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.

翻译：本文建立了独立随机变量序列的二次型与最大值之间的渐近独立性。基于这一理论结果，我们得到了二次型与最大值的渐近联合分布，可应用于高维检验问题。通过结合和型检验与最大值型检验，我们提出了针对单样本均值检验和两样本均值检验的Fisher组合检验。在这一新颖的一般框架下，我们放宽了现有文献中的若干强假设。蒙特卡洛模拟表明，我们提出的检验方法对稀疏数据和稠密数据均具有强稳健性。