Permutation support for multivariate techniques

Many writers have been concerned with the appropriateness of the use of standard tests, based on normal theory assumptions, when the underlying distribution is properly suspected of being non-normal. In this paper the multivariate generalization of the t-test is investigated. The first four permutation cumulants are determined for a statistic which is a simple function of Hotelling's T2. The derived permutation cumulants are applied to samples from bivariate normal, rectangular and double exponential populations. It is found that there is a mild disagreement between the nominal significance level of the test, based on the assumption that the underlying distribution is normal, and the actual permutation significance level, obtained by considering as equally likely all permutations of each sample from the rectangular and double exponential populations. A method of adjusting the test criterion when one suspects that the data do not come from a normal population is discussed. It involves fitting the parameters of a standard beta distribution by the first two permutation cumulants. It works reasonably well at all of the various significance levels investigated.