Robust functional inverse regression

Functional sufficient dimension reduction (FSDR) is a popular approach for supervised dimensionality reduction in regression settings, as it allows for the reduction of functional predictors to a lower-dimensional subspace without loss of information. However, most existing FSDR methods are vulnerable to heavy-tailedness or outliers, which are common in many real-world applications. To address this limitation, we propose a robust FSDR method that utilizes a functional pairwise spatial sign (PASS) operator. This approach is suitable for both completely observed functional data and sparsely observed longitudinal data. Our method provides a more robust approach to FSDR, by taking into account the spatial information of the data and assigning greater weights to the less outlier-prone observations. We also provide a convergence rate analysis of the estimator, demonstrating that our method yields a consistent estimate of the dimension reduction directions. The effectiveness of our proposed method is demonstrated through extensive simulations and real data analysis. Our method outperforms existing methods in terms of robustness and accuracy, making it a valuable tool for analyzing functional data across various applications.