Distance based approach for construction of rank tests

Critchlow (1992) proposed a construction based on distances which produces many familiar rank test statistics. The method allows the creation of families of statistics for standard nonparametric hypotheses, based on the same distance. The proposed test statistics are minimum interpoint distance between appropriate sets of permutations. These sets of permutations have group-theoretic descriptions, and the resulting rank statistic metrizes a quotient space of the permutation group. We enlarge the class of test statistics for five standard nonparametric hypothesis testing situations with new statistics based on Chebyshev's metric. Critchlow's construction is applied systematically to the two-sample and multi-sample location problems, the two-way layout problem, the one-sample location problem, and the problem of testing for trend. We also investigate some combinatorial and group theoretic properties of Chebyshev's metric that play a key role in the computation of the corresponding test statistics.