Parametric versus nonparametric goodness of fit: Another view

We consider chi-squared type tests for testing the hypothesis H 0 that a density f of observations X1,..., Xn lies in a parametric class of densities F. We consider a version of chi-squared type test using kernel estimates for the density. The main result is, following Liero, Läuter and Konakov (1998) the derivation of the asymptotic behavior of the power of the test 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 parametric estimator, the power of the test are studied.