Robust estimations of model parameters in simple linear regression model

In this thesis, the performance of the Least Squares (LS) estimators for model parameters in a simple linear regression model was examined under different scenarios. In the first scenario, various probability distributions that could serve as alternatives to the normal distribution, as well as several outlier models, were considered for the distribution of the error terms. In the second scenario, it was assumed that the error terms follow the right-skewed Gumbel distribution. Furthermore, in cases where outliers existed in the X-direction, the performance of LS estimators was compared with robust estimation methods commonly discussed in the literature, including the Least Absolute Deviation (LAD), Weighted Least Absolute Deviation (WLAD), and Least Median of Squares (LMS) estimators. To evaluate the efficiency of the estimation methods, the Monte-Carlo simulation technique was employed, and the Bias and Mean Squared Error (MSE) criteria were used as performance measures. To support the findings obtained from the simulation studies, a real dataset from the literature was also analyzed. The results of both the simulation and real data analyses revealed that the performance of LS estimators significantly deteriorates when the error terms deviate from normality or when the dataset contains outliers. In contrast, robust estimation methods were found to produce more stable and reliable results under such data contamination conditions. Moreover, the findings indicated that the LMS estimator exhibits greater robustness against assumption violations compared to other estimation methods when estimating model parameters. The WLAD estimator demonstrated the second-best performance after LMS, making it a reliable alternative to the LMS method. In conclusion, it is recommended to use the LMS robust estimation method instead of the ordinary LS estimation method for estimating model parameters in cases where the dataset contains outliers or the error terms do not have a normal distribution.