On principal Hessian directions for multivariate response regressions

We consider a multivariate response regression analysis with a vector of predictors. In this article, we develop the modification of principal Hessian directions based on principal components for estimating the central mean subspace without requiring a prespecified parametric model. We use a permutation test suggested by Cook and Yin (Aust New Z J Stat 43:147-199, 2001) for inference about the dimension. Simulation results and one real data are reported, and comparisons are made with four methods-most predictable variates, k-means inverse regression, optimal method of Yoo and Cook (Biometrika 94:231-242, 2007) and canonical correlation approach.