Testing the Deviation of a High-dimensional Mean

Abstract

This paper investigates the high-dimensional one-sample equivalence testing problem for a reference mean vector $\bbmu_0$: $H_0:\|\bbmu-\bbmu_0\|>d_0$ versus $H_1: \|\bbmu-\bbmu_0\|\leq d_0$ for a pre-specified length threshold $d_0>0$. In high-dimensional regimes, classical equivalence tests suffer from severe asymptotic breakdowns. To resolve this difficulty, we propose a novel test statistic based on a two-armed bandit (TAB) U-process, which leverages positive/negative feedback mechanisms from control theory. We rigorously establish the weak convergence of TAB U-process to a novel two-dimensional stochastic differential equation (SDE), termed the U-bandit SDE. By employing in-depth analysis of this SDE, we characterize the asymptotic size and power of the resulting test statistic. The theoretical framework is extended to the two-sample setting and validated through finite-sample simulations and an empirical analysis of microbiome compositions.