On approximating the distributions of goodness-of-fit test statistics based on the empirical distribution function: the case of unknown parameters

This paper discusses some problems possibly arising when approximating via Monte-Carlo simulations the distributions of goodness-of-fit test statistics based on the empirical distribution function. We argue that failing to re-estimate unknown parameters on each simulated Monte-Carlo sample — and thus avoiding to employ this information to build the test statistic — may lead to wrong, overly-conservative. Furthermore, we present some simple examples suggesting that the impact of this possible mistake may turn out to be dramatic and does not vanish as the sample size increases.