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

The supremum of the standardized empirical process is a promising statistic for testing whether the distribution function $F$ of i.i.d. real random variables 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 that an affine-linear transformation of the suprema converge in distribution to the Gumbel law as the sample size tends to infinity. This enables the construction of an asymptotic level-$\alpha$ test. However, the rate of convergence is extremely slow. As a consequence the probability of the type I error is much larger than $\alpha$ even for sample sizes beyond $10.000$. Now, the standardization 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 new test-statistic, for which we can derive the exact distribution (and the limit distribution) under the hypothesis. A comparison via a Monte-Carlo simulation shows that the new test is uniformly better than the Smirnov-test and an appropriately modified test due to \cite{r20}. Our methodology also works for the two-sided alternative $F \neq F_0$.