Generalized differentiation of probability functions: parameter dependent sets given by intersections of convex sets and complements of convex sets
The authors consider the probability functions defined on parameter dependent sets defined as intersections of convex sets and their complements (the general form of such a function is \(\varphi(x)=\mathbb{P}[\hat{g}(x,\xi)\leq 0]\)). They provide conditions under which the functions are sub-differentiable and a condition under which the function is smooth. All of this without any strong assumptions about the type of the distribution of the random vector \(\xi\) can have an arbitrary density with respect to the Lebesgue measure, bounded on compact sets. It is also not assumed that the set \(\{z: \hat{g}(x,z)\leq 0\}\) is bounded. The paper covers the situations known from the literature where some sources of non-smoothness may appear.
scientific article; zbMATH DE number 7490295
scientific article