Nonparametric versus parametric goodness of fit

We consider two tests for testing the hypothesis that a density f lies in a parametric class of densities. The first test is based on the integrated squared distance of the kernel density estimator from its hypothetical expectation, the second test is based on the maximal deviation of the kernel estimate on a grid. The unknown parameter is estimated by the maximum likelihood estimator. The main result is the derivation of the asymptotic behavior of the power of both tests under Pitman and "sharp peak" type alternatives. The connection of the rate of convergence of these local alternatives, the bandwidth of the kernel estimator, the parameter estimator and the power of both tests are studied and are compared. It turns out that under Pitman alternatives the L-2-test is always not worse than the L-infinity-test, but there exist sharp peak alternatives such that the L-infinity-test is better.