A unified approach to sufficient dimension reduction

Through investigating a recently introduced sufficient dimension reduction method with Hellinger index, this article shows that the generalized Hellinger index unifies three existing dimension reduction methods: kernel discriminant analysis, sliced regression and density minimum average variance estimation, with certain weight functions. The Hellinger index is then extended to regression models with multivariate responses. Furthermore, new algorithms based on Hellinger index to estimate the dual central subspaces and to enable variable selection for sparse models are proposed. Simulation studies and a real data analysis demonstrate the efficacy of the proposed approaches.