Conditional limit laws for goodness-of-fit tests

We study the conditional distribution of goodness of fit statistics of the Cram'{e}r–von Mises type given the complete sufficient statistics in testing for exponential family models. We show that this distribution is close, in large samples, to that given by parametric bootstrapping, namely, the unconditional distribution of the statistic under the value of the parameter given by the maximum likelihood estimate. As part of the proof, we give uniform Edgeworth expansions of Rao–Blackwell estimates in these models.