ABSTRACT
This chapter explains several applications of permutation tests for the general hypotheses. Permutation tests are based on resampling, but unlike the ordinary bootstrap, the samples are drawn without replacement. Permutation tests are often applied as a nonparametric test of the general hypothesis. Multivariate tests based on maximum likelihood depend on distributional assumptions about the underlying populations. Many of the procedures that are available for the multivariate two-sample problem require a computational approach for implementation. P. J. Bickel constructed a consistent distribution-free multivariate extension of the univariate Smirnov test by conditioning on the pooled sample. A multivariate test for equal distributions is based on nearest neighbors. The nearest neighbor tests are a type of test based on ordered distances between sample elements, which can be applied when the distributions are continuous. The multivariate rth nearest neighbor test can be implemented by an approximate permutation test.
