Quasi-Monte Carlo methods for Choquet integrals

TL;DR

This paper introduces quasi-Monte Carlo methods for numerically approximating Choquet integrals with capacities derived from distortion functions, providing error bounds based on integrand smoothness and discrepancy measures.

Contribution

It develops a novel quasi-Monte Carlo approach for Choquet integrals with explicit error bounds, extending numerical integration techniques to non-additive measures.

Findings

Derived error bounds using modulus of continuity and star discrepancy

Established explicit representations for step function integrals

Demonstrated the applicability of the method to Choquet integrals

Abstract

We propose numerical integration methods for Choquet integrals where the capacities are given by distortion functions of an underlying probability measure. It relies on the explicit representation of the integrals for step functions and can be seen as quasi-Monte Carlo methods in this framework. We give bounds on the approximation errors in terms of the modulus of continuity of the integrand and the star discrepancy.