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 in tegrated squared distance of the kernel density estimator from its hyp othetical expectation, the second test is based on the maximal deviati on of the kernel estimate on a grid. The unknown parameter is estimate d by the maximum likelihood estimator. The main result is the derivati on of the asymptotic behavior of the power of both tests under Pitman and ''sharp peak'' type alternatives. The connection of the rate of co nvergence of these local alternatives, the bandwidth of the kernel est imator, the parameter estimator and the power of both tests are studie d 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 s harp peak alternatives such that the L-infinity-test is better.