An adaptive $L_2$-type test for high-dimensional white noise

2026-09-27

AI Summary

This study addresses the applicability challenges in white noise testing for high-dimensional time series, where fixed or diverging dimensions induce distinct statistical behaviors. We propose an adaptive $L_2$-type test that reveals a phase transition phenomenon in the test statistic and introduces the first unified adaptive bootstrap procedure valid across different phases, effectively resolving the difficulty of identifying the operational regime in practice. Grounded in high-dimensional statistical theory, the proposed methodology is validated through extensive numerical experiments demonstrating its substantial superiority over existing methods in finite-sample settings. To facilitate practical implementation, the developed procedures have been integrated into the R package HDTSA.

Abstract

We propose a new $L_2$-type test for white noise which allows the dimension $p$ of the time series to either (i) be a fixed constant, or (ii) diverge with the sample size $n$. The proposed test statistic exhibits an interesting phase transition, following two different regimes of behavior: $p$ is fixed, and $p\rightarrow\infty$. Because identification of the operable regime is difficult, if not impossible in practice, we devise a novel adaptive bootstrap method to construct unified testing procedure across different phases. Numerical experiments confirm the good finite sample performance of the proposed adaptive $L_2$-type test in comparison to the existing methods in the literature. The proposed testing procedure has been implemented in R package HDTSA.

Problem

• high-dimensional white noise test
• L2-type test
• phase transition
• adaptive testing
• time series

Innovation

• L2-type test
• high-dimensional white noise
• phase transition
• adaptive bootstrap
• time series