HYPOTHESIS TESTING FOR HIGH-DIMENSIONAL TIME SERIES VIA SELF-NORMALIZATION
成果类型：
Article
署名作者：
Wang, Runmin; Shao, Xiaofeng
署名单位：
University of Illinois System; University of Illinois Urbana-Champaign
刊物名称：
ANNALS OF STATISTICS
ISSN/ISSBN：
0090-5364
DOI：
10.1214/19-AOS1904
发表日期：
2020
页码：
2728-2758
关键词：
CENTRAL-LIMIT-THEOREM
COVARIANCE-MATRIX
2-sample test
mean vector
Gaussian Approximation
regularized estimation
fewer observations
ASYMPTOTIC THEORY
Optimal Rates
heteroskedasticity
摘要：
Self-normalization has attracted considerable attention in the recent literature of time series analysis, but its scope of applicability has been limited to low-/fixed-dimensional parameters for low-dimensional time series. In this article, we propose a new formulation of self-normalization for inference about the mean of high-dimensional stationary processes. Our original test statistic is a U-statistic with a trimming parameter to remove the bias caused by weak dependence. Under the framework of nonlinear causal processes, we show the asymptotic normality of our U-statistic with the convergence rate dependent upon the order of the Frobenius norm of the long-run covariance matrix. The self-normalized test statistic is then constructed on the basis of recursive subsampled U-statistics and its limiting null distribution is shown to be a functional of time-changed Brownian motion, which differs from the pivotal limit used in the low-dimensional setting. An interesting phenomenon associated with self-normalization is that it works in the high-dimensional context even if the convergence rate of original test statistic is unknown. We also present applications to testing for bandedness of the covariance matrix and testing for white noise for high-dimensional stationary time series and compare the finite sample performance with existing methods in simulation studies. At the root of our theoretical arguments, we extend the martingale approximation to the high-dimensional setting, which could be of independent theoretical interest.