Hotelling $T^2$ -统计量的Cramér中偏差定理及其在整体检验中的应用
A Cramér moderate deviation theorem for Hotelling's $T^2$-statistic with applications to global tests
摘要：
在有限$(3+δ)$阶矩下证明了Hotelling $T^2$统计量的Cramer中偏差定理.结果被施加到大规模的测试上的均值向量的平等，并示出，测试的数量可以大到$e^{o(n^{1/3})}$之前的卡方分布校准变得不准确。作为中等偏差结果的一个应用，基于Hotelling的$T^2$统计量的最大值给出了m个均值向量相等的全局检验，并证明了其渐近零分布是极值I型分布.提出了一种新的零分布的中间近似，以改善极端分布近似的缓慢收敛速度。数值研究表明，新的测试程序工作良好，即使是一个小的样本量，并表现出良好的分析乳腺癌数据集。
英文摘要：
A Cramer moderate deviation theorem for Hotelling's $T^2$-statistic is proved under a finite $(3+δ)$th moment. The result is applied to large scale tests on the equality of mean vectors and is shown that the number of tests can be as large as $e^{o(n^{1/3})}$ before the chi-squared distribution calibration becomes inaccurate. As an application of the moderate deviation results, a global test on the equality of m mean vectors based on the maximum of Hotelling's $T^2$-statistics is developed and its asymptotic null distribution is shown to be an extreme value type I distribution. A novel intermediate approximation to the null distribution is proposed to improve the slow convergence rate of the extreme distribution approximation. Numerical studies show that the new test procedure works well even for a small sample size and performs favorably in analyzing a breast cancer dataset.
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