On the asymptotic distribution of (generalized) Lorenz transvariation measures

Abstract

A common problem associated with evaluating dominance relationships between distribution functions and their moments is the lack of resolution regarding the direction of dominance as a result of the functions crossing, prevalent in empirical applications. This paper proposes a method of examining the difference between (Generalized) Lorenz curves over the entire support of the variables, an idea first proposed by Anderson and Leo (On providing a complete ordering of non-combinable alternative prospects. University of Toronto Discussion Paper, 2017) and formalized by Anderson et al. (Somewhere between utopia and dystopia: choosing from multiple incomparable prospect. University of Toronto Discussion Paper, 2017) for the case of stochastic dominance. The method provides a way of ordering all the (Generalized) Lorenz curves under consideration. The paper also provides the exact limit distribution of these associated measures, which in consequence of the results due to Politis and Romano (Ann Stat 22(4):2031–2050, 1994), permits inference via subsampling, in lieu of the crossing of empirical (Generalized) Lorenz curves. We show that due to the relationship between the Lorenz curve and the Gini coefficient, the same can be said of the latter. An example is provided to demonstrate its application.