High-dimensional sparse generalized linear models (GLMs) have emerged in the setting that the number of samples and the dimension of variables are large, and even the dimension of variables grows faster than the number of samples. False discovery rate (FDR) control aims to identify some small number of statistically significantly nonzero results after getting the sparse penalized estimation of GLMs. Using the CLIME method for precision matrix estimations, we construct the debiased-Lasso estimator and prove the asymptotical normality by minimax-rate oracle inequalities for sparse GLMs. In practice, it is often needed to accurately judge each regression coefficient's positivity and negativity, which determines whether the predictor variable is positively or negatively related to the response variable conditionally on the rest variables. Using the debiased estimator, we establish multiple testing procedures. Under mild conditions, we show that the proposed debiased statistics can asymptotically control the directional (sign) FDR and directional false discovery variables at a pre-specified significance level. Moreover, it can be shown that our multiple testing procedure can approximately achieve a statistical power of 1. We also extend our methods to the two-sample problems and propose the two-sample test statistics. Under suitable conditions, we can asymptotically achieve directional FDR control and directional FDV control at the specified significance level for two-sample problems. Some numerical simulations have successfully verified the FDR control effects of our proposed testing procedures, which sometimes outperforms the classical knockoff method.
翻译:高度分散的广度线性模型(GLMS)出现在一个背景中,即样本数量和变量的维度是巨大的,甚至变量的维度比样本数量增长得快。虚假发现率(FDR)控制的目的是在获得对GLMS的微小的受限估计后,确定少量统计性和非零结果。我们使用CLIME方法进行精确矩阵估计,我们构建了低偏差-Lasso测算器,并证明对稀少的GLMS来说,微缩降压率或触角值不平等无常性。在实践中,常常需要准确地判断每个回归系数的正比和负偏差性,这决定了预测值变量与其余变量的可变数是否正或负相关。我们使用低偏差的测算器,我们建立了多种测试程序。在温度条件下,我们提出的降低偏差统计数据可以以静态控制方向(信号) FDDR和方向性发现变量在某种前的意义级别上有时要准确地判断每个回归系数的比值值。此外,我们还可以通过两种测算方法来测试我们的直径测测算。