On consistency and sparsity for sliced inverse regression in high dimensions

Abstract

We provide here a framework to analyze the phase transition phenomenon of slice inverse regression (SIR), a supervised dimension reduction technique introduced by Li [J. Amer. Statist. Assoc. 86 (1991) 316–342]. Under mild conditions, the asymptotic ratio $\rho=\lim p/n$ is the phase transition parameter and the SIR estimator is consistent if and only if $\rho=0$. When dimension $p$ is greater than $n$, we propose a diagonal thresholding screening SIR (DT-SIR) algorithm. This method provides us with an estimate of the eigenspace of $\operatorname{var}(\mathbb{E}[\boldsymbol{x}|y])$, the covariance matrix of the conditional expectation. The desired dimension reduction space is then obtained by multiplying the inverse of the covariance matrix on the eigenspace. Under certain sparsity assumptions on both the covariance matrix of predictors and the loadings of the directions, we prove the consistency of DT-SIR in estimating the dimension reduction space in high-dimensional data analysis. Extensive numerical experiments demonstrate superior performances of the proposed method in comparison to its competitors.