Abstract: This study considers testing for two-sample mean differences in high-dimensional temporally dependent data, which we then extend to the one-sample situation. To eliminate the bias caused by the temporal dependence in the time series observations, we propose a band-excluded U-statistic (BEU-statistic) to estimate the squared Euclidean distance between the two means that excludes cross-products of data vectors of temporally close time points. We derive the asymptotic normality of the BEU-statistic for the high-dimensional setting with “spatial” (column-wise) and temporal dependence. We also develop an estimator built on the kernel-smoothed cross-time covariances to estimate the variance of the BEU-statistic, facilitating a test procedure based on the standardized BEU-statistic. The proposed test is nonparametric and adaptive to a wide range of dependence and dimensionality, and has attractive power properties relative to those of a self-normalized test. A numerical simulation and a real-data analysis on the return and volatility of S&P 500 stocks before and after the 2008 financial crisis demonstrate the performance and utility of the proposed test.

Key words and phrases: High dimensionality, long-run variance estimation, L₂-type test, spatial and temporal dependence, U-statistics.