Penalized likelihood methods are fundamental to ultrahigh dimensional variable selection. How high dimensionality such methods can handle remains largely unknown. In this paper, we show that in the context of generalized linear models, such methods possess model selection consistency with oracle properties even for dimensionality of nonpolynomial (NP) order of sample size, for a class of penalized likelihood approaches using folded-concave penalty functions, which were introduced to ameliorate the bias problems of convex penalty functions. This fills a long-standing gap in the literature where the dimensionality is allowed to grow slowly with the sample size. Our results are also applicable to penalized likelihood with the L1-penalty, which is a convex function at the boundary of the class of folded-concave penalty functions under consideration. The coordinate optimization is implemented for finding the solution paths, whose performance is evaluated by a few simulation examples and the real data analysis.

中文翻译：

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NP 维非凹惩罚似然

惩罚似然方法是超高维变量选择的基础。这些方法能够处理多高的维度仍然很大程度上未知。在本文中，我们表明，在广义线性模型的背景下，对于一类使用折叠凹罚函数的惩罚似然方法，即使对于样本大小的非多项式（NP）阶维数，此类方法也具有与预言属性的模型选择一致性，其引入是为了改善凸罚函数的偏差问题。这填补了文献中长期存在的空白，即允许维度随着样本大小缓慢增长。我们的结果也适用于 L1 惩罚的惩罚似然，L1 惩罚是所考虑的折叠凹惩罚函数类边界上的凸函数。坐标优化是为了寻找解决方案路径而实现的，其性能通过一些仿真例子和真实数据分析来评估。