Goodness of fit tests, Critical values, Anderson - Darling statistic, Kolmogorov - Smirnov statistic, Kuiper statistic, Cramer - Von Mises statistic, Empirical distribution function, Monte-Carlo simulations

C12, C15, C63

This note 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 testing. Furthermore, we present a simple example suggesting that the impact of this possible mistake may turn out to be dramatic and does not vanish as the sample size increases.