Robust Liu estimator for regression based on an M-estimator

Consider the regression model y = beta(0)1 + X beta + epsilon. Recently, the Liu estimator, which is an alternative biased estimator <(beta)over cap>(L)(d) = (X'X + I)(-1)(X'X + dI)<(beta)over cap>(OLS), where 0 < d < 1 is a parameter, has been proposed to overcome multicollinearity. The advantage of <(beta)over cap>(L)(d) over the ridge estimator <(beta)over cap>(R)(k), is that <(beta)over cap>(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)over cap>(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)over cap>(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)over cap>(M), instead of the OLS estimator using the matrix (X'X + I)(-1)(X'X + dI).