Invariance of relative inverse function orderings under compositions of distributions
Let \(\mathcal{F}\) denote the class of continuous distributions on \([0,\infty)\) and let \(\mathbf{\Psi}\) be the class of continuous strictly increasing distribution functions on \([0,1]\). Let \(\leq_S\) be a preorder in \(\mathcal{F}\). The preorder \(\leq_S\) is said to be invariant under the class \(\mathbf{\Psi}\) if \(F\leq_S G\Rightarrow\psi F\leq_S\psi G\) for all \(\psi\in\mathbf{\Psi}\). The authors show that the function \(\lambda(t)=G^{-1}F(t)\), \(t>0\), is a maximal invariant under the group \(\mathbf{\Psi}\). As a result, the convex, star, superadditive, dispersive, and usual stochastic orders are invariant under that group. Some results about ordering the orbits and distances between ordered distributions are also given.
scientific article; zbMATH DE number 6086606