Robust Liu estimator for regression based on an M-estimator

Abstract

Consider the regression model y = beta 0 1 + Xbeta + epsilon. Recently, the Liu estimator, which is an alternative biased estimator beta L (d) = (X'X + I) -1 (X'X + dI)beta OLS , where 0>d>1 is a parameter, has been proposed to overcome multicollinearity . The advantage of beta L (d) over the ridge estimator beta R (k) is that beta L (d) is a linear function of d. Therefore, it is easier to choose d than to choose k in the ridge estimator. However, beta L (d) is obtained by shrinking the ordinary least squares (OLS) estimator using the matrix (X'X + I) -1 (X'X + dI) so that the presence of outliers in the y direction may affect the beta L (d) estimator. To cope with this combined problem of multicollinearity and outliers, we propose an alternative class of Liu-type M-estimators (LM-estimators) obtained by shrinking an M-estimator beta M , instead of the OLS estimator using the matrix (X'X + I) -1 (X'X + dI).

Adaptive Unified Biased Estimators of Parameters in Linear Model

“…However the least squares estimators are not robust for estimating β especially with multicollinearity and ill-conditioned design matrix. This problem leads to the development of the Stein [17] estimator, the ridge regression estimator [9] and the principal components estimator (see [7,16,22]), Panopoulos [15] did some comparison among several ridge estimators, Donatos and Michailidis [5,6] studied some small sample properties of ridge estimators and made a comparison with the least squares estimator, Choi and Hall [4] used the idea of ridge estimator in dealing with density estimation, Arslan and Billor [1] as well as Arnold and Stahlecker [3] investigated the ridge type estimators, Fu [8] further studied ridge estimator and applied it to a real data analysis, Inoue [10,11] studied the relative efficiency of double f -class generalized ridge and some related ridge estimators. For principal components estimators, Lin and Wei [13] studied the small sample properties of the principal components and Walker [20] Manuscript received January 20, 2003 investigated the influence diagnostics for fractional principal components estimators.…”

Robust Winsorized Shrinkage Estimators for Linear Regression Model

“…This data set contains 30 observations and four regressor variables namely X 1 , X 2 , X 3 , and X 4 . Arslan and Billor (2000) noted that the tobacco blends data suffers from the problem of multicollinearity and outliers simultaneously. The variance inflation factor (VIF) values for each term are 324.1412, 45.1728, 173.2577 and 138.1753.…”

Inferences Based on Robust Regression Estimators When There Is Multicolinearity

“…Numerous methods have been proposed and compared regarding how this issue might be addressed (e.g., Adegoke et al 2016;Arslan Billor, 2000;Ertaa et al, 2017;Kan et al, 2013;Lukman et al, 2014;Samkar & Alpu, 2010;Kan et al, 2013). The focus has been on minimizing mean squared error.…”