Reweighted Smooth Tests of Goodness of Fit

Starting from a smooth test for goodness of fit we develop a method to alter this test to a reweighted smooth test, which has greater power to a certain directional alternative. Whereas the original smooth test gives equal weight to the different directions in the space of alternatives, the reweighted smooth test makes a distinction in importance of the directions. The advantage of this test, as opposed to directional tests, is that the smooth character is largely maintained. We discuss both nested and non nested directional alternatives.