On High Dimensional Behaviour of Some Two-Sample Tests Based on Ball Divergence

TL;DR

This paper introduces two-sample tests based on ball divergence, analyzing their high-dimensional behavior, consistency, and power, especially in HDLSS settings, with theoretical proofs and empirical comparisons.

Contribution

It develops new high-dimensional two-sample tests based on ball divergence and establishes their consistency and optimality under various growth conditions.

Findings

Tests are consistent in HDLSS regime under regularity conditions.

Power of tests can approach one when sample sizes grow with dimension.

Proposed tests outperform existing methods in simulations and benchmarks.

Abstract

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.