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Zbl 1073.62547
Fan, Jianqing; Li, Runze
Variable selection via nonconcave penalized likelihood and its oracle properties. (English)
[J] J. Am. Stat. Assoc. 96, No. 456, 1348-1360 (2001). ISSN 0162-1459; ISSN 1537-274X/e

Summary: Variable selection is fundamental to high-dimensional statistical modeling, including nonparametric regression. Many approaches in use are stepwise selection procedures, which can be computationally expensive and ignore stochastic errors in the variable selection process. In this article, penalized likelihood approaches are proposed to handle these kinds of problems. The proposed methods select variables and estimate coefficients simultaneously. Hence they enable us to construct confidence intervals for estimated parameters. The proposed approaches are distinguished from others in that the penalty functions are symmetric, nonconcave on $(0, \infty)$, and have singularities at the origin to produce sparse solutions. Furthermore, the penalty functions should be bounded by a constant to reduce bias and satisfy certain conditions to yield continuous solutions. A new algorithm is proposed for optimizing penalized likelihood functions.\par The proposed ideas are widely applicable. They are readily applied to a variety of parametric models such as generalized linear models and robust regression models. They can also be applied easily to nonparametric modeling by using wavelets and splines. Rates of convergence of the proposed penalized likelihood estimators are established. Furthermore, with proper choice of regularization parameters, we show that the proposed estimators perform as well as the oracle procedure in variable selection; namely, they work as well as if the correct submodel were known. Our simulation shows that the newly proposed methods compare favorably with other variable selection techniques. Furthermore, the standard error formulas are tested to be accurate enough for practical applications.

MSC 2000:: *62J12 Generalized linear models
62G08 Nonparametric regression
62F12 Asymptotic properties of parametric estimators
62G20 Nonparametric asymptotic efficiency

Cited in: Zbl 1247.62246 Zbl 1227.62047 Zbl 1209.62131 Zbl 1247.62143 Zbl 1170.62034 Zbl 1170.62046 Zbl 1168.62064 Zbl 1167.62063 Zbl 1142.62027 Zbl 1133.62048 Zbl 1109.62058 Zbl 1092.62031

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