Worst-cases of distortion riskmetrics and weighted entropy with partial information

TL;DR

This paper investigates the worst-case scenarios for distortion risk metrics and weighted entropy when only partial distribution information like mean and variance is available, providing bounds and applications for various entropy-based measures.

Contribution

It introduces bounds for worst-case distortion risk metrics and weighted entropy under partial information, extending to multiple entropy and risk measure applications.

Findings

Derived worst-case bounds for distortion risk metrics.

Established worst-case bounds for weighted entropy.

Applied results to various entropy and risk measure examples.

Abstract

In this paper, we discuss the worst-case of distortion riskmetrics for general distributions when only partial information (mean and variance) is known. This result is applicable to general class of distortion risk measures and variability measures. Furthermore, we also consider worst-case of weighted entropy for general distributions when only partial information is available. Specifically, we provide some applications for entropies, weighted entropies and risk measures. The commonly used entropies include Gini functional, cumulative residual entropy, tail-Gini functional, cumulative Tsallis past entropy, extended Gini coefficient and so on. The risk measures contain some premium principles and shortfalls based on entropy. The shortfalls include the Gini shortfall, extended Gini shortfall, shortfall of cumulative residual entropy and shortfall of cumulative residual Tsallis entropy with order $\alpha$.