A Simple Two-Sample Test in High Dimensions Based on L2-Norm

doi:10.1080/01621459.2019.1604366

A Simple Two-Sample Test in High Dimensions Based on L2-Norm

2019-05-30

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DOI：10.1080/01621459.2019.1604366

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Testing the equality of two means is a fundamental inference problem. For high-dimensional data, the Hotelling's T-2-test either performs poorly or becomes inapplicable. Several modifications have been proposed to address this issue. However, most of them are based on asymptotic normality of the null distributions of their test statistics which inevitably requires strong assumptions on the covariance. We study this problem thoroughly and propose an L-2-norm based test that works under mild conditions and even when there are fewer observations than the dimension. Specially, to cope with general nonnormality of the null distribution we employ the Welch-Satterthwaite chi(2)-approximation. We derive a sharp upper bound on the approximation error and use it to justify that chi(2)-approximation is preferred to normal approximation. Simple ratio-consistent estimators for the parameters in the chi(2)-approximation are given. Importantly, our test can cope with singularity or near singularity of the covariance which is commonly seen in high dimensions and is the main cause of nonnormality. The power of the proposed test is also investigated. Extensive simulation studies and an application show that our test is at least comparable to and often outperforms several competitors in terms of size control, and the powers are comparable when their sizes are. for this article are available online.

Keywords:

High-dimensional data

Hotelling's T2-test

chi 2-type mixtures

Welch-Satterthwaite chi 2-approximation

A Simple Two-Sample Test in High Dimensions Based on L2-Norm

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