The development of high-dimensional white noise test is important in both statistical theories and applications, where the dimension of the time series can be comparable to or exceed the length of the time series. This paper proposes several distribution-free tests using the rank based statistics for testing the high-dimensional white noise, which are robust to the heavy tails and do not quire the finite-order moment assumptions for the sample distributions. Three families of rank based tests are analyzed in this paper, including the simple linear rank statistics, non-degenerate U-statistics and degenerate U-statistics. The asymptotic null distributions and rate optimality are established for each family of these tests. Among these tests, the test based on degenerate U-statistics can also detect the non-linear and non-monotone relationships in the autocorrelations. Moreover, this is the first result on the asymptotic distributions of rank correlation statistics which allowing for the cross-sectional dependence in high dimensional data.

翻译：高维白噪声检验的发展在统计理论及应用中都至关重要，其中时间序列的维度可能达到或超过序列长度。本文利用基于秩的统计量提出了若干无分布检验方法，用于检测高维白噪声。这些方法对重尾分布具有稳健性，且无需样本分布满足有限阶矩假设。本文分析了三类基于秩的检验，包括简单线性秩统计量、非退化U统计量和退化U统计量。针对每类检验，我们建立了其渐近零分布并证明了速率最优性。在这些检验中，基于退化U统计量的检验还能检测自相关中的非线性和非单调关系。此外，本文首次给出了允许高维数据截面相关的秩相关统计量的渐近分布结果。