Comparison of nonparametric goodness of fit tests

We consider two tests for testing the hypothesis that a density lies in a parametric class of densities and compare them by means of simulation. Both considered tests are based on the integrated squared distance of the kernel density estimator from its hypothetical expectation. However, different kernels are used. The unknown parameter will be replaced by its maximum-likelihood-estimation (m.l.e.). The power of both tests will be examined under local alternatives. Although both tests are asymptotically equivalent, it will be shown that there is a difference between the power of both tests when a finite number of random variables is used. Furthermore it will be shown that asymptotically equivalent approximations of the power can differ significantly when finite sample sizes are used.