Some recent advances in sufficient dimension reduction

Sufficient dimension reduction (SDR) is a main stream of the dimension reduction technique in modern statistical research. It aims at replacing the original predictor X in the data with some reduced predictor β^<roman>T</roman>X without loss of information for modeling the response Y. In this paper, we briefly review the literature of SDR, including its basic theory and major types of methods such as the inverse regression methods, the minimum average variance estimation, and the semi-parametric methods based on estimating equations. We also review some recent advances including the sparse and non-sparse SDR methods under high-dimensional settings, the application of SDR in causal inference, and nonlinear SDR.