Nonparametric two-sample tests for increasing convex order

Given two independent samples of non-negative random variables with unknown distribution functions $F$ and $G$, respectively, we introduce and discuss two tests for the hypothesis that $F$ is less than or equal to $G$ in increasing convex order. The test statistics are based on the empirical stop-loss transform, critical values are obtained by a bootstrap procedure. It turns out that for the resampling a size switching is necessary. We show that the resulting tests are consistent against all alternatives and that they are asymptotically of the given size $\alpha$. A specific feature of the problem is the behavior of the tests `inside' the hypothesis, where $F\not=G$. We also investigate and compare this aspect for the two tests.