Partially parametric testing

Suppose a smooth test of goodness of fit has been applied to assess the validity of a parametric analysis, such as an analysis of variance. If the smooth test rejects the distributional assumption, the original parametric model can be replaced by the order k smooth alternative that is the basis of the smooth test. Here we demonstrate that basing the analysis on such an alternative, when it is consistent with the data, may result not only in a valid analysis, but also in a test with greater power than for the original parametric analysis. Several examples are given, including the one sample t-test, the one-way analysis of variance and randomised complete block designs.