  Motivated by the prevalence of high dimensional low sample size datasets in
modern statistical applications, we propose a general nonparametric framework,
Direction-Projection-Permutation (DiProPerm), for testing high dimensional
hypotheses. The method is aimed at rigorous testing of whether lower
dimensional visual differences are statistically significant. Theoretical
analysis under the non-classical asymptotic regime of dimension going to
infinity for fixed sample size reveals that certain natural variations of
DiProPerm can have very different behaviors. An empirical power study both
confirms the theoretical results and suggests DiProPerm is a powerful test in
many settings. Finally DiProPerm is applied to a high dimensional gene
expression dataset.
