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Many tests have been proposed to remedy the classical Hotelling's $T^2$ testin the "large $p$, small $n$" paradigm, but the existence of an optimalsum-of-squares type test has not been explored. This paper shows that undercertain 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 withthe data transformation by $\Sigma^{\eta}$ for $\eta \in (-\infty, \infty)$. Toextend the result to the case of unknown $\Sigma^{-1}$, we propose aNeighborhood-Assisted Hotelling's $T^2$ statistic obtained by replacing theinverse of sample covariance matrix in the classical Hotelling's $T^2$statistic with a regularized covariance estimator. Utilizing a regressionmodel, we establish its asymptotic normality under mild conditions. We showthat the proposed test is able to match the performance of the populationHotelling's $T^2$ test under certain conditions, and thus possesses certainoptimality. Moreover, it can adaptively attain the best power by empiricallychoosing a neighborhood size to maximize its signal-to-noise ratio. Simulationexperiments and case studies are given to demonstrate the empirical performanceof the proposed test.

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