Fan Jianqing, Lv Jinchi
Princeton University and University of Southern California.
IEEE Trans Inf Theory. 2011 Aug;57(8):5467-5484. doi: 10.1109/TIT.2011.2158486.
Penalized likelihood methods are fundamental to ultra-high 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 Non-Polynomial (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 L(1)-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.
惩罚似然方法是超高维变量选择的基础。这类方法能够处理多高的维度在很大程度上仍然未知。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惩罚函数的惩罚似然方法，即使对于样本量的非多项式（NP）阶维度，这些方法也具有与神谕性质一致的模型选择一致性。引入折叠凹惩罚函数是为了改善凸惩罚函数的偏差问题。这填补了文献中一个长期存在的空白，即允许维度随样本量缓慢增长的情况。我们的结果也适用于具有L(1)惩罚的惩罚似然，L(1)惩罚在所考虑的折叠凹惩罚函数类的边界处是一个凸函数。通过坐标优化来实现求解路径，其性能通过一些模拟示例和实际数据分析进行评估。
