The Sparse Laplacian Shrinkage Estimator for High-Dimensional Regression

Jian Huang; Shuangge Ma; Hongzhe Li; Cun-Hui Zhang

doi:10.1214/11-AOS897

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The Sparse Laplacian Shrinkage Estimator for High-Dimensional Regression

Journal article

Open access

Peer reviewed

The Sparse Laplacian Shrinkage Estimator for High-Dimensional Regression

Jian Huang, Shuangge Ma, Hongzhe Li and Cun-Hui Zhang

Annals of statistics, Vol.39(4), pp.2021-2046

2011

DOI: 10.1214/11-AOS897

PMCID: PMC3217586

PMID: 22102764

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Abstract

We propose a new penalized method for variable selection and estimation that explicitly incorporates the correlation patterns among predictors. This method is based on a combination of the minimax concave penalty and Laplacian quadratic associated with a graph as the penalty function. We call it the sparse Laplacian shrinkage (SLS) method. The SLS uses the minimax concave penalty for encouraging sparsity and Laplacian quadratic penalty for promoting smoothness among coefficients associated with the correlated predictors. The SLS has a generalized grouping property with respect to the graph represented by the Laplacian quadratic. We show that the SLS possesses an oracle property in the sense that it is selection consistent and equal to the oracle Laplacian shrinkage estimator with high probability. This result holds in sparse, high-dimensional settings with p ≫ n under reasonable conditions. We derive a coordinate descent algorithm for computing the SLS estimates. Simulation studies are conducted to evaluate the performance of the SLS method and a real data example is used to illustrate its application.

variable selection

Graphical structure

penalized regression

minimax concave penalty

high-dimensional data

oracle property

Details

Title: Subtitle: The Sparse Laplacian Shrinkage Estimator for High-Dimensional Regression
Creators: Jian Huang - Department of Statistics and Biostatistics Rutgers University Piscataway, NJ 08854
Shuangge Ma - Department of Statistics and Biostatistics Rutgers University Piscataway, NJ 08854
Hongzhe Li - Department of Statistics and Biostatistics Rutgers University Piscataway, NJ 08854
Cun-Hui Zhang - Department of Statistics and Biostatistics Rutgers University Piscataway, NJ 08854
Resource Type: Journal article
Publication Details: Annals of statistics, Vol.39(4), pp.2021-2046
DOI: 10.1214/11-AOS897
PMID: 22102764
PMCID: PMC3217586
NLM abbreviation: Ann Stat
ISSN: 0090-5364
eISSN: 2168-8966
Grant note: R01 CA127334-04 || CA / National Cancer Institute : NCI R01 CA120988-04 || CA / National Cancer Institute : NCI
Language: English
Date published: 2011
Academic Unit: Statistics and Actuarial Science
Record Identifier: 9983985969602771

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