We discuss the testing problem of homogeneity of the marginal distributions of a continuous bivariate distribution based on a paired sample with possibly missing components (missing completely at random). Applying the well-known two-sample Crámer–von-Mises distance to the remaining data, we determine the limiting null distribution of our test statistic in this situation. It is seen that a new resampling approach is appropriate for the approximation of the unknown null distribution. We prove that the resulting test asymptotically reaches the significance level and is consistent. Properties of the test under local alternatives are pointed out as well. Simulations investigate the quality of the approximation and the power of the new approach in the finite sample case. As an illustration we apply the test to real data sets.

中文翻译：

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使用可能不完整的配对数据测试连续双变量分布的边缘同质性

我们讨论了基于配对样本的连续双变量分布的边缘分布同质性的测试问题，该样本可能缺少组件（完全随机缺失）。将众所周知的两样本 Cramer-von-Mises 距离应用于剩余数据，我们确定了在这种情况下测试统计量的极限零分布。可以看出，一种新的重采样方法适用于未知零分布的近似。我们证明结果检验渐近地达到显着性水平并且是一致的。还指出了在本地替代方案下的测试特性。模拟研究了近似的质量和新方法在有限样本情况下的能力。作为说明，我们将测试应用于真实数据集。