On consistency and sparsity for sliced inverse regression in high dimensions

Qian Lin Harvard University Zhigen Zhao Temple University Jun S. Liu Harvard University

Statistics Theory and Methods mathscidoc:1701.333182

Distinguished Paper Award in 2017

Annals of statistics
We provide here a framework to analyze the phase transition phenomenon of slice inverse regression (SIR), a supervised dimension reduction technique introduced by \cite{Li:1991}. Under mild conditions, the asymptotic ratio ρ=limp/n is the phase transition parameter and the SIR estimator is consistent if and only if ρ=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 eigen-space of the covariance matrix of the conditional expectation var(E[x|y]). The desired dimension reduction space is then obtained by multiplying the inverse of the covariance matrix on the eigen-space. 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.
Sliced inverse regression, phase transition, high dimensional statistics
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  • Accepted at Feb 22, 2017
  title={On consistency and sparsity for sliced inverse regression in high dimensions},
  author={Qian Lin, Zhigen Zhao, and Jun S. Liu},
  booktitle={Annals of statistics},
Qian Lin, Zhigen Zhao, and Jun S. Liu. On consistency and sparsity for sliced inverse regression in high dimensions. In Annals of statistics. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170121193117869457117.
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