基于柯西组合的随机投影检验用于两样本均值

中文摘要 AI 辅助

当维度超过样本量时，高维两样本均值检验具有挑战性。Lopes等人（2011）提出的随机投影方法通过将数据映射到较低维空间来解决这一困难，在该空间中可应用Hotelling的$T^2$统计量，同时保留有用的协方差信息，并在变量表现出不可忽略的协方差结构时提高检验功效。然而，单一投影检验可能对实现的投影矩阵敏感，而现有的多重投影程序通常依赖重抽样或模拟进行校准，理论理解有限。此外，基于投影的Hotelling检验可能对稀疏均值差异不敏感。为解决这些局限性，我们提出了一种柯西组合随机投影检验（CRPT），该方法在多次独立随机投影后应用Hotelling的$T^2$检验，并通过柯西变换组合投影后的$p$值。所提出的方法保留了随机投影方法纳入协方差信息的能力，同时减少了对任何单一投影的依赖。在高斯假设下，我们建立了所提出统计量的零假设尾部行为，并进一步研究了其在适当备择假设下的渐近功效。为提高对稀疏备择假设的敏感性，我们进一步开发了CRPT的功效增强版本。通过模拟研究和真实数据分析来检验所提出程序的性能和实践适用性。

英文摘要

High-dimensional two-sample mean testing is challenging when the dimension exceeds the sample size. The random projection method proposed by Lopes et al. (2011) addresses this difficulty by mapping the data to a lower dimension space where Hotelling's $T^2$ statistic can be applied, while retaining useful covariance information and gaining power when the variables exhibit non-negligible covariance structure. However, single projection tests may be sensitive to the realized projection matrix, whereas existing multiple projection procedures often rely on resampling or simulation for calibration, with limited theoretical understanding. Moreover, projection-based Hotelling tests may be less sensitive to sparse mean differences. To address these limitations, we propose a Cauchy-combined random projection test (CRPT), which applies Hotelling's $T^2$ test after multiple independent random projections and combines the projected $p$-values through the Cauchy transformation. The proposed method retains the ability of random projection methods to incorporate covariance information while reducing reliance on any single projection. Under the Gaussian assumption, we establish the null tail behavior of the proposed statistic and further investigate its asymptotic power under suitable alternatives. To improve sensitivity to sparse alternatives, we further develop a power-enhanced version of CRPT. Simulation studies and real data analysis are conducted to examine the performance and practical applicability of the proposed procedures.

发表机构

• School of Sciences, Chang’an University（长安大学理学院）
• Center for Statistics and Data Science, Beijing Normal University（北京师范大学统计与数据科学中心）
• Center for Applied Statistics and School of Statistics, Renmin University of China（中国人民大学应用统计中心及统计学院）

机构由 AI 辅助整理，请以论文原文为准。