Generalized Lorenz dominance orders

TL;DR

This paper extends discrete majorization theory using non-normalized Lorenz curves and introduces generalized dominance orders, providing new theorems and methods for transforming arrays via elementary impact increases.

Contribution

It generalizes the Muirhead theorem by incorporating elementary impact increases and local increases, broadening the framework of majorization theory.

Findings

Proves that arrays can be transformed via elementary impact increases while maintaining generalized dominance.

Shows that ordered arrays can be derived from dominated arrays through elementary impact increases.

Extends majorization theory to non-normalized Lorenz curves with new theoretical results.

Abstract

We extend the discrete majorization theory by working with non-normalized Lorenz curves. Then we prove two generalizations of the Muirhead theorem. These not only use elementary transfers but also local increases. Together these operations are described as elementary impact increases. The first generalization shows that if an array X is dominated, in the generalized sense, by an array Y then Y can be derived from X by a finite number of elementary impact increases and this in such a way that each step transforms an array into a new one which is strictly larger in the generalized majorization sense. The other one shows that if the dominating array, Y, is ordered decreasingly then elementary impact increases starting from the dominated array, X, lead to the dominating one. Here each step transforms an array to a new one for which the decreasingly ordered version dominates the previous one and is dominated by Y.