The intermediates take it all: Asymptotics of higher criticism statistics and a powerful alternative based on equal local levels
The intermediates take it all: Asymptotics of higher criticism statistics and a powerful alternative based on equal local levels
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中间体占据一切：更高批评统计的渐近和基于平等地方水平的强大替代方案
DOI：
10.1002/bimj.201300255
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发表时间：
2015
影响因子：
1.7
通讯作者：
H. Finner
中科院分区：
文献类型：
--
作者：
V. Gontscharuk;S. Landwehr;H. Finner
高批评（HC）统计可以看作是著名的科尔莫戈洛夫-斯米尔诺夫统计的标准化版本，有着悠久的历史，可以追溯到七十年代中期。最初，HC 统计量与拟合优度 (GOF) 检验结合使用，但最近在检验高维数据中的全局零假设方面引起了一些关注。人们对 HC 的持续兴趣似乎是受到与该统计数据相关的一系列良好渐近特性的启发。例如，与 Kolmogorov-Smirnov 检验不同，基于 HC 统计量的 GOF 检验已知在中等尾部中渐近敏感，因此它适用于检测稀疏混合模型中信号的存在。然而，有关 HC 统计量渐近行为的一些问题仍然悬而未决。我们重点关注其中两个问题，即为什么特定的中间范围对于基于 HC 统计的 GOF 测试至关重要，以及为什么 HC 分布向极限分布的收敛极其缓慢。此外，HC 统计量的渐近和有限行为的不一致促使我们提供一种新的 HC 检验，它比原始 HC 检验具有更好的有限性质，同时显示相同的渐近性。该测试的动机是与原始 HC 测试相关的所谓局部水平的渐近行为。通过数值计算和模拟，我们表明新的 HC 测试通常比普通混合物模型中的原始 HC 测试更强大。
The higher criticism (HC) statistic, which can be seen as a normalized version of the famous Kolmogorov–Smirnov statistic, has a long history, dating back to the mid seventies. Originally, HC statistics were used in connection with goodness of fit (GOF) tests but they recently gained some attention in the context of testing the global null hypothesis in high dimensional data. The continuing interest for HC seems to be inspired by a series of nice asymptotic properties related to this statistic. For example, unlike Kolmogorov–Smirnov tests, GOF tests based on the HC statistic are known to be asymptotically sensitive in the moderate tails, hence it is favorably applied for detecting the presence of signals in sparse mixture models. However, some questions around the asymptotic behavior of the HC statistic are still open. We focus on two of them, namely, why a specific intermediate range is crucial for GOF tests based on the HC statistic and why the convergence of the HC distribution to the limiting one is extremely slow. Moreover, the inconsistency in the asymptotic and finite behavior of the HC statistic prompts us to provide a new HC test that has better finite properties than the original HC test while showing the same asymptotics. This test is motivated by the asymptotic behavior of the so‐called local levels related to the original HC test. By means of numerical calculations and simulations we show that the new HC test is typically more powerful than the original HC test in normal mixture models.