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：统计深度函数用于度量空间中的元素相对于某个分布的离群性或中心性。它是一个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与Singh（1993）提出了一种基于深度秩的多元两样本检验方法。本文致力于提升相关检验统计量的功效，并拓展其在函数型数据中的适用性。在此过程中，我们构建了一个对两样本具有更自然对称性的检验统计量。我们推导了所提统计量的零假设渐近性质，同时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有限样本下的表现，并基于海洋漂流浮标的年度温度曲线数据集进行了实际数据分析。