Data-driven slicing for dimension reduction in regressions: A likelihood-ratio approach

To efficiently estimate the central subspace in sufficient dimension reduction, response discretization via slicing its range is one of the most used methodologies when inverse regression-based methods are applied. However, existing slicing schemes are almost all ad hoc and not widely accepted. Thus, how to define data-driven schemes with certain optimal properties is a longstanding problem in this field. The research described herewith is then two-fold. First, we introduce a likelihood-ratio-based framework for dimension reduction, subsuming the popularly used methods including the sliced inverse regression, the sliced average variance estimation and the likelihood acquired direction. Second, we propose a regularized log likelihood-ratio criterion to obtain a data-driven slicing scheme and derive the asymptotic properties of the estimators. A simulation study is carried out to examine the performance of the proposed method and that of existing methods. A data set concerning concrete compressive strength is also analyzed for illustration and comparison.