Dimension Reduction for Extreme Regression via Contour Projection

In extreme regression problems, a primary objective is to infer extreme values of the response given a set of predictors.The high dimensionality and heavy-tailedness of the predictors limit the applicability of classical tools for inferring conditional extremes.In this paper, we focus on the central extreme subspace (CES), whose existence and uniqueness are guaranteed under fairly mild conditions.By projecting the data onto the CES, the dimension of the predictors is reduced while all the information for inferring conditional extremes is retained, which effectively addresses the high dimensionality issue.We propose the novel COPES method to estimate the CES by utilizing contour projection.Notably, COPES is robust against heavy-tailed predictors.The theoretical justification for the consistency of COPES is established.Overall, our proposal not only extends the toolkit for extreme regression but also broadens the scope of dimension reduction techniques.The effectiveness of our proposal is demonstrated through extensive simulation studies and an application to Chinese stock market data.