A Multiplier Bootstrap Test for High-Dimensional Two-Sample Means with Non-Asymptotic Guarantees

Testing the equality of two mean vectors in high-dimensional settings is a fundamental challenge, particularly in the "large $p$, small $n$" regime. Traditional tests based on the $L_2$ norm often rely on asymptotic normality or chi-square approximations, which frequently suffer from substantial size distortion due to slow convergence or require restrictive assumptions on the covariance structure. To address these limitations, we propose a Gaussian multiplier bootstrap procedure for the two-sample $L_2$-norm test that avoids the explicit estimation of the inverse covariance matrix. A key theoretical contribution is the derivation of a non-asymptotic Berry-Esseen bound for the test statistic over high-dimensional Euclidean balls, providing a rigorous guarantee for the Gaussian approximation error in finite samples. Furthermore, we establish the consistency of the multiplier bootstrap estimator under mild conditions. Numerical simulations demonstrate that the proposed method maintains accurate size control and achieves robust power against dense alternatives, outperforming existing asymptotic methods in settings with heteroscedasticity or complex covariance dependence. Finally, a systematic comparison between $L_2$- and $L_\infty$-norm-based tests is provided, elucidating their respective power advantages under dense versus sparse alternative hypotheses.