Goodness-of-fit: a comparison of parametric bootstrap and exact conditional tests

We study goodness of fit tests for exponential families. We compare, via Monte Carlo simulations, the powers of exact conditional tests based on co-sufficient samples (samples from the conditional distribution given the sufficient statistic) and approximate unconditional tests based on the parametric bootstrap. We use the Gibbs sampler to generate the co-sufficient samples. The gamma and von Mises families are investigated, and the Cramer-von Mises and Watson test statistics are applied. The results of this study show that those two tests have very similar powers even for samples of very small size, such as n=5 for the gamma family and n=10 for the von Mises family.