A Blockwise Bootstrap-Based Two-Sample Test for High-Dimensional Time Series

TL;DR

This paper introduces a new statistical test for comparing high-dimensional time series data, using a bootstrap method and novel theoretical results.

Contribution

The paper presents a novel high-dimensional central limit theorem for α-mixing sequences and improves convergence rates under exponential tails.

Findings

A blockwise bootstrap method is used to compute critical values for the test.

The proposed test is effective for detecting change points in high-dimensional time series.

Numerical experiments show the method's effectiveness and advantages.

Abstract

We propose a two-sample testing procedure for high-dimensional time series. To obtain the asymptotic distribution of our ℓ∞-type test statistic under the null hypothesis, we establish high-dimensional central limit theorems (HCLTs) for an α-mixing sequence. Specifically, we derive two HCLTs for the maximum of a sum of high-dimensional α-mixing random vectors under the assumptions of bounded finite moments and exponential tails, respectively. The proposed HCLT for α-mixing sequence under bounded finite moments assumption is novel, and in comparison with existing results, we improve the convergence rate of the HCLT under the exponential tails assumption. To compute the critical value, we employ the blockwise bootstrap method. Importantly, our approach does not require the independence of the two samples, making it applicable for detecting change points in high-dimensional time series. Numerical results emphasize the effectiveness and advantages of our method.

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