In order to obtain quantitative results about the influence of outliers on tests, their maximum size and minimum power over certain neighborhoods are evaluated asymptotically. The neighborhoods are defined in terms of $\varepsilon$-contamination and total variation, the tests considered are based on statistics $n^{-\frac{1}{2}} \sum^n_{i=1} IC(x_i), IC$ a rather arbitrary function. Furthermore, the unique $IC^\ast$ is determined that leads to a maximin test with respect to this subclass of tests. A comparison with the likelihood ratio of least favorable pairs shows that the test based on $n^{-\frac{1}{2}}\sum^n_{i=1} IC^\ast(x_i)$ is in fact maximin among all tests at a given level. Tests based on $(M)$-statistics are also considered.