A Neighborhood-Assisted Hotelling's $T^2$ Test for High-Dimensional Means

Many tests have been proposed to remedy the classical Hotelling's $T^2$ test in the "large $p$, small $n$" paradigm, but the existence of an optimal sum-of-squares type test has not been explored. This paper shows that under certain conditions, the population Hotelling's $T^2$ test with the known $\Sigma^{-1}$ attains the best power among all the $L_2$-norm based tests with the data transformation by $\Sigma^{\eta}$ for $\eta \in (-\infty, \infty)$. To extend the result to the case of unknown $\Sigma^{-1}$, we propose a Neighborhood-Assisted Hotelling's $T^2$ statistic obtained by replacing the inverse of sample covariance matrix in the classical Hotelling's $T^2$ statistic with a regularized covariance estimator. Utilizing a regression model, we establish its asymptotic normality under mild conditions. We show that the proposed test is able to match the performance of the population Hotelling's $T^2$ test under certain conditions, and thus possesses certain optimality. Moreover, it can adaptively attain the best power by empirically choosing a neighborhood size to maximize its signal-to-noise ratio. Simulation experiments and case studies are given to demonstrate the empirical performance of the proposed test.