An Approximation for the Power Function of a Semi-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(x), belongs to a given family of distribution functions F. We propose a method for deriving approximate values of the power of a weighted Cram′er-von Mises type test of goodness of fit. Our method relies on Karhunen-Lo`eve [K.L] expansions on (0, 1) for the weighted a Brownian bridges.