Statistical depth functions provide measures of the outlyingness, or centrality, of the elements of a space with respect to a distribution. It is a nonparametric concept applicable to spaces of any dimension, for instance, multivariate and functional. Liu and Singh (1993) presented a multivariate two-sample test based on depth-ranks. We dedicate this paper to improving the power of the associated test statistic and incorporating its applicability to functional data. In doing so, we obtain a more natural test statistic that is symmetric in both samples. We derive the null asymptotic of the proposed test statistic, also proving the validity of the testing procedure for functional data. Finally, the finite sample performance of the test for functional data is illustrated by means of a simulation study and a real data analysis on annual temperature curves of ocean drifters is executed.

翻译：统计深度函数提供了一种衡量空间中元素相对于分布的外向性或中心性的度量。这是一种适用于任意维度空间（例如多元空间和函数空间）的非参数概念。Liu和Singh（1993）提出了一种基于深度排序的多元双样本检验方法。本文致力于提升相关检验统计量的功效，并将其适用性扩展至函数型数据。通过这一改进，我们获得了一个在双样本间具有对称性的更自然检验统计量。我们推导了所提检验统计量的零假设渐近分布，并证明了该检验方法对函数型数据的有效性。最后，通过模拟研究展示了该检验在函数型数据中的有限样本表现，并通过对海洋漂流器年温度曲线的实际数据分析进行了验证。