Open Access
October 2020 Hypothesis testing for high-dimensional time series via self-normalization
Runmin Wang, Xiaofeng Shao
Ann. Statist. 48(5): 2728-2758 (October 2020). DOI: 10.1214/19-AOS1904


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.


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Runmin Wang. Xiaofeng Shao. "Hypothesis testing for high-dimensional time series via self-normalization." Ann. Statist. 48 (5) 2728 - 2758, October 2020.


Received: 1 December 2018; Revised: 1 September 2019; Published: October 2020
First available in Project Euclid: 19 September 2020

MathSciNet: MR4152119
Digital Object Identifier: 10.1214/19-AOS1904

Primary: 60K35 , 62H15
Secondary: 62G10 , 62G20

Keywords: high-dimensional inference , nonlinear time series , Pivotal , self-normalization , U-statistic

Rights: Copyright © 2020 Institute of Mathematical Statistics

Vol.48 • No. 5 • October 2020
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