Distance weighted directional regression for Fréchet sufficient dimension reduction

Analysis of non-Euclidean data accumulated from human longevity studies, brain functional network studies, and many other areas has become an important issue in modern statistics. Fréchet sufficient dimension reduction aims to identify dependencies between non-Euclidean object-valued responses and multivariate predictors while simultaneously reducing the dimensionality of the predictors. We introduce the distance weighted directional regression method for both linear and nonlinear Fréchet sufficient dimension reduction. We propose a new formulation of the classical directional regression method in sufficient dimension reduction. The new formulation is based on distance weighting, thus providing a unified approach for sufficient dimension reduction with Euclidean and non-Euclidean responses, and is further extended to nonlinear Fréchet sufficient dimension reduction. We derive the asymptotic normality of the linear Fréchet directional regression estimator and the convergence rate of the nonlinear estimator. Simulation studies are presented to demonstrate the empirical performance of the proposed methods and to support our theoretical findings. The application to human mortality modeling and diabetes prevalence analysis show that our proposal can improve interpretation and out-of-sample prediction.