An approximation for the power function of a non-parametric test of fit

We consider in this paper goodness-of-fit tests of the null hypothesis that the underlying d.f. of a sample F(.), belongs to a given family of distribution functions F. We propose a method for deriving approximate values of the power of a weighted Cramér-von Mises type test of goodness of fit. Our method relies on Karhunen-Loève [K.L] expansions on (0, 1] for weighted Brownian bridges.