Robust Group Identification and Variable Selection in Sliced Inverse Regression Using Tukey's Biweight Criterion and Ball Covariance

Abstract

The SSIR-PACS is a group identification and a model-free variable selection method under sufficient dimension reduction (SDR) settings. It combined the Pairwise Absolute Clustering and Sparsity (PACS) with sliced inverse regression (SIR) methods to produce solutions with sparsity and the ability of group identification. However, the SSIR-PACS depends on classical estimates for dispersion and location, squared loss function, and non-robust weights for outliers. In this paper, a robust version of SSIR-PACS (RSSIR-PACS) is proposed. We replaced the squared loss by the criterion of Tukey's biweight. Also, the non-robust weights to outliers, which depend on Pearson’s correlations, are substituted with robust weights based on recently developed ball correlation. Moreover, the estimates of the mean and covariance matrix are substituted by the median and ball covariance, respectively. The RSSIR-PACS is robust to outliers in both the response and covariates. According to the results of simulations, RSSIR-PACS produces very good results. If the outliers are existing, the efficacy of RSSIR-PACS is considerably better than the efficacy of the competitors. In addition, a robust criteria to estimate the structural dimension d is proposed. The RSSIR-PACS makes SSIR-PACS practically feasible. Also, we employed real data to demonstrate the utility of RSSIR-PACS.