Adjusting inverse regression for predictors with clustered distribution

TL;DR

This paper introduces adjusted inverse regression methods for sufficient dimension reduction that model the predictor distribution under a mixture model assumption, effectively handling clustered data and improving upon traditional methods.

Contribution

The paper develops novel inverse regression techniques that accommodate clustered predictor distributions by modeling their conditional mean and variance under a mixture model, bridging inverse regression and localized SDR methods.

Findings

Methods are $\

they are $\

fully recover the desired reduced predictor

Abstract

A major family of sufficient dimension reduction (SDR) methods, called inverse regression, commonly require the distribution of the predictor $X$ to have a linear $E(X|\beta^\mathsf{T}X)$ and a degenerate $\mathrm{var}(X|\beta^\mathsf{T}X)$ for the desired reduced predictor $\beta^\mathsf{T}X$. In this paper, we adjust the first and second-order inverse regression methods by modeling $E(X|\beta^\mathsf{T}X)$ and $\mathrm{var}(X|\beta^\mathsf{T}X)$ under the mixture model assumption on $X$, which allows these terms to convey more complex patterns and is most suitable when $X$ has a clustered sample distribution. The proposed SDR methods build a natural path between inverse regression and the localized SDR methods, and in particular inherit the advantages of both; that is, they are $\sqrt{n}$-consistent, efficiently implementable, directly adjustable under the high-dimensional settings, and fully recovering the desired reduced predictor. These findings are illustrated by simulation studies and a real data example at the end, which also suggest the effectiveness of the proposed methods for nonclustered data.