Mean Tests For High-dimensional Time Series
Shuyi Zhang, Song Chen, Yumou Qiu · Statistica Sinica · 2023
This paper considers testing for two-sample mean difference with high-dimensional temporally dependent data, which is later extended to the one-sample situation.To eliminate the bias caused by the temporal dependence among the time series observations, a band-excluded U-statistic (BEU) is proposed to estimate the squared Euclidean distance between the two means, which excludes cross-products of data vectors among temporally close time points.The asymptotic normality of the BEU statistic is derived under the high-dimensional setting with "spatial" (column-wise) and temporal dependence.An estimator built on the kernel smoothed cross-time covariances is developed to estimate the variance of the BEU-statistic, which facilitates 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 a self-normalized test.Numerical simulation and a real data analysis on the return and volatility of S&P 500 stocks before and after the 2008 financial crisis are conducted to demonstrate the performance and utility of the proposed test.