Adaptive test for mean vectors of high-dimensional time series data with factor structure

Statistical inference of high-dimensional time series data is of increasing interest in various fields such as social sciences and biology. In this article, we consider the problem of testing the equality of high-dimensional mean vectors in the approximate factor model, which allows for time series dependence among distinct observations and more flexible dependence within observations. We propose a data-adaptive test based on the factor-adjusted data rather than on the directly observed data. By combining the tests with different norms, the proposed test adapts to various alternative scenarios and thus overcomes the shortcomings of the tests based either on $L_2$-norm or $L_{\infty }$-norm. Multiplier bootstrap method is utilized to approximate the true underlying distribution of the proposed test statistics. Theoretical analysis shows that the proposed test enjoys desirable properties. Besides, we conduct thorough numerical study to compare the empirical performance of the proposed test with some state-of-the-art tests. A real stock market data set is analyzed to show the empirical usefulness of the proposed test.