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In this article, we propose some two-sample tests based on Ball Divergence and investigate their high dimensional behaviour. Under appropriate regularity conditions, we establish the consistency of these tests in High Dimension, Low Sample Size (HDLSS) asymptotic regime, where the dimension of the data grows to infinity while the sample sizes from the two distributions remain fixed. We also show that our tests are minimax rate optimal over a certain class of alternatives. We further study the behaviour of these tests in situations, where the sample sizes increase with the dimension. In such situations, we establish the consistency of our tests under relatively weaker conditions. In this context, we provide an example to show that even when there are no consistent tests in the HDLSS regime, the powers of the proposed tests can converge to unity if the sample sizes increase with the dimension at an appropriate rate. We analyze several simulated and benchmark datasets to compare the performance of our tests with the state-of-the-art methods that can be used for testing the equality of two high dimensional probability distributions. |