The categorical Gini correlation proposed by Dang et al. is a dependence measure to characterize independence between categorical and numerical variables. The asymptotic distributions of the sample correlation under dependence and independence have been established when the dimension of the numerical variable is fixed. However, its asymptotic behavior for high dimensional data has not been explored. In this paper, we develop the central limit theorem for the Gini correlation in the more realistic setting where the dimensionality of the numerical variable is diverging. We then construct a powerful and consistent test for the $K$-sample problem based on the asymptotic normality. The proposed test not only avoids computation burden but also gains power over the permutation procedure. Simulation studies and real data illustrations show that the proposed test is more competitive to existing methods across a broad range of realistic situations, especially in unbalanced cases.

翻译：Dang等人提出的分类Gini相关性是一种用于刻画分类变量与数值变量之间独立性的依赖度量。在数值变量维度固定的情况下，已有研究建立了样本相关性在依赖与独立条件下的渐近分布。然而，其在高维数据中的渐近行为尚未得到探索。本文在更具现实意义的假设下——即数值变量维度发散——推导了Gini相关性的中心极限定理。基于该渐近正态性，我们为K样本问题构建了一个强大且一致的检验方法。所提检验不仅避免了计算负担，而且在功效上优于置换检验方法。仿真研究与真实数据实例表明，在广泛的实际场景中（尤其是不平衡情形下），所提检验相比现有方法更具竞争力。