Ming-Yen Cheng, National Taiwan University

"A simple and adaptive two-sample test in high dimensions"

High-dimensional data are commonly encountered nowadays. Testing the equality of two means is a fundamental problem in the inference. But the conventional Hotelling's $T^2$ 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, they all use normal approximation to the null distributions of their test statistics, thus they all require strong regularity conditions. We study this issue thoroughly and propose an $L^2$-norm based test that works under milder conditions and even when there are fewer observations than the dimension. In particular, to cope with possible non-normality of the null distribution, we employ the Welch-Satterthwaite $\chi^2$-approximation. Simple ratio-consistently estimators for the parameters in the approximation distribution are given. While 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. The approximate and asymptotical powers of the proposed test are also investigated. Simulation studies and real data applications show that our test has a better size controlling than a benchmark test, while the powers are comparable when their sizes are comparable.