Nonparametric Statistics: Asymptotics

Abstract

Nonparametric methods are procedures of statistical inference for models without precise assumptions about the distribution of the sample. A typical case is the two-sample test problem, where samples from two populations are observed, and the hypothesis that two populations have the same distribution is tested. Various types of test procedures have been proposed, their distributions under the hypothesis are obtained from the permutation distribution, that is, the conditional distribution given the set of values of the two samples combined. In most cases, small sample exact distribution is complicated to compute, but asymptotic approximations based on the central limit theorem for the permutation distribution can be applied. Also, power of the tests can be calculated from the normal approximation of the test statistic under the contiguous alternatives, and the relative efficiency of the test is defined and calculated. There are classes of tests with high relative efficiency. Also optimal parametric test can be regarded as approximate nonparametric tests in this context. Similar theory obtains for other classes of nonparametric tests and for interval estimation procedures derived from them.