Robust dimension reduction using sliced inverse median regression

Dimension reduction is a useful technique when working with high-dimensional predictors, as meaningful data visualizations and graphical analyses using fewer predictors can be achieved. We propose a new non-iterative and robust against extreme values estimation of the effective dimension reduction (e.d.r) subspace, which is based on the estimation of the conditional median function of the predictors given the response. The existing literature on robust estimation of the e.d.r subspace relies on iterative algorithms, such as the composite quantile minimum average variance estimation and the sliced regression. Compared with these existing robust dimension reduction methods, the new method avoids iterations by directly estimating the e.d.r subspace and has better finite sample performance. It is shown that the inverse Tukey and Oja median regression curve falls into the e.d.r subspace, and that its directions can be estimated $$\sqrt{n}$$ n -consistently.