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Title: A neighborhood-assisted Hotelling's T2 test for high-dimensional means Authors:  Jun Li - Kent State University (United States) [presenting]
Yumou Qiu - Iowa State University (United States)
Abstract: Many tests have been proposed to remedy the classical Hotelling's $T^2$ test in the ``large $p$, small $n$'' paradigm, but how to incorporate data dependence in the sum-of-squares type test to enhance the power has not been explored. We show 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 transformed 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 $T^2$ (NEAT) 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. Without any structural assumption on $\Sigma$, the proposed NEAT test is able to enhance the power by incorporating variable dependence through an adaptively chosen neighborhood, and thus more powerful than other tests without utilizing dependence. An optimal neighborhood size selection procedure is proposed to maximize the power of the NEAT test via maximizing the signal-to-noise ratio. As a special case, our results demonstrate that if $\Sigma$ happens to satisfy a certain bandable structure, the neighborhood exploration procedure leads to an optimal test that matches the population Hotelling's $T^2$ test.