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

Jinyuan Chang, Jing He, Weiming Li, Chen Lin

arXiv 27 Sep 2026 · Mathematics — Statistics Theory

arXiv:2609.33418 · PDF · Extracted main text

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.

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