Higher criticism to compare two large frequency tables, with sensitivity to possible rare and weak differences
用高阶批判检验比较大型频数表中的稀疏微弱差异AI 解读 ↓
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用高阶批判检验比较大型频数表中的稀疏微弱差异
- 研究问题
- 当两个大型频数表仅在未知且较少的类别上存在微弱差异时，如何检测这些差异，并刻画检验有无渐近功效的边界？
- 方法与思路
- 作者将各类别精确二项检验所得的P值汇入高阶批判检验，借助中等偏差分析研究稀疏程度、扰动强度与渐近功效。
- 材料陈述的贡献
- 摘要称，作者区分低计数与高计数情形并刻画功效区域；高计数下的相变曲线在形式上与双样本正态均值模型相同。
- 阅读关注点
- 应核对稀疏备择与扰动尺度的精确定义、两类计数情形的划分，以及最大功效和无功效结论各自需要的渐近条件。
- 阅读边界
- 仅据版本未注明的索引摘要；扰动尺度的数学排版需查原文，不能重构公式，也无法判断有限样本检验水平和实际功效。
原文摘要
We adapt Higher Criticism (HC) to the comparison of two frequency tables which may—or may not—exhibit moderate differences between the tables in some unknown, relatively small subset out of a large number of categories. Our analysis of the power of the proposed HC test quantifies the rarity and size of assumed differences and applies moderate deviations-analysis to determine the asymptotic powerfulness/powerlessness of our proposed HC procedure. Our analysis considers the null hypothesis of no difference in underlying generative model against a rare/weak perturbation alternative, in which the frequencies of N1−β out of the N categories are perturbed by r(logN)/2n in the Hellinger distance; here, n is the size of each sample. Our proposed Higher Criticism (HC) test for this setting uses P-values obtained from N exact binomial tests. We characterize the asymptotic performance of the HC-based test in terms of the sparsity parameter β and the perturbation intensity parameter r. Specifically, we derive a region in the (β,r)-plane where the test asymptotically has maximal power, while having asymptotically no power outside this region. Our analysis distinguishes between cases in which the counts in both tables are low, versus cases in which counts are high, corresponding to the cases of sparse and dense frequency tables. The phase transition curve of HC in the high-counts regime matches formally the curve delivered by HC in a two-sample normal means model.