Abstract: Sufficient dimension reduction often resorts to inverse regression, and most inverse regression methods rely on slicing a quantitative response. The choice of a particular slicing scheme is critical, but there are no current methods in the literature about how to select an optimal slicing scheme. We consider two popular slicing-based methods, namely, the sliced inverse regression and the sliced average variance estimation. By recasting the eigen-decomposition problem as a trace-optimization problem, we propose a penalized criterion for choosing an optimal slicing scheme. A dynamic programming algorithm is developed for numerical optimization. The theoretical properties are studied under mild conditions. Simulation examples show that our methods compare favorably with existing methods. An illustrative data analysis is also presented.

Key words and phrases: Nonlinear least squares, quantile slicing, optimal number of slices, SAVE, SIR, trace maximization.