Robust Diagnostics and Estimation in Heteroscedastic Regression Model in the Presence of Outliers

The violation of the assumption of homoscedasticity in OLS method, usually called heteroscedasticity, gravely misleads the inferential statistics. The current study has considered the situation when outliers occur in heteroscedastic data. Hence, the main focus of this research is to take remedial measures on the violation of the assumption of homoscedasticity in the presence of outliers. This thesis also concerns on the normality assumption of the errors of regression model in the presence of outliers. It is now evident that outliers have great impact on the existing normality tests, heteroscedasticity tests, and the estimators for heteroscedastic model. We propose the Robust Rescaled Moment (RRM) test for testing the normality of the regression residuals when there is an evidence of outlier(s). The results of the study signify that the RRM test offers substantial improvements over other existing tests in the presence of outliers. For the detection of heteroscedasticity in the presence of outliers, a modified version of the classical Goldfeld-Quandt (MGQ) test is proposed which is most powerful than the classical tests of heteroscedasticity. Most statistics practitioners assume that the forms of the heteroscedastic error structures are known which may lead to inefficient estimates if it is not correctly specified. In this respect, a Leverage Based Near-Neighbor (LBNN) method is proposed, where prior information on the structure of the heteroscedastic error is not required. The findings indicate that the LBNN is very efficient for correcting the problem of heteroscedastic errors with unknown structure. We also examine the effect of outliers on the existing remedial measures of heteroscedasticity. Hence, in this thesis, a one step M-type of Robust Weighted Least Squares Method (RWLS) and the Two-Step Robust Weighted Least Squares (TSRWLS) are developed. Finally, the new robust wild bootstrap techniques which are resistant to outliers are proposed. The proposed techniques are based on the weighted residuals which incorporated the MM estimator, robust location, robust scale and the bootstrap sampling schemes of Wu (1986) and Liu (1988). All procedures, in this thesis, are examined by using real data and Monte Carlo simulation studies. The comparative studies among the classical and proposed robust methods reveal that all the proposed robust methods outperform the classical methods.

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