Consistency and Unbiasedness of Certain Nonparametric Tests

It is shown that there exist strictly unbiased and consistent tests for the univariate and multivariate two- and k-sample problem, for the hypothesis of independence, and for the hypothesis of symmetry with respect to a given point. Certain new tests for the univariate two-sample problem are discussed. The large sample power of these tests and of the Mann-Whitney test are obtained by means of a theorem of Hoeffding. There is a discussion of the problem of tied observations.