## The Annals of Statistics

### 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.

#### Article information

Source
Ann. Statist., Volume 46, Number 2 (2018), 580-610.

Dates
Revised: January 2017
First available in Project Euclid: 3 April 2018

https://projecteuclid.org/euclid.aos/1522742430

Digital Object Identifier
doi:10.1214/17-AOS1561

Mathematical Reviews number (MathSciNet)
MR3782378

Zentralblatt MATH identifier
06870273

#### Citation

Lin, Qian; Zhao, Zhigen; Liu, Jun S. On consistency and sparsity for sliced inverse regression in high dimensions. Ann. Statist. 46 (2018), no. 2, 580--610. doi:10.1214/17-AOS1561. https://projecteuclid.org/euclid.aos/1522742430

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#### Supplemental materials

• Supplement to “On the consistency and sparsity for sliced inverse regression for high dimensions”. In the supplement, we prove the rest of the results stated in the paper.