.

V. Yohai and R. Zamar , High breakdown-point of estimates of regression by means of the minimization of an efficient scale, Journal of the American Statistical Association, Vol.83, 1988.

V. Yohai , High breakdown point and high efficiency robust estimates for regression, The Annals of Statistics, Vol. 15 , 1987.

J. Yoonsuh, Multiple predicting K-fold cross-validation for model selection, Journal of Nonparametric Statistics, Vo.30, 2108.

A. Zellner, An Efficient Method of Estimating Seemingly Unrelated Regressions and Tests for Aggregation Bias, Journal of American Statistics. Assoc., Vol. 5, 1962.

How to Cite
Omara, T. (2021). Robust Liu-Type Estimator for SUR Model. Statistics, Optimization & Information Computing, 9(3), 607-617.
Research Articles