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The supremum of the standardized empirical process is a promising statisticfor testing whether the distribution function $F$ of i.i.d. real randomvariables is either equal to a given distribution function $F_0$ (hypothesis)or $F \ge F_0$ (one-sided alternative). Since \cite{r5} it is well-known thatan affine-linear transformation of the suprema converge in distribution to theGumbel law as the sample size tends to infinity. This enables the constructionof an asymptotic level-$\alpha$ test. However, the rate of convergence isextremely slow. As a consequence the probability of the type I error is muchlarger than $\alpha$ even for sample sizes beyond $10.000$. Now, thestandardization consists of the weight-function $1/\sqrt{F_0(x)(1-F_0(x))}$.Substituting the weight-function by a suitable random constant leads to a newtest-statistic, for which we can derive the exact distribution (and the limitdistribution) under the hypothesis. A comparison via a Monte-Carlo simulationshows that the new test is uniformly better than the Smirnov-test and anappropriately modified test due to \cite{r20}. Our methodology also works forthe two-sided alternative $F \neq F_0$.

Exact and asymptotic goodness-of-fit tests based on the maximum and its location of the empirical process