3992

Jianqing Fana & Runze Lia

Journal of the American Statistical Association

Taylor & Francis

2015

Hard thresholding , LASSO , Nonnegative garrote , Penalized likelihood , Oracle estimator , SCAD , Soft thresholding

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 coefŽ cients simultaneously. Hence they enable us to construct conŽ dence intervals for estimated parameters. The proposed approaches are distinguished from others in that the penalty functions are symmetric, nonconcave on 401ˆ5, 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. 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.