Analysis of high-dimensional one group repeated measures designs

We propose a novel one sample test for repeated measures designs and derive its limit distribution for the situation where both the sample size n as well as the dimension d of the observations go to infinity. This covers the high-dimensional case with d > n. The tests are based on a quadratic form which involve new unbiased and dimension-stable estimators of different traces of the underlying unrestricted covariance structure. It is shown that the asymptotic distribution of the statistic may be standard normal, standardized chi(2)-distributed, or even of weighted chi(2)-form in some situations. To this end, we suggest an approximation technique which is asymptotically valid in the first two cases and provides an accurate approximation for the latter. We motivate and illustrate the application with a sleep lab data set and also discuss the practical meaning of d -> infinity in case of repeated measures designs. It turns out that the limit behaviour depends on how the number of repeated measures is increased which is crucial for application.