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Abstract:
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High-dimensional data are commonly encountered nowadays. Testing the equality of two means is a fundamental problem. The conventional Hotelling's test performs poorly or becomes inapplicable in high dimensions. Several modifications have been proposed to address this challenging issue and shown to perform well. However, many of them use normal approximation to the null distributions, thus they require strong regularity assumptions on the underlying covariance structure. We study this issue and propose an L2-norm based test that works under much milder conditions and even when there are fewer observations than the dimension. In particular, we employ the Welch-Satterthwaite approximation and ratio-consistently estimators for the parameters in the approximation distribution. While many existing tests are not, our test is adaptive to singularity or near singularity of the unknown covariance structure, which is commonly seen in high dimensions and has great impact on the shape of the null distribution. Simulation studies and real data applications show that our test has a much better size controlling than some existing tests, while the powers are comparable their sizes are comparable.
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