Block-Independent Likelihood Ratio Testing for High-Dimensional Mean Vectors with Applications to Matrix-Variate Data

TL;DR

This paper introduces the Block Independent Likelihood Ratio Test (BILT), a new high-dimensional mean vector test that relaxes independence assumptions, offering better power and error control in correlated data settings.

Contribution

The paper proposes BILT, a novel test for high-dimensional means that generalizes DLRT by assuming block independence, with proven asymptotic properties and superior performance.

Findings

BILT maintains Type I error control across various covariance structures.

BILT achieves higher power than DLRT in simulations.

Application to ADNI data demonstrates practical utility.

Abstract

Testing the equality of two high-dimensional mean vectors is a fundamental problem in multivariate analysis. While the classical Hotelling's $T^2$ test is optimal in low-dimensional settings, it fails when the dimension $p$ is comparable to or exceeds the sample size $n$. Several extensions, including the Diagonal Likelihood Ratio Test (DLRT), have been proposed under the working independence assumption among variables. However, such an assumption can lead to a substantial loss of power when correlations are present. In this paper, we propose a new test, the Block Independent Likelihood Ratio Test (BILT), which generalizes DLRT by relaxing the working independence assumption to a block independence assumption. We establish its asymptotic normality of the null distribution of the BILT statistic for 'increasing $p$ with small $n$' under mild regularity conditions. We further analyze the asymptotic power of BILT under a local alternatives. Extensive simulation studies show that BILT maintains Type I error control and achieves substantially higher power than DLRT across a wide range of covariance structures. An application to the Alzheimer's Disease Neuroimaging Initiative (ADNI) dataset further demonstrates the application of BILT to testing mean differences between two matrix-variate populations.