Asymptotic distributions of some test criteria for the mean vector with fewer observations than the dimension
Asymptotic distributions of some test criteria for the mean vector with fewer observations than the dimension
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DOI：
10.1016/j.jmva.2013.01.008
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发表时间：
2013-04
期刊：
影响因子：
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通讯作者：
Shota Katayama;Y. Kano;M. Srivastava
中科院分区：
文献类型：
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作者：
Shota Katayama;Y. Kano;M. Srivastava
关于高维数据的平均向量的假设检验问题已经被许多作者研究过。他们提出了几个检验标准，并在一定的限制条件下得到了它们的渐近分布，当样本大小和维数都趋于无穷大时。实际上，这些作者使用的条件排除了总体协方差矩阵特征值激增的典型情况，例如，具有复合对称结构的总体协方差矩阵（方差相同；协方差相同）。在本文中，我们放宽了它们的条件以包含这些重要的情况，得到了具有中等尖峰特征值的总体协方差矩阵的非标准渐近分布，即正态分布与卡方分布的卷积，以及具有相当尖峰特征值的总体协方差矩阵的卡方分布的卷积形式的渐近分布。
The problem of hypothesis testing concerning the mean vector for high dimensional data has been investigated by many authors. They have proposed several test criteria and obtained their asymptotic distributions, under somewhat restrictive conditions, when both the sample size and the dimension tend to infinity. Indeed, the conditions used by these authors exclude a typical situation where the population covariance matrix has spiked eigenvalues, as for instance, the population covariance matrix with the compound symmetry structure (the variances are the same; the covariances are the same). In this paper, we relax their conditions to include such important cases, obtaining rather non-standard asymptotic distributions which are the convolution of normal and chi-squared distributions for the population covariance matrix with moderate spiked eigenvalues, and obtaining the asymptotic distributions in the form of convolutions of chi-square distributions for the population covariance matrix with quite spiked eigenvalues.