论文标题
最大测试的最佳测试，用于检测使用高斯或较重尾巴的稀疏信号
Optimality of the max test for detecting sparse signals with Gaussian or heavier tail
论文作者
论文摘要
高维测试中的一个基本问题是全球无效测试：测试无效的$ n $假设是否同时存在。最大测试将最小的$ n $边际P值作为测试统计量，以其简单性和鲁棒性而广泛流行。但是，它相对于其他测试的理论性能受到质疑。在全球测试问题的高斯序列版本中，Donoho和Jin（2004）发现了一个所谓的“弱，稀疏”渐近状态，在该制度中，当所有非零信号强度相同时，较高的批评和Berk-Jones测试就具有比最大测试更好的最大检测边界。我们研究了一个更通用的模型，其中非编号是从通用分布中得出的，并表明，最大测试的检测边界在“弱，稀疏”状态下是最佳的，只要分布的尾巴不比高斯较轻。此外，我们从理论上和模拟中表明，当非零含量的分布具有多项式尾巴时，对Donoho和Jin（2004）的更高批评的更高批评可能具有非常低的功率。
A fundamental problem in high-dimensional testing is that of global null testing: testing whether the null holds simultaneously in all of $n$ hypotheses. The max test, which uses the smallest of the $n$ marginal p-values as its test statistic, enjoys widespread popularity for its simplicity and robustness. However, its theoretical performance relative to other tests has been called into question. In the Gaussian sequence version of the global testing problem, Donoho and Jin (2004) discovered a so-called "weak, sparse" asymptotic regime in which the higher criticism and Berk-Jones tests achieve a better detection boundary than the max test when all of the nonzero signal strengths are identical. We study a more general model in which the non-null means are drawn from a generic distribution, and show that the detection boundary for the max test is optimal in the "weak, sparse" regime, provided that the distribution's tail is no lighter than Gaussian. Further, we show theoretically and in simulation that the modified higher criticism of Donoho and Jin (2004) can have very low power when the distribution of non-null means has a polynomial tail.
