Doubly weighted m-estimator of linear regression model parameters

Ordinary Least Squares (OLS) estimator produces the Best Linear Unbiased Estimate (BLUE) of the parameter of linear regression model if the assumptions of normality and constant variance of the error terms are satisfied. However, the assumption of the constant error terms across the entire observations is frequently violated by real life data. Due to the failure of OLS estimator for contaminated data, robust alternatives such as Least Absolute Deviation (LAD) method and M estimators are proposed. M-estimators are robust to outliers in the y-direction but fail for x outliers. To obtain M-estimator that is robust to outliers in both directions, weights were applied to two loss functions L_x and L_y to remove the effect of outliers in y and x directions. The method handles both the simple and multiple linear regression models and yields set of solutions that are unbiased and efficient. Comparative analysis of the performance of the proposed method with the existing methods indicates that the method competes favourably and are particularly more robustand efficient than other estimators considered when outliers lie on the X-direction and on both X and Y directions. The finite sample performance of the proposed method is studied using MonteCarlo simulation.

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