Fast Algorithm of Generalized-S (GS) Estimator for Robust Estimation Multiple Linear Regression Parameter.

Abstract

Generalized-S (GS) estimator is a robust estimator for regression model parameter based on robust scale estimate. GS estimates of regression model parameters are yielded by minimization of robust estimates of pairwise residual difference scales. Hence, GS estimator could be seen as a generalization of S estimator which deals with minimization of robust estimates of residual scales. By viewing GS estimator as generalization of S estimator, one could find the fact that the former has high breakdown point property as the later does. On the other hand, it has high efficiency property while the later does not. Meanwhile, several fast iterative computational methods for high breakdown robust estimators of regression parameters have been developed. The methods were called fast algorithms. There are three approaches have been proposed i.e. fast algorithm for LTS estimator, S estimator, and τ estimator. These methods have been applied for multiple linear regression model parameter estimation. This fact leads to the notion of developing a fast algorithm of GS estimator for multiple regression model parameter in this research in order to get another comprehensive method for high breakdown and high efficiency robust estimator beside fast τ approach. The algorithm is then applied to simulation data contaminated with outliers and the results yielded then will be compared with the ones produced by fast algorithm for S estimator and OLS to investigate its efficiency by looking at RMSE of the estimates resulted in various condition by considering several proportions, locations, and scales of outliers, as well as generating models. Finally, in all cases which involve outlier contamination considered, fast algorithm of GS consistently shows the better results by yielding the smallest RMSE value for each case. This fact indicates that fast algorithm of GS has higher efficiency than that of fast S. As addition, in case where the data are not contaminated with outliers, fast algorithm of GS gives the results that relatively close to OLS does, while fast S does not.

Penduga Generalized-S (GS) adalah suatu penduga kekar parameter regresi berdasarkan dugaan kekar skala. Penduga GS dapat dipandang sebagai perluasan penduga S karena penduga GS diperoleh dari minimasi dugaan M skala selisih sisaan (M estimates of residual differences scales) sedangkan penduga S didapatkan dari minimasi dugaan M skala sisaan (M estimates of residuals scales). Hal ini menyebabkan penduga GS memiliki efisiensi yang tinggi sementara penduga S memiliki efisiensi yang rendah. Untuk data yang tidak menyimpang begitu jauh dari asumsi normalitas, penduga GS memberikan hasil yang mendekati penduga kuadrat terkecil, namun tidak demikian halnya dengan penduga S.