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Consistent tuning parameter selection in high dimensional sparse linear regression. (English) Zbl 1216.62103
Summary: An exhaustive search as required for traditional variable selection methods is impractical in high dimensional statistical modeling. Thus, to conduct variable selection, various forms of penalized estimators with good statistical and computational properties, have been proposed during the past two decades. The attractive properties of these shrinkage and selection estimators, however, depend critically on the size of regularization which controls model complexity. We consider the problem of consistent tuning parameter selection in high dimensional sparse linear regression where the dimension of the predictor vector is larger than the size of the sample. First, we propose a family of high dimensional Bayesian Information Criteria (HBIC), and then investigate the selection consistency, extending the results of the extended Bayesian Information Criterion (EBIC), of J. Chen and Z. Chen [Biometrika 95, No. 3, 795–771 (2008; Zbl 1437.62415)] to ultra-high dimensional situations. Second, we develop a two-step procedure, the SIS + AENET, to conduct variable selection in $$p>n$$ situations. The consistency of tuning parameter selection is established under fairly mild technical conditions. Simulation studies are presented to confirm theoretical findings, and an empirical example is given to illustrate the use in the internet advertising data.

##### MSC:
 62J05 Linear regression; mixed models 62F15 Bayesian inference 62H12 Estimation in multivariate analysis 65C60 Computational problems in statistics (MSC2010) 62G20 Asymptotic properties of nonparametric inference
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