In this article, we present a nonparametric method for the general two-sample problem involving functional random variables modelled as elements of a separable Hilbert space ${\cal H}$. First, we present a general recipe based on linear projections to construct a measure of dissimilarity between two probability distributions on ${\cal H}$. In particular, we consider a measure based on the energy statistic and present some of its nice theoretical properties. A plug-in estimator of this measure is used as the test statistic to construct a general two-sample test. Large sample distribution of this statistic is derived both under null and alternative hypotheses. However, since the quantiles of the limiting null distribution are analytically intractable, the test is calibrated using the permutation method. We prove the large sample consistency of the resulting permutation test under fairly general assumptions. We also study the efficiency of the proposed test by establishing a new local asymptotic normality result for functional random variables. Using that result, we derive the asymptotic distribution of the permuted test statistic and the asymptotic power of the permutation test under local contiguous alternatives. This establishes that the permutation test is statistically efficient in the Pitman sense. Extensive simulation studies are carried out and a real data set is analyzed to compare the performance of our proposed test with some state-of-the-art methods.

翻译：本文提出了一种非参数方法，用于处理涉及函数型随机变量的一般双样本问题，其中随机变量建模为可分希尔伯特空间${\cal H}$中的元素。首先，我们提出了一种基于线性投影的通用方案，用于构造${\cal H}$上两个概率分布之间的相异度量。特别地，我们考虑了一种基于能量统计量的度量，并展示了其一些良好的理论性质。该度量的插件估计量被用作检验统计量，以构建一般的双样本检验。我们在原假设和备择假设下推导了该统计量的大样本分布。然而，由于极限零分布的分位数在解析上难以处理，因此通过置换方法对检验进行校准。我们证明了在相当一般的假设下，所得置换检验的大样本一致性。通过建立函数型随机变量的新局部渐近正态性结果，我们进一步研究了所提检验的效率。利用该结果，我们在局部邻接备择假设下推导了置换检验统计量的渐近分布以及置换检验的渐近势，从而证明了该置换检验在皮特曼意义下具有统计效率。我们进行了广泛的模拟研究，并分析了一个真实数据集，以将所提检验的性能与一些前沿方法进行比较。