Abstract: For effective dimension reduction (e.d.r.) in regression, the sliced inverse regression (SIR) is used to detect detailed structures of conditional distributions and reduce the dimensionality of covariates in a nonparametric manner. Subsequent analysis can then be based on features of a lower dimension, which assists with model interpretation and increases the estimation efficiency. The concept of e.d.r. has led to the framework of sufficient dimension reduction (SDR), with promising developments in various fields. Here, we first review the SIR and other estimation methods for SDR when a complete random sample with finite-dimensional covariates is available. Then, we discuss extensions and applications to cases with more complicated structures, including high-dimensional data and two types of incomplete data. Lastly, we emphasize the importance of SDR in modern statistical applications, and explain how conventional SDR methods need to adapt to different data structures in order to ensure good performance.

Key words and phrases: Dimension reduction, high-dimensional data, semiparametric statistics.