Abstract: We propose an optimal family of estimators in sufficient dimension reduction using a Fourier transform based on a quadratic discrepancy function. Our proposed approach has advantages over existing methods in that it avoids the slicing scheme of a response variable and easily handles multivariate responses. We further develop four sub-optimal estimators: degenerated and special estimators for computational efficiency and simplicity, and robust and its degenerated estimators for a less restrictive condition for estimation and inference. Marginal and conditional hypothesis tests for the predictors and dimensions are also obtained. Simulation studies and a real-data analysis illustrate the efficacy of our proposed methods.

Key words and phrases: Fourier transform, minimum discrepancy, sufficient dimension reduction.