Electronic Journal of Statistics

Thresholding-based iterative selection procedures for model selection and shrinkage

Yiyuan She

Full-text: Open access

Abstract

This paper discusses a class of thresholding-based iterative selection procedures (TISP) for model selection and shrinkage. People have long before noticed the weakness of the convex l1-constraint (or the soft-thresholding) in wavelets and have designed many different forms of nonconvex penalties to increase model sparsity and accuracy. But for a nonorthogonal regression matrix, there is great difficulty in both investigating the performance in theory and solving the problem in computation. TISP provides a simple and efficient way to tackle this so that we successfully borrow the rich results in the orthogonal design to solve the nonconvex penalized regression for a general design matrix. Our starting point is, however, thresholding rules rather than penalty functions. Indeed, there is a universal connection between them. But a drawback of the latter is its non-unique form, and our approach greatly facilitates the computation and the analysis. In fact, we are able to build the convergence theorem and explore theoretical properties of the selection and estimation via TISP nonasymptotically. More importantly, a novel Hybrid-TISP is proposed based on hard-thresholding and ridge-thresholding. It provides a fusion between the l0-penalty and the l2-penalty, and adaptively achieves the right balance between shrinkage and selection in statistical modeling. In practice, Hybrid-TISP shows superior performance in test-error and is parsimonious.

Article information

Source
Electron. J. Statist., Volume 3 (2009), 384-415.

Dates
First available in Project Euclid: 29 April 2009

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1241011807

Digital Object Identifier
doi:10.1214/08-EJS348

Mathematical Reviews number (MathSciNet)
MR2501318

Zentralblatt MATH identifier
1326.62158

Subjects
Primary: 62J07: Ridge regression; shrinkage estimators 62J05: Linear regression

Keywords
Sparsity nonconvex penalties thresholding model selection & shrinkage lasso ridge SCAD

Citation

She, Yiyuan. Thresholding-based iterative selection procedures for model selection and shrinkage. Electron. J. Statist. 3 (2009), 384--415. doi:10.1214/08-EJS348. https://projecteuclid.org/euclid.ejs/1241011807


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