Sparse Sliced Inverse Regression via Lasso. Issue 528 (2nd October 2019)

• Record Type: Journal Article
• Authors: Lin, Qian; Zhao, Zhigen; Liu, Jun S.
• Is Part Of: Journal of the American Statistical Association. Volume 114:Issue 528(2019)
• Page Start: 1726
• Page End: 1739
• Publication Date: 2019-10-02
• Subjects: Dimension reduction -- High dimensional statistics -- Minimax -- Theory of large deviation; Statistics -- Periodicals; Statistics -- Periodicals; Statistiques -- Périodiques; États-Unis -- Statistiques -- Périodiques; 519.5
• DOI: 10.1080/01621459.2018.1520115
• Languages: English
• ISSNs: 0162-1459

Abstract

Abstract: For multiple index models, it has recently been shown that the sliced inverse regression (SIR) is consistent for estimating the sufficient dimension reduction (SDR) space if and only if ρ = lim p n = 0, where p is the dimension and n is the sample size. Thus, when p is of the same or a higher order of n, additional assumptions such as sparsity must be imposed in order to ensure consistency for SIR. By constructing artificial response variables made up from top eigenvectors of the estimated conditional covariance matrix, we introduce a simple Lasso regression method to obtain an estimate of the SDR space. The resulting algorithm, Lasso-SIR, is shown to be consistent and achieves the optimal convergence rate under certain sparsity conditions when p is of order o ( n 2 λ 2 ), where λ is the generalized signal-to-noise ratio. We also demonstrate the superior performance of Lasso-SIR compared with existing approaches via extensive numerical studies and several real data examples. Supplementary materials for this article are available online.