Contour projected dimension reduction

TL;DR

This paper introduces contour projection-based dimension reduction methods for regression, defining new subspace concepts and demonstrating their advantages, especially with heavy-tailed predictor distributions, through theoretical analysis and empirical studies.

Contribution

It develops a novel dimension reduction framework using contour projection, defining the central and generalized contour subspaces, and proposes CP-based methods with proven theoretical properties.

Findings

CP methods outperform non-CP approaches with heavy-tailed distributions

Theoretical properties of CP-based estimators are established

Empirical example demonstrates practical usefulness

Abstract

In regression analysis, we employ contour projection (CP) to develop a new dimension reduction theory. Accordingly, we introduce the notions of the central contour subspace and generalized contour subspace. We show that both of their structural dimensions are no larger than that of the central subspace Cook [Regression Graphics (1998b) Wiley]. Furthermore, we employ CP-sliced inverse regression, CP-sliced average variance estimation and CP-directional regression to estimate the generalized contour subspace, and we subsequently obtain their theoretical properties. Monte Carlo studies demonstrate that the three CP-based dimension reduction methods outperform their corresponding non-CP approaches when the predictors have heavy-tailed elliptical distributions. An empirical example is also presented to illustrate the usefulness of the CP method.