稀疏方案下的全局检验：ANOVA，多重比较和更高的批评
Global testing under sparse alternatives: ANOVA, multiple comparisons and the higher criticism
摘要：
检验线性模型中回归系数子集的显著性，是统计分析的一个主要内容，至少可以追溯到Fisher的工作，他引入了方差分析（ANOVA）。我们研究这个问题的假设下，系数向量是稀疏的，在现代高维设置中的常见情况。假设我们有$p$协变量，并且在替代方案下，响应仅取决于$p^{1-α}$的顺序，即$0\leα\le1$。在中等稀疏水平下，即$0\leα\le1/2$，我们证明了方差分析在某些条件下是最优的。在强稀疏约束下，即$α>1/2$下，情况不再如此。在这种情况下，通常首选多重比较程序，我们在$α\geq3/4$时确定其最优性。然而，这两个非常流行的方法是次优的，有时无能为力，在中等强度的稀疏性下，$1/2<α<3/4$。我们提出了一种基于更高的批评，是强大的在整个范围$α>1/2$的方法。这种最优性属性适用于各种设计，包括经典的（平衡的）多路设计和更现代的“$p>n$“设计中出现的遗传学和信号处理。除了标准的固定效应模型，我们建立了一个随机效应模型的回归向量的非零系数呈正态分布的类似结果。
英文摘要：
Testing for the significance of a subset of regression coefficients in a linear model, a staple of statistical analysis, goes back at least to the work of Fisher who introduced the analysis of variance (ANOVA). We study this problem under the assumption that the coefficient vector is sparse, a common situation in modern high-dimensional settings. Suppose we have $p$ covariates and that under the alternative, the response only depends upon the order of $p^{1-α}$ of those, $0\leα\le1$. Under moderate sparsity levels, that is, $0\leα\le1/2$, we show that ANOVA is essentially optimal under some conditions on the design. This is no longer the case under strong sparsity constraints, that is, $α>1/2$. In such settings, a multiple comparison procedure is often preferred and we establish its optimality when $α\geq3/4$. However, these two very popular methods are suboptimal, and sometimes powerless, under moderately strong sparsity where $1/2<α<3/4$. We suggest a method based on the higher criticism that is powerful in the whole range $α>1/2$. This optimality property is true for a variety of designs, including the classical (balanced) multi-way designs and more modern "$p>n$" designs arising in genetics and signal processing. In addition to the standard fixed effects model, we establish similar results for a random effects model where the nonzero coefficients of the regression vector are normally distributed.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
