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February 2021 On the optimality of sliced inverse regression in high dimensions
Qian Lin, Xinran Li, Dongming Huang, Jun S. Liu
Ann. Statist. 49(1): 1-20 (February 2021). DOI: 10.1214/19-AOS1813

Abstract

The central subspace of a pair of random variables $(y,\boldsymbol{x})\in \mathbb{R}^{p+1}$ is the minimal subspace $\mathcal{S}$ such that $y\perp\!\!\!\!\!\perp \boldsymbol{x}|P_{\mathcal{S}}\boldsymbol{x}$. In this paper, we consider the minimax rate of estimating the central space under the multiple index model $y=f(\boldsymbol{\beta }_{1}^{\tau }\boldsymbol{x},\boldsymbol{\beta }_{2}^{\tau }\boldsymbol{x},\ldots,\boldsymbol{\beta }_{d}^{\tau }\boldsymbol{x},\epsilon )$ with at most $s$ active predictors, where $\boldsymbol{x}\sim N(0,\boldsymbol{\Sigma })$ for some class of $\boldsymbol{\Sigma }$. We first introduce a large class of models depending on the smallest nonzero eigenvalue $\lambda $ of $\operatorname{var}(\mathbb{E}[\boldsymbol{x}|y])$, over which we show that an aggregated estimator based on the SIR procedure converges at rate $d\wedge ((sd+s\log (ep/s))/(n\lambda ))$. We then show that this rate is optimal in two scenarios, the single index models and the multiple index models with fixed central dimension $d$ and fixed $\lambda $. By assuming a technical conjecture, we can show that this rate is also optimal for multiple index models with bounded dimension of the central space.

Citation

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Qian Lin. Xinran Li. Dongming Huang. Jun S. Liu. "On the optimality of sliced inverse regression in high dimensions." Ann. Statist. 49 (1) 1 - 20, February 2021. https://doi.org/10.1214/19-AOS1813

Information

Received: 1 April 2018; Revised: 1 November 2018; Published: February 2021
First available in Project Euclid: 29 January 2021

Digital Object Identifier: 10.1214/19-AOS1813

Subjects:
Primary: 62J02
Secondary: 62H25

Rights: Copyright © 2021 Institute of Mathematical Statistics

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Vol.49 • No. 1 • February 2021
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