Large-scale multiple two-sample {\em Student}'s $t$ testing problems often arise from the statistical analysis of scientific data. To detect components with different values between two mean vectors, a well-known procedure is to apply the Benjamini and Hochberg (B-H) method and two-sample {\em Student}'s $t$ statistics to control the false discovery rate (FDR). In many applications, mean vectors are expected to be sparse or asymptotically sparse. When dealing with such type of data, {\em can we gain more power than the standard procedure such as the B-H method with Student's $t$ statistics while keeping the FDR under control?} The answer is positive. By exploiting the possible sparsity information in mean vectors, we present an uncorrelated screening-based (US) FDR control procedure, which is shown to be more powerful than the B-H method. The US testing procedure depends on a novel construction of screening statistics, which are asymptotically uncorrelated with two-sample {\em Student}'s $t$ statistics. The US testing procedure is different from some existing {\em testing following screening} methods (Reiner, et al., 2007; Yekutieli, 2008) in which independence between screening and testing is crucial to control the FDR, while the independence often requires additional data or splitting of samples. An inappropriate splitting of samples may result in a loss rather than an improvement of statistical power. Instead, the uncorrelated screening US is based on the original data and does not need to split the samples. Theoretical results show that the US testing procedure controls the desired FDR asymptotically.
Numerical studies are conducted and indicate that the proposed procedure works quite well.
Two-sample testing in non-sparse high-dimensional linear models
在分析高维模型时，模型参数的稀疏性是一个常见但通常不可取的假设。本文研究了如下两个样本检验问题：给定两个由两个高维线性模型生成的样本，检验这两个线性模型的回归系数是否相同.我们提出了一个名为TIERS（TestIng Equality of Regression Slopes的缩写）的框架，它解决了两个样本的测试问题，而不对回归参数的稀疏性做任何假设。TIERS通过将两个样本进行卷积来构建新的模型，使得原始假设转化为新的矩条件。一个自规范化的结构，然后开发形成一个时刻测试。我们为开发的框架提供了严格的理论。在非常弱的条件下的特征协方差，我们表明，所提出的测试在控制I型错误的准确性是强大的缺乏稀疏的功能和重尾的误差分布，即使当样本量是小于的特征尺寸。此外，我们讨论了极大极小最优性和效率的性质所提出的测试。仿真分析表明，我们的测试具有良好的有限样本性能。在推导测试时，我们还开发了独立感兴趣的工具。该测试是建立在一种新的估计，称为自动自适应Dantzig的（ADDS），它不仅自动选择一个适当的规模的误差项，但也结合了先验信息。为了有效地近似检验统计量的临界值，我们开发了一种新的高维插件的方法，补充了高斯逼近论的最新进展。
