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We consider the two-sample mean test for high-dimensional time series data. Two test statistics are constructed. The first test statistics is based on the L2 norm which is a banded version of U-Statistics to estimate the sum of squared difference between the population means while controlling the temporal dependence. The L2 test can be viewed as a generalization of the two-sample test of Chen and Qin (2010) for independent data. This test is proved to be asymptotically unbiased and powerful for dense and weak signals. The second test, in terms of the thresholding test, can enhance the power of the test for sparse signals. The test statistic is constructed by reducing the variation of the statistic and increasing the signal to noise ratio.
The limiting distributions of the two test statistics are derived under the null and alternative hypotheses. Both of the proposed methods are free of distribution assumption, and are adaptive to temporal dependence. It also does not require specific relationship between sample size and dimension. Simulation study is carried out to compare the performance of the proposed test with other existing methods.
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