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August 2005 Variable selection using MM algorithms
David R. Hunter, Runze Li
Ann. Statist. 33(4): 1617-1642 (August 2005). DOI: 10.1214/009053605000000200


Variable selection is fundamental to high-dimensional statistical modeling. Many variable selection techniques may be implemented by maximum penalized likelihood using various penalty functions. Optimizing the penalized likelihood function is often challenging because it may be nondifferentiable and/or nonconcave. This article proposes a new class of algorithms for finding a maximizer of the penalized likelihood for a broad class of penalty functions. These algorithms operate by perturbing the penalty function slightly to render it differentiable, then optimizing this differentiable function using a minorize–maximize (MM) algorithm. MM algorithms are useful extensions of the well-known class of EM algorithms, a fact that allows us to analyze the local and global convergence of the proposed algorithm using some of the techniques employed for EM algorithms. In particular, we prove that when our MM algorithms converge, they must converge to a desirable point; we also discuss conditions under which this convergence may be guaranteed. We exploit the Newton–Raphson-like aspect of these algorithms to propose a sandwich estimator for the standard errors of the estimators. Our method performs well in numerical tests.


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David R. Hunter. Runze Li. "Variable selection using MM algorithms." Ann. Statist. 33 (4) 1617 - 1642, August 2005.


Published: August 2005
First available in Project Euclid: 5 August 2005

zbMATH: 1078.62028
MathSciNet: MR2166557
Digital Object Identifier: 10.1214/009053605000000200

Primary: 62J12 , 65C20

Keywords: AIC , BIC , EM algorithm , Lasso , MM algorithm , oracle estimator , penalized likelihood , SCAD

Rights: Copyright © 2005 Institute of Mathematical Statistics


Vol.33 • No. 4 • August 2005
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