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.

中文翻译：

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基于等级的高维白噪声测试

高维白噪声测试的发展在统计理论和应用中都具有重要意义，其中时间序列的维度可以与时间序列的长度相当或超过。本文提出了几种使用基于秩的统计量来测试高维白噪声的无分布测试，这些测试对重尾具有鲁棒性，并且不需要对样本分布进行有限阶矩假设。本文分析了三类基于秩的检验，包括简单的线性秩统计量、非退化 U 统计量和退化 U 统计量。为这些测试的每个族建立了渐近零分布和速率最优性。在这些测试中，基于退化U统计量的检验还可以检测出自相关中的非线性和非单调关系。此外，这是秩相关统计量的渐近分布的第一个结果，它允许高维数据中的横截面依赖性。