The enhancement of partial robust m-regression (PRM) and split sample bootstrap (SSB) for high dimensional data with outliers / Mazni Mohamad

Partial Least Squares regression (PLSR) is a regression technique that is commonly used to analyse the relationship between variables in high dimensional data. PLSR also offers good solution to multicollinearity problem in regression analysis. Hence, PLSR is a very powerful tool in dealing with multivariate data especially that of high dimension. Multivariate data is however often contains outliers. The presence of outliers in a data set may cause the regression parameter estimates become imprecise; hence, lead us to making such an invalid conclusion. Several robust PLSR methods have been proposed by researchers to cater this outlying problem. One of those is known as Partial Robust M-regression (PRM). This method is very much emphasizes on the robust starting values and the weights to be used. Reason for such emphasis is to ensure that more protection can be given against both vertical outliers and high leverage points. This study focuses on the enhancement of PRM method by introducing several PRM-based methods. Altogether there are five PRM-based methods that have been proposed in this research which are Winsorized Mean PRM- based method (PRMWM), Tukey Bisquare PRM-based method (PRMBS), Hampel PRM-based method (PRMH), the integrated Winsorized Mean and Tukey Bisquare PRM-based method (PRMWMBS) and the integrated Winsorized Mean and Hampel method (PRMWMH). The performances of all methods are assessed through both numerical examples and simulation studies under various outlying conditions. In most conditions, the PRMWM, PRMBS and PRMH outperform the original PRM. However, the integrated approach seems not to be a good idea as the two methods proposed in this study which are PRMWMBS and PRMWMH are not performing well compared to PRM. In short, this study perceives that PRMWM is the best method among all proposed methods because it consistently outperforms other methods. In order to further assess the performance of the methods, this study introduced a new bootstrap method for regression models with high dimensional data which is Weighted Split Sample Bootstrap (WSSB). The accuracy of this bootstrap technique has been measured and compared to that of the Split Sample Bootstrap (SSB). Results indicate that WSSB outperforms SSB. Therefore, WSSB is then used to compare the performance of PRMWM as compared to PRM via numerical examples and simulation studies. In general, it can be seen that PRMWM outperforms other methods as it produces the smallest prediction error values. the same time, the establishment of these models can fill the gap in Islamic derivative study and promote the growth of Islamic financial products.