  We propose a new approach to the problem of high-dimensional multivariate
ANOVA via bootstrapping max statistics that involve the differences of sample
mean vectors. The proposed method proceeds via the construction of simultaneous
confidence regions for the differences of population mean vectors. It is suited
to simultaneously test the equality of several pairs of mean vectors of
potentially more than two populations. By exploiting the variance decay
property that is a natural feature in relevant applications, we are able to
provide dimension-free and nearly-parametric convergence rates for Gaussian
approximation, bootstrap approximation, and the size of the test. We
demonstrate the proposed approach with ANOVA problems for functional data and
sparse count data. The proposed methodology is shown to work well in
simulations and several real data applications.
