In this paper, test statistics for repeated measures design are introduced when the dimension is large. By large dimension is meant the number of repeated measures and the total sample size grow together but either one could be larger than the other. Asymptotic distribution of the statistics are derived for the equal as well as unequal covariance cases in the balanced as well as unbalanced cases. The asymptotic framework considered requires proportional growth of the sample sizes and the dimension of the repeated measures in the unequal covariance case. In the equal covariance case, one can grow at much faster rate than the other. The derivations of the asymptotic distributions mimic that of Central Limit Theorem with some important peculiarities addressed with sufficient rigor. Consistent and unbiased estimators of the asymptotic variances, which make efficient use of all the observations, are also derived. Simulation study provides favorable evidence for the accuracy of the asymptotic approximation under the null hypothesis. Power simulations have shown that the new methods have comparable power with a popular method known to work well in low-dimensional situation but the new methods have shown enormous advantage when the dimension is large. Data from Electroencephalograph (EEG) experiment is analyzed to illustrate the application of the results.

中文翻译：

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不等协方差矩阵的高维多元重复测量分析

本文引入了维数较大时重复测量设计的检验统计量。大维度是指重复测量的数量和总样本量一起增长，但其中一个可能大于另一个。统计量的渐近分布是针对平衡和不平衡情况下的相等和不等协方差情况导出的。在协方差不等的情况下，所考虑的渐近框架要求样本大小和重复测量的维度成比例增长。在协方差相等的情况下，一个可以比另一个更快地增长。渐近分布的推导模仿了中心极限定理的推导，其中一些重要的特性得到了足够的严格处理。渐近方差的一致且无偏估计量，还导出了有效利用所有观察结果的方法。仿真研究为原假设下渐近逼近的准确性提供了有利的证据。功效模拟表明，新方法与已知在低维情况下运行良好的流行方法具有相当的功效，但新方法在维数较大时显示出巨大的优势。分析来自脑电图 (EEG) 实验的数据以说明结果的应用。功效模拟表明，新方法与已知在低维情况下效果良好的流行方法具有相当的功效，但新方法在维数较大时显示出巨大的优势。分析来自脑电图 (EEG) 实验的数据以说明结果的应用。功效模拟表明，新方法与已知在低维情况下效果良好的流行方法具有相当的功效，但新方法在维数较大时显示出巨大的优势。分析来自脑电图 (EEG) 实验的数据以说明结果的应用。