  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.
