对称条件下的极值概率问题与Hotelling的T^2检验
Extremal Probabilistic Problems and Hotelling's T^2 Test Under Symmetry Condition
摘要：
我们考虑任意d维样本的霍特林T^2统计量。如果抽样不太确定或不均匀，则在零均值假设下，T^2在分布上趋于χ^2_d。我们证明了由Efron引入的正交对称条件的检验可以被构造，它与基于χ^2_d的检验没有本质的区别，同时不仅适用于大的随机齐次样本，而且适用于所有的多维样本。主要的断言有形式的不等式，而不是极限定理;这些不等式是确切的代表某些极值问题的解决方案。让我们也提到一个辅助结果，它本身可能是有趣的：χ_d-（d-1）^{1/2}在d中的分布减少到它的极限N（0，1/2）。
英文摘要：
We consider Hotelling's T^2 statistic for an arbitrary d-dimensional sample. If the sampling is not too deterministic or inhomogeneous, then under zero means hypothesis, T^2 tends to χ^2_d in distribution. We show that a test for the orthant symmetry condition introduced by Efron can be constructed which does not essentially differ from the one based on χ^2_d and at the same time is applicable not only for large random homogeneous samples but for all multidimensional samples without exceptions. The main assertions have the form of inequalities, not that of limit theorems; these inequalities are exact representing the solutions to certain extremal problems. Let us also mention an auxiliary result which itself may be of interest: χ_d-(d-1)^{1/2} decreases in distribution in d to its limit N(0,1/2).
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