期刊
ANNALS OF STATISTICS
卷 33, 期 4, 页码 1617-1642出版社
INST MATHEMATICAL STATISTICS
DOI: 10.1214/009053605000000200
关键词
AIC; BIC; EM algorithm; LASSO; MM algorithm; penalized likelihood; oracle estimator; SCAD
资金
- NIDA NIH HHS [P50 DA010075-100008, P50 DA010075] Funding Source: Medline
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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