On splines approximation for sliced average variance estimation

To avoid the inconsistency and slow convergence rate of the slicing estimator of the sliced average variance estimation (SAVE), particularly in the continuous response cases, we suggest B-spline approximation that can make the estimator n consistent and keeps the spirit of easy implementation that the slicing estimation shares. Compared with kernel estimation that has been used in the literature, B-spline approximation is of higher accuracy and is easier to implement. To estimate the structural dimension of the central dimension reduction space, a modified Bayes information criterion is suggested, which makes the leading term and the penalty term comparable in magnitude. This modified criterion can help to enhance the efficacy of estimation. The methodologies and theoretical results are illustrated through an application to the horse mussel data and simulation comparisons with existing methods by simulations.