This book deals with the permutation approach to a variety of univariate and multivariate problems of hypothesis testing in a nonparametric framework. The great majority of univariate problems may be usefully and effectively solved within standard parametric or nonparametric methods as well, although in relatively mild conditions their permutation counterparts are generally asymptotically as good as the best parametric ones. Moreover, it should be noted that permutation methods are essentially of a nonparametrically exact nature in a conditional context. In addition, there are a number of parametric tests the distributional behavior of which is only known asymptotically. Thus, for most sample sizes of practical interest, the relative lack of efficiency of permutation solutions may sometimes be compensated by the lack of approximation of parametric asymptotic counterparts. Moreover, when responses are normally distributed and there are too many nuisance parameters to estimate and remove, due to the fact that each estimate implies a reduction of the degrees of freedom in the overall analysis, it is possible for the permutation solution to become better than its parametric counterpart (see, for example, Chapter 6). In addition, assumptions regarding the validity of parametric methods (such as normality and random sampling) are rarely satisfied in practice, so that consequent inferences, when not improper, are necessarily approximated, and their approximations are often difficult to assess.
KeywordsNull Distribution Global Test Permutation Distribution Global Null Hypothesis Intersection Hypothesis
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