High-Dimensional  Multivariate Repeated Measures Analysis with Unequal Covariance Matrices

Abstract
In this talk, multivariate tests for repeated measures design are introduced for  high dimensional situation. By high dimension is meant the dimension of the multivariate observations and the total sample size grow together but either one could be larger than the other. Asymptotic distribution of the test 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 divergence of the sample sizes and the dimension of the repeated measures in the unequal covariance case.  In the equal covariance case, one can diverge at much faster rate than the other.  Consistent and unbiased estimators of the asymptotic variances, which make efficient use of all the  observations, are also derived.  Extension of the results to higher-order asymptotics and under fairly general conditions will also be discussed.  Simulation study provides favorable evidence for the accuracy of the asymptotic approximation under the null hypothesis.  In low-dimensional situation, power simulations have shown that the new methods have comparable power with a popular method known to work well in low dimension but the new methods have shown enormous advantage when the dimension is large. Data from Electroencephalograph (EEG) experiment will be analyzed to illustrate the application of the results.