Generalized differentiation of probability functions: parameter dependent sets given by intersections of convex sets and complements of convex sets
type: Item
instance of: scholarly article
title: Generalized differentiation of probability functions: parameter dependent sets given by intersections of convex sets and complements of convex sets
zbMATH Open document ID: 1493.90117
author: Wim van Ackooij
author: Pedro Pérez-Aros
published in: Applied Mathematics and Optimization
publication date: 2022-03-15T00:00:00Z
review text: 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.
reviewed by: Marcin Anholcer
Mathematics Subject Classification ID: 90C15
zbMATH DE Number: 7490295
zbMATH Keywords: probability functions
zbMATH Keywords: sub-differentiability
zbMATH Keywords: spherical radial-like decomposition
MaRDI profile type: Publication
full work available at URL: s00245-022-09844-5
OpenAlex ID: W4220945801
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DOI: 10.1007/S00245-022-09844-5
description: scientific article; zbMATH DE number 7490295