A Neighborhood-Assisted Hotelling's T^2 Test for High-Dimensional Means
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This paper aims to revive the classical Hotelling's T^2 test in the "large p, small n" paradigm. A Neighborhood-Assisted Hotelling's T^2 statistic is proposed to replace 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 with a known covariance under certain conditions, and thus possesses certain optimality. Moreover, the test has the ability to attain its best power possible by adjusting a neighborhood size to unknown structures of population mean and covariance matrix. An optimal neighborhood size selection procedure is proposed to maximize the power of the Neighborhood-Assisted T^2 test via maximizing the signal-to-noise ratio. Simulation experiments and case studies are given to demonstrate the empirical performance of the proposed test.
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