Although the Hotelling's T2$T^2$$T^2$ test has been a widely used test for hypothesis testing problems on the mean vectors, it is not well defined when the data dimension is larger than the sample size. Dempster's non-exact test, as a remedy for the Hotelling's T2$T^2$$T^2$ test, is known to be more powerful than the Hotelling's T2$T^2$$T^2$ test and is well defined even when the dimension is much larger than the sample size. However, Dempster's non-exact test will lose power when the variances of the covariates are different. In this paper, we propose a standardized Dempster's non-exact test for the classical mean testing problem. The proposed test is more powerful for data with heteroscedastic features and is applicable to the high-dimensional case. An approximate distribution of the test statistic has been established, and to better control the type I error rate when the sample size is small, we further constructed a Monte Carlo version of the proposed standardized Dempster's non-exact test. Various simulation studies and a real data application were conducted with comparison to other popular tests. The numerical results showed that while the type I error rates were well controlled, the testing power of our proposed test was generally higher than those of other tests.

中文翻译：

---

高维均值向量的标准化 Dempster 非精确检验

虽然霍特林的吨2${吨}^2$$T^2$test 是一种广泛用于均值向量上的假设检验问题的检验，当数据维度大于样本量时，它的定义并不明确。登普斯特的非精确检验，作为霍特林氏症的补救措施吨2${吨}^2$$T^2$测试，已知比霍特林的测试更强大吨2${吨}^2$$T^2$即使维度远大于样本量，也能很好地定义。但是，当协变量的方差不同时，Dempster 的非精确检验将失去功效。在本文中，我们针对经典均值检验问题提出了一种标准化的 Dempster 非精确检验。所提出的测试对于具有异方差特征的数据更强大，并且适用于高维情况。已经建立了检验统计量的近似分布，为了在样本量较小时更好地控制 I 类错误率，我们进一步构建了所提出的标准化 Dempster 非精确检验的蒙特卡罗版本。进行了各种模拟研究和真实数据应用，并与其他流行测试进行了比较。