In a linear regression model, the ordinary least squares (OLS) method is considered the best method to estimate the regression parameters if the assumptions are met. However, if the data does not satisfy the underlying assumptions, the results will be misleading. The violation for the assumption of constant variance in the least squares regression is caused by the presence of outliers and heteroscedasticity in the data. This assumption of constant variance (homoscedasticity) is very important in linear regression in which the least squares estimators enjoy the property of minimum variance. Therefor e robust regression method is required to handle the problem of outlier in the data. However, this research will use the weighted least square techniques to estimate the parameter of regression coefficients when the assumption of error variance is violated in the data. Estimation of WLS is the same as carrying out the OLS in a transformed variables procedure. The WLS can easily be affected by outliers. To remedy this, We have suggested a strong technique for the estimation of regression parameters in the existence of heteroscedasticity and outliers. Here we apply the robust regression of M-estimation using iterative reweighted least squares (IRWLS) of Huber and Tukey Bisquare function and resistance regression estimator of least trimmed squares to estimating the model parameters of state-wide crime of united states in 1993. The outcomes from the study indicate the estimators obtained from the M-estimation techniques and the least trimmed method are more effective compared with those obtained from the OLS.

In the multiple linear regression analysis, the ridge regression estimator is often used to address the problem of multicollinearity. Besides multicollinearity, outliers also constitute a problem in the multiple linear regression analysis. We propose a new biased estimators of the robust ridge regression called the Ridge Least Median Squares (RLMS) estimator in the presence of multicollinearity and outliers. For this purpose, a simulation study is conducted in order to see the difference between the proposed method and the existing methods in terms of their effectiveness measured by the mean square error. In our simulation studies the performance of the proposed method RLMS is examined for different number of observations and the different percentage of outliers. The results of different illustrative cases are presented. This paper also provides the results of the RLMS on a real-life data set. The results show that RLMS is better than the existing methods of Ordinary Least Squares (OLS), Ridge Least Absolute Value (RLAV) and Ridge Regression (RR) in the presence of multicollinearity and outliers.

The comparison between several robust ridge regression estimators in the presence of multicollinearity and multiple outliers AIP Conference Proceedings 1613, 388 (2014) Abstract. This study presents an improvement to robust ridge regression estimator. We proposed two methods Bisquare ridge least trimmed squares (BRLTS) and Bisquare ridge least absolute value (BRLAV) based on ridge least trimmed squares (RLTS) and ridge least absolute value (RLAV), respectively. We compared these methods with existing estimators, namely ordinary least squares (OLS) and Huber ridge regression (HRID) using three criteria: Bias, Root Mean Square Error (RMSE) and Standard Error (SE) to estimate the parameters coefficients. The results of Bisquare ridge least trimmed squares (BRLTS) and Bisquare ridge least absolute value (BRLAV) are compared with existing methods using real data and simulation study. The empirical evidence shows that the results obtain from the BRLTS are the best among the three estimators followed by BRLAV with the least value of the RMSE for the different disturbance distributions and degrees of multicollinearity.

The presence of heteroscedasticity, multicollinearity and outliers are classical problems of data within the linear regression framework. This research is a proposal of new methods which can be a potential candidate for weighted robust wild bootstrap regression as well as the multicollinearity robust regression model with outliers' pattern based on Latin root. This proposal arises as a logical combination of principles used in the Latin root, wild bootstrap sampling procedure of Wu and Liu. The weighted robust GM-estimator of Krasker and Welsch (1982) with initial MM-estimator of Yohai (1987) and S-estimator of Rousseeuw and Yohai (1984) together with two different weighting procedures of Hampel's and Andrews sin weighted function are considered in the analysis. This paper investigate the nonresistance of weighted robust wild bootstrap (WRW Boot) regression and our proposed method for resistance to multicollinearity, outliers and heteroscedasticity error variance. The use of modified weighted robust wild bootstrap methods (WRW Boot) based on Latin root with multicollinearity and outlier diagnostic method yields more reliable trend estimations. From numerical example and simulation study, the resulting of the modified weighted robust wild bootstrap methods based on Latin root with multicollinearity and outlier diagnostic method (WRW Boot) is efficient than other estimators, using Standard Error (SE) and the Root Mean Squared Error criterion for numerical example and simulation study respectively for many combinations of error distribution and degree of multicollinearity.