A Bayes-Motivated Quadratic-Form Test for High-Dimensional Mean Testing

TL;DR

This paper introduces a Bayesian-inspired quadratic-form test for high-dimensional mean differences, effective in small samples, heterogeneous variances, and robust to distribution misspecification.

Contribution

It develops a novel two-sample mean test based on the Bayes factor suitable for high-dimensional data with growing dimension and demonstrates its superior performance.

Findings

Performs well with heterogeneous variances

Maintains control of type I error in small samples

Effective for detecting sparse and non-sparse mean differences

Abstract

We propose a two-sample mean test based on the Bayes factor with non-informative priors, specifically designed for scenarios where the dimension $p$ grows with the sample size $n$ with a linear rate $p/n \to c_1 \in (0, \infty)$. We establish the asymptotic normality of the test statistic and the asymptotic power. Through extensive simulations, we demonstrate that the proposed test performs competitively against several existing methods, particularly when the marginal variances of the individual features are heterogeneous and when the sample size is small. Furthermore, our test remains robust under distribution misspecification. The proposed method not only effectively detects both sparse and non-sparse differences in mean vectors but also maintains a well-controlled type I error rate, even in small-sample scenarios. We also demonstrate the performance of our proposed test using the small round blue cell tumors (SRBCT) dataset.