  In this article, we propose some two-sample tests based on ball divergence
and investigate their high dimensional behavior. First, we study their behavior
for High Dimension, Low Sample Size (HDLSS) data, and under appropriate
regularity conditions, we establish their consistency in the HDLSS regime,
where the dimension of the data grows to infinity while the sample sizes from
the two distributions remain fixed. Further, 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 involving two normal distributions to prove 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. This rate is obtained by establishing the minimax rate
optimality of our tests over a certain class of alternatives. Several simulated
and benchmark data sets are analyzed to compare the performance of these
proposed tests with the state-of-the-art methods that can be used for testing
the equality of two high-dimensional probability distributions.
