  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 $F \neq F_0$.
