Abstract: Two recent streams of two-sample tests for high-dimensional data are the sum-of-squares-based and supremum-based tests. The former is powerful against dense differences in two population means, and the latter is powerful against sparse differences. However, the level of sparsity and signal strength are often unknown, in practice, making it unclear which type of test to use. Here, we propose an adaptive weighted component test that provides good power against a variety of alternative hypotheses with unknown sparsity levels and va rying signal strengths. The basic idea is to first allocate different weights to components with varying magnitudes in a sum-of-squares-based test, and then to combine multiple weighted component tests to make the underlying test adaptive to different sparsity levels of the mean differences. We examine the asymptotic properties of the proposed test, and use numerical comparisons to demonstrate the superior performance of the proposed test across a spectrum of situations.

Key words and phrases: High-dimensional test, Huber’s weight function, testing equality of mean vectors, weighted components.