A ROBUST ESTIMATION PROCEDURE IN LINEAR REGRESSION IN THE PRESENCE OF OUTLIERS
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2018-07
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Abstract
The best method that can be used in regression analysis under some assumptions is Ordinary Least Squares (OLS) method. Such assumptions include; constant variance, normality and uncorrelated error terms among others. Violation of these assumptions such as the presence of outliers may have a large effect on the ordinary least square estimates. The existing robust techniques; Least Absolute (LA) estimator, Huber Maximum (H-M)estimator, Bi-Square Maximum (B-M)estimator, M-Maximum (MM) estimator, Least Median Square (LMS)estimator, Least Trimmed Square (LTS) estimator and S estimator (S) cope with these unusual observations. However, they may be sensitive to the presence of outliersespecially when they are in both directions of X and Y. The aim of this study therefore was to obtain a procedure in estimating parameters in linear regression model that is robust in the presence of outliers. The objectives were to: (i) obtain the best method in linear regression analysis among the existing robust methods using efficiency and breakdown point; (ii) propose an efficient procedure in estimation of parameters in Linear Regression in the presence of outliers with high breakdown points; and (iii) validate and compare the proposed procedure with the existing ones.
An estimation method for the classical linear regression was proposedusing regularization methods that allow one to handle a variety of inferential problems. Specifically, each outlying point in the data is estimated using case-specific parameter through Ridge regression approach and penalized estimators were suggested when the number of parameters in the model is more than the number of observed data points. The proposed estimation methodwas compared with the existing methods usingsimulated data with varying proportion and magnitude of outliers in all directions. The results from simulation were validated using real-life datasets. The performances were assessedusing breakdown point and efficiency.
The findings of the study were that:
(i) the strength and weakness of any robust estimator strongly depend on the percentage, magnitude and direction of the outlier;
(ii) among the existing robust estimators, MM is the best in terms of unidirectional outlier (X or Y direction);
(iii) the proposed method was found to be better in dealing with outliers in all directions (X or Y or both); and
(iv) robust estimators cannot improve the adequacy property of linear regression model.
The study concluded that the proposed method is a better procedure in estimation of parameters in linear regression in the presence of outliers due to its high efficiency and breakdown point of up to 50%. The proposed method is therefore preferred and recommended whenever there is high proportion of outliers and the direction of outliers is unknown.
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ROBUST ESTIMATION PROCEDURE, LINEAR REGRESSION, PRESENCE, OUTLIERS, Ordinary Least Squares (OLS)