A consistent test of equality of distributions for Hilbert-valued random elements
针对取值于可分希尔伯特空间的两个独立随机元素，提出一种基于自助法的检验方法，判断其分布是否相等，适用于函数型数据分析等场景。
Two independent random elements taking values in a separable Hilbert space are considered. The aim is to develop a test with bootstrap calibration to check whether they have the same distribution or not. A transformation of both random elements into a new separable Hilbert space is considered so that the equality of expectations of the transformed random elements is equivalent to the equality of distributions. Thus, a bootstrap test procedure to check the equality of means can be used in order to solve the original problem. It will be shown that both the asymptotic and bootstrap approaches proposed are asymptotically correct and consistent. The results can be applied, for example, in functional data analysis. In practice, the test can be solved with simple operations in the original space without applying the mentioned transformation, which is used only to guarantee the theoretical results. Empirical results and comparisons with related methods support and complement the theory.