A nonparametric test for the two-sample problem based on order statistics

We study a nonparametric test procedure based on order statistics for testing the null hypothesis of equality of two continuous distributions. The exact null distribution of the proposed test statistic is obtained using an enumeration method and a novel combinatorial argument. A recurrence relation for the probability generating function and a sequential approach for computing the mean and variance of the distribution are given. Critical values and characteristics of the distribution for selected small sample sizes are presented. For the Lehmann alternative family, the exact power function of the new test is derived, and its power performance is examined. We also study the power performance of the proposed test under the location-shift and scale-shift alternatives using Monte Carlo simulations and observe its superior performance when compared to commonly used nonparametric tests under various scenarios. A generalization of the proposed procedure for unequal sample sizes is discussed. An illustrative example and some concluding remarks are provided.

本文研究一种基于次序统计量（order statistics）的非参数检验方法，用于检验两个连续分布相等的原假设。本文通过枚举法与全新的组合论证方法，推导得到所提出检验统计量的精确原分布。本文给出了该分布概率生成函数的递推关系式，以及用于计算其均值与方差的序贯方法。本文呈现了针对若干选定小样本量情形下的分布临界值与分布特性。针对莱曼备择族（Lehmann alternative family），本文推导了该新检验的精确势函数，并对其势性能进行了分析。本文还通过蒙特卡洛模拟（Monte Carlo simulations），研究了所提检验在位置漂移与尺度漂移备择假设下的势性能，并发现相较于若干常用非参数检验方法，该检验在多种场景下均表现更优。本文还讨论了将所提检验方法推广至样本量不等情形的方案。文末给出了一个示例说明与若干结论性评述。