Multivariate integration of functions depending explicitly on the minimum and the maximum of the variables

TL;DR

This paper presents a new method for multivariate integration of functions involving minimum and maximum of variables, simplifying calculations of aggregation function metrics and expected values.

Contribution

It introduces an alternative integral expression replacing min and max with single variables, aiding in computing aggregation metrics and expectations.

Findings

Simplified integral expression for functions of min and max

Application to compute orness and andness averages

Method for calculating expected values and distribution functions

Abstract

By using some basic calculus of multiple integration, we provide an alternative expression of the integral $$ \int_{]a,b[^n} f(\mathbf{x},\min x_i,\max x_i) d\mathbf{x}, $$ in which the minimum and the maximum are replaced with two single variables. We demonstrate the usefulness of that expression in the computation of orness and andness average values of certain aggregation functions. By generalizing our result to Riemann-Stieltjes integrals, we also provide a method for the calculation of certain expected values and distribution functions.