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：针对来自两个非负随机变量的样本，我们提出了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下支配另一个变量的原假设。这些检验基于Lorenz P-P图（LPP），即一个分布的非缩放Lorenz曲线的逆函数与另一个分布的非缩放Lorenz曲线的复合函数。当且仅当占优条件被违反时，LPP会超过恒等函数，这为构造检验统计量提供了相当简单的方法——检验统计量定义为恒等函数与LPP之间差值的泛函。我们在原假设下确定了此类检验统计量的随机上界，并推导了其极限分布（可通过Bootstrap程序近似）。我们还在一组相对宽松的条件下（允许依赖样本和独立样本）证明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。