On Testing Homological Equivalence

In this article, we develop a test to check whether the support of the unknown distribution generating the data is homologically equivalent to the support of some specified distribution. Similarly, it is also checked 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 are established. Moreover, some simulation studies are conducted when the specified population distributions are uniform distribution over circle and 3-D torus, which indicate that the proposed tests are performing well. Furthermore, the practicability of the tests are shown on two well-known real data sets also.