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.

In the presence of multicollinearity, the ordinary least squares (OLS) estimators, aside from BLUE (best linear unbiased estimator), lose efficiency and fail to achieve minimum variance. In addition, these estimators are highly sensitive to outliers in the response direction. To overcome these limitations, robust estimation techniques are often integrated with shrinkage methods. This study proposes a new class of Kibria Ridge M-estimators specifically developed to simultaneously address multicollinearity and outlier contamination. A comprehensive Monte Carlo simulation study is conducted to evaluate the performance of the proposed and existing estimators. Based on the mean squared error criterion, the proposed Kibria Ridge M-estimators consistently outperform the traditional ridge-type estimators under varying parameter settings. Furthermore, the practical applicability and superiority of the proposed estimators are validated using the Tobacco and Anthropometric datasets. Overall, the new proposed estimators demonstrate good performance, offering robust and efficient alternatives for regression modeling in the presence of multicollinearity and outliers.

This paper addresses new exponential estimators for population mean in case of non-response on both the study and the concomitant variables using simple random sampling. The expressions for theoretical bias and mean square error of new estimators are derived up to first-order approximation and comparisons are made with the existing estimators. The proposed estimators are observed more efficient as compared to the considered estimators in the literature. For instance, the classical [4] unbiased estimator, the estimator of [9], and other existing estimators under the explained conditions. The theoretical results are supported numerically by using real-life data sets, under the criteria of bias, mean square error, percent relative efficiency and mathematical conditions. It is also clear from the numerical results that the suggested exponential estimators performed better than the estimators in the literature.