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The methods of two-parameter ridge and ordinary ridge regression are very sensitive to the presence of the joint problem of multicollinearity and outliers in the y-direction. To overcome this problem, modified robust ridge M-estimators are proposed. The new estimators are then compared with the existing ones by means of extensive Monte Carlo simulations. According to mean squared error (MSE) criterion, the new estimators outperform the least square estimator, ridge regression estimator, and two-parameter ridge estimator in many considered scenarios. Two numerical examples are also presented to illustrate the simulation results.

In multiple linear regression models, the multicollinearity problem mostly occurs when the explanatory variables are correlated among each other. It is well known that when the multicollinearity exists, the variance of the ordinary least square estimator is unstable. As a remedy, Liu in [1] developed a new method of estimation with biasing parameter d. In this paper, we have introduced a new method to estimate the biasing parameter in order to mitigate the problem of multicollinearity. The proposed method provides the class of estimators that are based on quantile of the regression coefficients. The performance of the new estimators is compared with the existing estimators through Monte Carlo simulation, where mean squared error and mean absolute error are considered as evaluation criteria of the estimators. Portland cement and US Crime data is used as an application to illustrate the benefit of the new estimators. Based on simulation and numerical study, it is concluded that the new estimators outperform the existing estimators in certain situations including high and severe cases of multicollinearity. 95% mean prediction interval of all the estimators is also computed for the Portland cement data. We recommend the use of new method to practitioners when the problem of high multicollinearity exists among the explanatory variables.

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Two-parameter ridge regression is a widely used method in the last two decades to circumvent the problem of multicollinearity. Ridge parameter k plays an important role in such situations. Several methods are available in the literature for the estimation of ridge parameter. For high multicollinearity, the available methods do not perform well in terms of mean square error. In this article, we propose some new estimators for the ridge parameter. Based on simulation study, our new estimators generally perform better than ordinary least squares estimator, ridge regression estimator and two-parameter ridge regression estimator in many considered scenarios especially for high multicollinearity. In addition, the new estimators also perform well for some non-normal error distributions. Finally, two real-life examples are used to illustrate the application of the proposed estimator.

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