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TITLE: Projection Test for High-Dimensional Mean Vectors with Optimal Direction
ABSTRACT:
Testing the population mean is fundamental in statistical inference. When the dimensionality of a population is high, traditional Hotelling's T2 test becomes practically infeasible. In this paper, we propose a new testing method for high-dimensional mean vectors. The new method projects the original sample to a lower-dimensional space and carries out a test with the projected sample. We derive the theoretical optimal direction with which the projection test possesses the best power under alternatives. We further propose an estimation procedure for the optimal direction, so that the resulting test is an exact $t$-test under the normality assumption and an asymptotic chi-square test with 1 degree of freedom without the normality assumption. Monte Carlo simulation studies show that the new test can be much more powerful than the existing methods, while it also well retains Type I error rate.
