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