Given samples from two non-negative random variables, we propose a new class of nonparametric tests for the null hypothesis that one random variable dominates the other with respect to second-order stochastic dominance. These tests are based on the Lorenz P-P plot (LPP), which is the composition between the inverse unscaled Lorenz curve of one distribution and the unscaled Lorenz curve of the other. The LPP exceeds the identity function if and only if the dominance condition is violated, providing a rather simple method to construct test statistics, given by functionals defined over the difference between the identity and the LPP. We determine a stochastic upper bound for such test statistics under the null hypothesis, and derive its limit distribution, to be approximated via bootstrap procedures. We also establish the asymptotic validity of the tests under relatively mild conditions, allowing for both dependent and independent samples. Finally, finite sample properties are investigated through simulation studies.

翻译：针对两个非负随机变量的样本，我们提出一类新的非参数检验方法，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。该类检验基于洛伦兹P-P图（LPP），该图由某一分布的反向非尺度化洛伦兹曲线与另一分布的非尺度化洛伦兹曲线复合而成。当且仅当占优条件被违反时，LPP超过恒等函数，从而为构建检验统计量提供了一种简便方法——通过定义在恒等函数与LPP之差上的泛函实现。我们确定了原假设下此类检验统计量的随机上界，并推导其极限分布（可通过bootstrap程序近似）。同时，在相对温和的条件下（允许相依样本与独立样本），我们建立了检验的渐近有效性。最后，通过模拟研究考察了有限样本性质。