Dimension reduction for nonelliptically distributed predictors

TL;DR

This paper introduces a reformulation of dimension reduction techniques that removes the need for predictors to have elliptical or normal distributions, maintaining statistical properties like consistency and normality.

Contribution

It proposes a new approach based on the 'central solution space' that relaxes distributional assumptions in dimension reduction methods.

Findings

New methods perform well in simulations.

Methods are applicable to nonelliptically distributed predictors.

Comparable or improved results on real data.

Abstract

Sufficient dimension reduction methods often require stringent conditions on the joint distribution of the predictor, or, when such conditions are not satisfied, rely on marginal transformation or reweighting to fulfill them approximately. For example, a typical dimension reduction method would require the predictor to have elliptical or even multivariate normal distribution. In this paper, we reformulate the commonly used dimension reduction methods, via the notion of "central solution space," so as to circumvent the requirements of such strong assumptions, while at the same time preserve the desirable properties of the classical methods, such as $\sqrt{n}$-consistency and asymptotic normality. Imposing elliptical distributions or even stronger assumptions on predictors is often considered as the necessary tradeoff for overcoming the "curse of dimensionality," but the development of this paper shows that this need not be the case. The new methods will be compared with existing methods by simulation and applied to a data set.