We consider the following test problems:
a) Testing, whether a density function f of i.i.d. r.v.’s X1, ..., Xn belongs to a given parametric class of functions, that is
The test statistics for all these problems are based on a quadratic distance Qn of a nonparametric estimator of the function of interest, i.e. a kernel density estimator n, a kernel estimator of the conditional variance n and smoothed estimators of the cell probabilities nj, respectively, from the smoothed hypothetical functions. Using limit theorems for Qn we derive asymptotic -tests and investigate the power under local alternatives. Differences and common features and problems of these approaches are considered, comparisons to other test procedures are discussed.