Ridge regression is used to get precise estimates by introducing some bias when the problem of multicollinearity is present in the model. Biasing parameter plays a vital role in this bias-variance tradeoff. Several methods are available in literature for the estimation of biasing parameter. For high noise and multicollinearity, available methods do not perform well in terms of their mean square error. In this article, we proposed a new estimator and compared its performance with some other available estimators. Our suggested estimator is a class of estimators based on quantile of ratio of error variance to the canonical coefficients. Based on simulation study, our proposed estimator outperforms other estimators especially when multicollinearity and error variance are high. Finally, a real-life example is used to illustrate the application of the proposed estimator.

In linear regression model, the ordinary least square and ridge regression estimators are sensitive to outliers in y-direction. In this article, we proposed two new robust quantile-based ridge and ridge m-estimators (QR and QRM) to deal with multicollinearity and outliers in y-direction. A simulation study has been conducted to compare the performance of the estimators. Based on mean square error criterion, it is shown that QR and QRM estimators outperform other considered estimators in many evaluated instances. An application is given to illustrate the performance of proposed estimators.

The ordinary least squares and ridge regression estimators in a multiple linear regression model with multicollinearity and y-direction outliers lead to unfavorable results. In order to mitigate such situation, the available literature provides few ridge M-estimators to get precise estimates. The ridge parameter, k, plays a vital role in a bias-variance tradeoff for these estimators. However, for high signal-to-noise ratio and multicollinearity with y-direction outliers, the available methods may not perform well in terms of their mean squared error. In this article, we propose a new quantile based ridge M-estimator. The new estimator gives an automated choice of quantile probability of ridge parameter according to the level of noise and multicollinearity. Based on a simulation study, the new estimator outperforms the ordinary least square estimator, ridge estimator, and other considered ridge M-estimators especially for high multicollinearity, significant error variance, and y-direction outliers. Besides normal distribution, new estimator also performs well for heavy-tailed error distribution. Finally, two real-life examples are used to illustrate the application of the proposed estimator.

Ridge regression is used to circumvent the problem of multicollinearity among predictors and many estimators for ridge parameter $k$ are available in the literature. However, if the level of collinearity among predictors is high, the existing estimators also have high mean square errors (MSE). In this paper, we consider some existing and propose new estimators for the estimation of ridge parameter k. Extensive Monte Carlo simulations as well as a real-life example are used to evaluate the performance of proposed estimators based on the MSE criterion. The results show the superiority of our proposed estimators compared to the existing estimators.

The ordinary least squared and ridge regression estimators in a linear regression model are sensitive to outliers in the y-variable. In such situations, ridge M-estimators are widely used which are robust to the y-variable outliers and overcome the multicollinearity problem. Similar to ridge regression, to lower the mean square error (MSE) of ridge M-estimators, it is crucial to select the most robust ridge parameter. The performance of existing estimators for the estimation of robust ridge parameters deteriorates when the degree of multicollinearity, error variance, and y-variable outliers increases from moderate to high. In this paper, some new robust ridge M-estimators have been proposed. The efficiency of the new estimators has been compared through a Monte Carlo simulation study. Based on the MSE criterion, the new estimators outperform existing estimators. A numerical example has been provided to illustrate the simulation results.