ABSTRACT

For high-dimensional two-sample Behrens–Fisher problems, several non-scale-invariant and scale-invariant tests have been proposed. Most of them impose strong assumptions on the underlying group covariance matrices so that their test statistics are asymptotically normal. However, in practice, these assumptions may not be satisfied or hardly be checked so that these tests may not be able to maintain the nominal size well in practice. To overcome this difficulty, in this paper, a normal reference scale-invariant test is proposed and studied. It works well by neither imposing strong assumptions on the underlying group covariance matrices nor assuming their equality. It is shown that under some regularity conditions and the null hypothesis, the proposed test and a chi-square-type mixture have the same normal and non-normal limiting distributions. It is then justifiable to approximate the null distribution of the proposed test using that of the chi-square-type mixture. The distribution of the chi-square type mixture can be well approximated by the Welch–Satterthwaite chi-square-approximation with the approximation parameter consistently estimated from the data. The asymptotic power of the proposed test is established. Numerical results demonstrate that the proposed test has much better size control and power than several well-known non-scale-invariant and scale-invariant tests.

摘要

对于高维二维样本 Behrens-Fisher 问题，已经提出了几种非标度不变和标度不变检验。他们中的大多数对基础组协方差矩阵强加了强假设，因此他们的检验统计量是渐近正态的。然而，在实践中，这些假设可能不被满足或几乎没有被检查，因此这些测试可能无法在实践中很好地保持标称尺寸。为了克服这一困难，本文提出并研究了一种正态参考尺度不变检验。它既不对基础组协方差矩阵强加强假设也不假设它们相等，因此效果很好。结果表明，在某些规律性条件和零假设下，所提出的检验和卡方型混合具有相同的正态和非正态极限分布。然后，使用卡方混合类型来近似拟议检验的零分布是合理的。卡方类型混合的分布可以通过 Welch–Satterthwaite 卡方近似很好地近似，其近似参数从数据中一致估计。建立了所提出的检验的渐近幂。数值结果表明，所提出的测试比几个众所周知的非尺度不变和尺度不变测试具有更好的尺寸控制和功率。建立了所提出的检验的渐近幂。数值结果表明，所提出的测试比几个众所周知的非尺度不变和尺度不变测试具有更好的尺寸控制和功率。建立了所提出的检验的渐近幂。数值结果表明，所提出的测试比几个众所周知的非尺度不变和尺度不变测试具有更好的尺寸控制和功率。