Abstract: We propose some two-sample tests based on ball divergence and investigate their high dimensional behaviour. First, we consider the High Dimension, Low Sample Size (HDLSS) setup. Under appropriate regularity conditions, we establish the consistency of these tests in the HDLSS regime, where the dimension grows to infinity while the sample sizes from the two distributions remain fixed. Next, we show that these conditions can be relaxed when the sample sizes also increase with the dimension, and in such cases, consistency can be proved even for shrinking alternatives. We use a simple example to show that even when there are no consistent tests in the HDLSS regime, the proposed tests can be consistent if the sample sizes increase with the dimension at an appropriate rate. This rate is obtained by establishing the minimax rate optimality of these tests over a certain class of alternatives. Several simulated and benchmark data sets are analyzed to compare the empirical performance of these tests with some state-of-the-art methods available for testing the equality of two high dimensional distributions.

Key words and phrases: Ball divergence, energy statistics, high dimensional asymptotics, minimax rate optimality, permutation tests, shrinking alternatives.