On the Weak Consistency of Permutation Tests

Consistency of some nonparametric tests with real variables has been studied by several authors under the assumption that population variance is finite and/or in the presence of some violations of the data exchangeability between samples. Since main inferential conclusions of permutation tests concern the actual dataset, where sample sizes are held fixed, we consider the notion of consistency in the weak version (in probability). Here, we characterize weak consistency of permutation tests assuming population mean is finite and without assuming existence of population variance. Moreover, since permutation test statistics do not require to be standardized, we do not assume that data are homoscedastic in the alternative. Several application examples to mostly used test statistics are discussed. A simulation study and some hints for robust testing procedures are also presented.