On Approximating the Distributions of goodness-of-Fit Test Statistics Based on the Empirical Distribution Function: the Case of Unknown Parameters

This note discusses some problems possibly arising when approximating via MonteCarlo 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.