On some high‐dimensional two‐sample tests based on averages of inter‐point distances

Over the last two decades, several two‐sample tests based on averages of inter‐point distances have been proposed in the literature. Most of these tests are based on the Euclidean distance, and they can be used even when the dimension of the data is much larger than the sample size. But these tests can produce poor results in high‐dimensional set‐ups even when the two distributions differ widely in their scatters and shapes. To overcome these limitations, we modify some tests by replacing the Euclidean distance with a new class of distance functions. The high‐dimensional consistency of these modified tests is established under appropriate regularity conditions. Numerical studies are also carried out to demonstrate the usefulness of the proposed methods.