Robust functional sliced inverse regression

Functional data are infinite-dimensional statistical objects which pose significant challenges to both theorists and practitioners. To avoid the stringent constraints for parametric methods and low convergence rate for nonparametric methods, many functional dimension reduction methods have received attention in the functional data analysis literature, which, if desired, can be combined with low dimensional nonparametric regression in a later step. However, as far as we know that all of the functional dimension reduction methods are based on the classical estimates of the first and second moments of the data, and therefore sensitive to outliers. In the present paper, we propose a robust functional dimension reduction method by replacing the classical estimates with robust ones in the functional sliced inverse regression (FSIR). This leads to procedures which maintain the clever estimation scheme of the original FSIR method but can cope better with outliers. A comparison with FSIR is also made through simulation studies to show the robustness of the robust functional sliced inverse regression (RFSIR). As applications, the Orange juice data and the Tecator data are analyzed by using the proposed RFSIR method.