Testing for multi-dimensional white noise is an important subject in statistical inference. Such test in the high-dimensional case becomes an open problem waiting to be solved, especially when the dimension of a time series is comparable to or even greater than the sample size. To detect an arbitrary form of departure from high-dimensional white noise, a few tests have been developed. Some of these tests are based on max-type statistics, while others are based on sum-type ones. Despite the progress, an urgent issue awaits to be resolved: none of these tests is robust to the sparsity of the serial correlation structure. Motivated by this, we propose a Fisher's combination test by combining the max-type and the sum-type statistics, based on the established asymptotically independence between them. This combination test can achieve robustness to the sparsity of the serial correlation structure,and combine the advantages of the two types of tests. We demonstrate the advantages of the proposed test over some existing tests through extensive numerical results and an empirical analysis.
翻译:在统计推论中,多维白噪音测试是一个重要课题。在高维情况下,这种测试成为有待解决的公开问题,特别是当时间序列的尺寸与样本大小相当或甚至更大时。为了检测出一种任意的脱离高维白噪音的形式,已经开发了一些测试。其中一些测试基于最高类型统计,而另一些测试则基于总类型统计。尽管取得了进步,但一个紧迫的问题仍有待解决:这些测试中没有一个对串联关联结构的广度是强大的。为此,我们提议根据确定的最高类型和总类型统计数据之间的零星独立性,进行渔业者的混合测试。这种组合测试可以使序列关联结构的宽度变得稳健,并将两种类型的测试的优势结合起来。我们通过大量的数字结果和实验性分析,展示了拟议测试对一些现有测试的优势。