Robust Liu estimator for regression based on an M-estimator

Consider the regression model y = ß01 + Xß + ?. Recently, the Liu estimator, which is an alternative biased estimator ß^L(d) = (X'X + I)-1 (X'X + dI)ß^OLS, where 0 < d < 1 is a parameter, has been proposed to overcome multicollinearity. The advantage of ß^L(d) over the ridge estimator ß^R(k) is that ß^L(d) is a linear function of d. Therefore, it is easier to choose d than to choose k in the ridge estimator. However, ß^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 ß^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 ß^M, instead of the OLS estimator using the matrix (X'X + I)-1 (X'X + dI)