In this article, we propose a one-sample test to check whether the support of the unknown distribution generating the data is homologically equivalent to the support of some specified distribution or not OR using the corresponding two-sample test, one can test whether the supports of two unknown distributions are homologically equivalent or not. In the course of this study, test statistics based on the Betti numbers are formulated, and the consistency of the tests is established under the critical and the supercritical regimes. Moreover, some simulation studies are conducted and results are compared with the existing methodologies such as Robinson's permutation test and test based on mean persistent landscape functions. Furthermore, the practicability of the tests is shown on two well-known real data sets also.

翻译：本文提出了一种单样本检验方法，用于判断生成数据的未知分布支撑集是否与特定分布的支撑集同调等价；或通过相应的双样本检验，检验两个未知分布的支撑集是否同调等价。研究过程中，基于贝蒂数构建了检验统计量，并在临界和超临界条件下建立了检验的一致性。此外，开展了仿真研究，并将结果与现有方法（如罗宾逊置换检验和基于平均持续景观函数的检验）进行了比较。同时，通过在两个知名真实数据集上的应用，展示了该检验的实用性。