A research on robust regression methods

In many regression applications, the distribution of the error is assumed to be normal and Least Squares (LS) method is used for parameter estimation. However, in practice, even if the distribution of errors is assumed to be normal, residuals are not generally normally distributed. If the data contains outlier (s) or there are observations (s) which suspected to be outlier, the assumption of normality is violated and parameter estimates which made by using the OLS, will biased. Many statisticians used robust methods for parameter estimates when such problems occur. One of these methods is the M-Estimation Method, a generalized version of the Maximum Likelihood (ML) Estimation method. However, traditional M-Estimators can not achieve a good solution if the data set has skewness and excess kurtosis. In this thesis, using the relationship between Pearson Differential Equation and Influence Function (IF), M-Estimation method is proposed for datasets that are follow Pearson Type VI (PVI) distribution. The advantage of this method is that, while the traditional M-Estimators do not take into account the skewness and kurtosis values of the data set, the Pearson Differential Equation takes into account these values and generate dynamic solutions for different skewness and kurtosis values. Objective, Influence and Weight functions are obtained by using the Probability Density Function (PDF) of the PVI distribution. In addition, the tail properties of PVI distribution are examined and its behavior in simulation studies is observed. By using the Weight Function, the Iteratively Re- Weighted Least Squares Estimation Method (IRWLS) is used to estimate regression parameters. The performance of the proposed method is compared with other M-Estimators in terms of Total Absolute Deviation (TAB) and Mean Square Error (MSE) criteria by using simulation studies with different scenarios and real data sets. Keywords: M- Estimation Method, Robust Regression, Pearson Type VI Distribution, Influence Function, Iteratively Re- Weighted Least Squares Method.