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.

翻译：针对两个非负随机变量的样本，我们提出了一族检验方法，用于检验一个随机变量在二阶随机占优意义上支配另一个随机变量的原假设。检验统计量定义为恒等映射与洛伦兹P-P图之差的泛函，其中洛伦兹P-P图由一种分布的逆未缩放洛伦兹曲线与另一种分布的未缩放洛伦兹曲线复合而成。我们确定了原假设下这类检验统计量的上界，推导了其极限分布，并通过Bootstrap过程进行近似。随后在相对温和的条件下建立了检验的渐近有效性，并通过模拟研究了有限样本性质。结果表明，我们的检验方法可作为基于累积分布函数积分差值的经典方法的有效替代——后者需有界支撑且在某些情形下难以检测与原假设的偏离。