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Abstract:
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Although many methods have been tried to extend the classical Hotelling's T Square test under high dimensional settings, their test statistics are constructed without incorporating data dependence, because the sample covariance matrix is not invertible when the dimension is larger than the sample size. To still incorporate the advantageous effect of dependence among variables, we propose a novel Neighborhood-Assisted (NA) test, which replaces the sample covariance in Hotelling's statistic with the regularized estimator through banding the Cholesky factor (Bickel and Levina, 2008). Asymptotical distribution of the proposed test statistic is derived, and a testing procedure is constructed. It has been shown that this NA test is actually robust to a wide range of dependent structure, which does not rely on any structural assumption of the unknown covariance matrix. Simulations and case studies are made to demonstrate the performance of the proposed NA test.
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