A new class of nonparametric tests for second-order stochastic dominance based on the Lorenz P-P plot

TL;DR

This paper introduces a novel nonparametric testing method for second-order stochastic dominance using the Lorenz P-P plot, offering advantages over traditional approaches especially with unbounded distributions.

Contribution

The paper develops a new family of tests based on the Lorenz P-P plot, providing asymptotic validity and improved finite sample performance for stochastic dominance testing.

Findings

Test statistics bounds under null hypothesis derived

Limit distribution approximated via bootstrap

Simulation results show improved detection capabilities

Abstract

Given samples from two non-negative random variables, we propose a family of tests for the null hypothesis that one random variable stochastically dominates the other at the second order. Test statistics are obtained as functionals of the difference between the identity and the Lorenz P-P plot, defined as the composition between the inverse unscaled Lorenz curve of one distribution and the unscaled Lorenz curve of the other. We determine upper bounds for such test statistics under the null hypothesis and derive their limit distribution, to be approximated via bootstrap procedures. We then establish the asymptotic validity of the tests under relatively mild conditions and investigate finite sample properties through simulations. The results show that our testing approach can be a valid alternative to classic methods based on the difference of the integrals of the cumulative distribution functions, which require bounded support and struggle to detect departures from the null in some cases.