A hyperbolic divergence based nonparametric test for two-sample multivariate distributions

Two-sample hypothesis testing, as a fundamental problem in statistical inference, seeks to detect the difference between two probability measures and has numerous real-world applications. Current test procedures for multivariate two-sample problems typically rely on angles and lengths in a Euclidean space, or lengths in a unit hypersphere after representing data with the spherical model. This article introduces a hyperbolic divergence based on hyperbolic lengths in hyperbolic geometry, as well as a subsequent nonparametric approach to testing the multivariate two-sample problem. We investigate the properties of our test procedure and discover that our hyperbolic divergence statistic is strongly consistent and consistent against all other alternatives; we also demonstrate that its limit distribution is an infinite mixture of ${\chi }^2$ distributions under the null hypothesis and a normal distribution under the alternative hypothesis. To calculate the $P$-value, we employ the permutation method. Furthermore, in numerical studies, we compare our method with several nonparametric procedures under various distributional assumptions and alternatives. We discover that our test procedure has some advantages when the distributions' complex correlation structures differ. Finally, we examine one real data set to show how our method can be used to test two-sample heterogeneity.