Abstract: A considerable amount of research in the literature has focused on quantifying the effect of extreme observations on classical methods for estimating the Central Subspace (CS) for regression through the study of influence functions and their sample estimates. Alternatively, a method that is inherently robust to data contamination is also important and desirable for the increased reliability in the estimation of the CS without relying on the identification and removal of influential values. To this end, we develop a new method that is innately resistant to outlying observations in recovering a dimension reduction subspace for regression based on the R´enyi divergence. In addition to deriving the theoretical Influence Function (IF), the Sample Influence Function (SIF) values are directly utilized to provide new powerful and efficient methods for both estimating the dimension of the CS and selecting an optimal level of the tuning parameter to decrease the impact of extreme observations. The model-free approach is detailed theoretically, its performance investigated through simulation, and the application in practice is demonstrated through a real data analysis.

Keywords words and phrases: Bootstrap dimension estimation, robustness, sample influence function, sufficient dimension reduction, tuning parameter estimation.