  Testing for the equality of two high-dimensional distributions is a
challenging problem, and this becomes even more challenging when the sample
size is small. Over the last few decades, several graph-based two-sample tests
have been proposed in the literature, which can be used for data of arbitrary
dimensions. Most of these test statistics are computed using pairwise Euclidean
distances among the observations. But, due to concentration of pairwise
Euclidean distances, these tests have poor performance in many high-dimensional
problems. Some of them can have powers even below the nominal level when the
scale-difference between two distributions dominates the location-difference.
To overcome these limitations, we introduce a new class of dissimilarity
indices and use it to modify some popular graph-based tests. These modified
tests use the distance concentration phenomenon to their advantage, and as a
result, they outperform the corresponding tests based on the Euclidean distance
in a wide variety of examples. We establish the high-dimensional consistency of
these modified tests under fairly general conditions. Analyzing several
simulated as well as real data sets, we demonstrate their usefulness in high
dimension, low sample size situations.
