高维中柯西组合 Hettmansperger-Randles 位置检验

中文摘要 AI 辅助

高维均值检验在强相关和重尾分布下具有挑战性，因为经典的 Hotelling 统计量是病态的，而基于样本矩的正则化版本对异常值仍然敏感。本文针对椭圆分布下的单样本均值推断，开发了一个稳健的正则化 Hotelling 框架。所提出的 HRST 统计量将 Hettmansperger-Randles 空间位置估计器与迹归一化收缩 Tyler 散度矩阵的逆相结合，并通过岭-预解随机矩阵理论而非稀疏协方差或精度矩阵假设来校准所得的二次型。在 $p/n\to y\in(0,\infty)$ 的范围内，允许消失的下谱边和有限秩发散尖峰，我们为每个收缩水平建立了可行的渐近正态性，推导了具有显式非中心参数的局部备择分布，并证明了有限收缩网格上的联合高斯极限。这些结果证明了一个跨收缩水平的自适应柯西组合检验的合理性。模拟和一项配对肿瘤-正常基因表达研究表明，HRST 在重尾分布下保持检验水平并提高稳健性，同时在强相关下保持功效。

英文摘要

High-dimensional mean testing is challenging under strong dependence and heavy tails, since classical Hotelling statistics are ill posed and regularized versions based on sample moments remain sensitive to outliers. This paper develops a robust regularized Hotelling framework for one-sample mean inference under elliptical distributions. The proposed HRST statistic combines a Hettmansperger-Randles spatial location estimator with the inverse of a trace-normalized shrinkage Tyler scatter matrix, and calibrates the resulting quadratic form through ridge-resolvent random-matrix theory rather than sparse covariance or precision-matrix assumptions. In the regime $p/n\to y\in(0,\infty)$, allowing a vanishing lower bulk edge and finite-rank diverging spikes, we establish feasible asymptotic normality for each shrinkage level, derive local-alternative distributions with explicit noncentrality parameters, and prove joint Gaussian limits over finite shrinkage grids. These results justify an adaptive Cauchy combination test across shrinkage levels. Simulations and a paired tumor-normal gene-expression study show that HRST maintains size and improves robustness under heavy-tailed distributions while retaining power under strong correlation.

发表机构

• School of Mathematical Sciences, Tiangong University（天津大学数学科学学院）
• School of Statistics and Data Science, Nankai University（南开大学统计与数据科学学院）
• Faculty of Arts and Sciences, Beijing Normal University（北京师范大学文理学院）

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