独立随机变量的二次型与最大值的渐近独立性及其在高维检验中的应用
Asymptotic Independence of the Quadratic form and Maximum of Independent Random Variables with Applications to High-Dimensional Tests
AI总结：
本文证明了独立随机变量二次型与最大值的渐近独立性，据此提出结合 sum-type 与 max-type 的 Fisher 组合检验用于高维均值检验，放宽了现有假设并在模拟中对稀疏与密集数据均表现出强稳健性。
摘要：
本文建立了一列独立随机变量的二次型与最大值之间的渐近独立性。基于这一理论结果，我们求得了二次型与最大值的渐近联合分布，并可将其应用于高维检验问题。通过结合 sum-type 检验与 max-type 检验，我们提出了用于单样本均值检验和双样本均值检验的 Fisher 组合检验。在这一新颖的通用框架下，现有文献中的若干强假设被放宽。Monte Carlo 模拟表明，我们提出的检验对稀疏数据和密集数据均具有很强的稳健性。
英文摘要：
This paper establishes the asymptotic independence between the quadratic form and maximum of a sequence of independent random variables. Based on this theoretical result, we find the asymptotic joint distribution for the quadratic form and maximum, which can be applied into the high-dimensional testing problems. By combining the sum-type test and the max-type test, we propose the Fisher's combination tests for the one-sample mean test and two-sample mean test. Under this novel general framework, several strong assumptions in existing literature have been relaxed. Monte Carlo simulation has been done which shows that our proposed tests are strongly robust to both sparse and dense data.
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