Problem statement: Many regression estimators have been used to remedy multicollinearity problem. The ridge estimator has been the most popular one. However, the obtained estimate is biased. Approach: In this stuyd, we introduce an alternative shrinkage estimator, called modified unbiased ridge (MUR) estimator for coping with multicollinearity problem. This estimator is obtained from Unbiased Ridge Regression (URR) in the same way that Ordinary Ridge Regression (ORR) is obtained from Ordinary Least Squares (OLS). Properties of MUR estimator are derived. Results: The empirical study indicated that the MUR estimator is more efficient and more reliable than other estimators based on Matrix Mean Squared Error (MMSE).Conclusion: In order to solve the multicollinearity problem, the MUR estimator was recommended.

The multicollinearity in multiple linear regression models and the existence of influential data points are common problems. These problems exert undesirable effects on the least squares estimators. So, it is very important to introduce some alternative biased estimators of the robust ridge regression to overcome the influence of these problems simultaneously. In this paper, alternative biased robust regression estimator is defined by mixing the ridge estimation technique into the robust least median squares estimation to obtain the Ridge Least Median Squares (RLMS). The efficiency of the combined estimator (RLMS) is compared with some existing regression estimators, which namely, the Ordinary Least Squares (LS); Ridge Regression (RR) and Ridge Least Absolute Deviation (RLAD). The numerical results of this study show that, the RLMS regression estimator is more efficient than other estimators, based on, Bias and mean squared error criteria for many combinations of influential data points and degree of multicollinearity.

In regression analysis, many statistical tests have been proposed to find out whether the error term is normally distributed or not. These statistical tests are typically constructed using OLS residuals. However, since diagnostic tests for normality are very sensitive to outliers, they have a zero breakdown value. In this paper, we attempt to investigate the effects of using residuals from robust regression replacing OLS residuals in test statistics for normality. We study a modified Jarque-Bera test statistic for normality based on robust regression residuals. The asymptotic distribution of this robustified normality test is derived, and its breakdown property is discussed.

The multicollinearity in multiple linear regression models and the existence of leverage data points are common problems. These problems exert undesirable effects on the least squares estimators. So, it would seem important to combine methods of estimation designed to deal with these problems simultaneously. In this paper, alternative biased robust regression estimator is defined by mixing the ridge estimation technique into the robust least trimmed squares estimation to obtain the Ridge Least Trimmed Squares (RLTS). The efficiency of the combined estimator(RLTS) is compared with some existing regression estimators, which namely, the Ordinary Least Squares (LS); Ridge Regression (RR) and Ridge Least Absolute Deviation(RLAD). The numerical results of this study show that, the RLTS regression estimator is more efficient than other estimators, based on, Bias and mean squared error criteria for many combinations of leverage data points and degree of multicollinearity.