@MISC{Fan99variableselection, author = {Jianqing Fan and Runze Li}, title = {Variable Selection via Penalized Likelihood}, year = {1999} }

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Abstract

Variable selection is vital to statistical data analyses. Many of procedures in use are stepwise selection procedures, which can be expensive in computation and ignore stochastic errors in the variable selection process of previous steps. An automatic and simultaneous variable selection procedure can be obtained by using a penalized likelihood method. In traditional linear models, the best subset selection and stepwise deletion methods coincide with a penalized least-squares method when design matrices are orthonormal. In this paper, we propose a few new approaches to selecting variables for linear models, robust regression models and generalized linear models based on a penalized likelihood approach. A family of thresholding functions are proposed. The LASSO proposed by Tibshirani (1996) is a member of the penalized least-squares with the L 1 -penalty. A smoothly clipped absolute deviation (SCAD) penalty function is introduced to ameliorate the properties of L 1 -penalty. A ...

...alues of #z#. See Figure 2. The solution is continuous in #z# only when the minimum of + p # ()over # 0is attained at 0, as shown in Figure 2. When p # (##)=## # as in the bridge regression (Frank a=-=nd Friedman, 19-=-93), the solution is continuous only when q # 1. However, when q>1, the minimum of + p # # ()iszeroand hence it does not correspond to a thresholding rule. The only continuous solution with a thresho...

...ariance a. We computed the Bayesian risk via numerical integration. Figure 3(a) depicts the Bayesian risk as a function of a under the squared loss, for the universal thresholding = # 2log(d) (see Do=-=noho and Johnstone, 1994-=-) with d =20; 40; 60 and 100, and Figure 3(b) is for d =512,1024, 2048 and 4096 . From Figures 3(a) and 3(b), the Bayesian risks achieve their minimums when a # 3:7. It can be seen from these twosgure...