This paper takes a different look on the problem of testing the mutual independence of the components of a high-dimensional vector. Instead of testing if all pairwise associations (e.g. all pairwise Kendall's $\tau$) between the components vanish, we are interested in the (null)-hypothesis that all pairwise associations do not exceed a certain threshold in absolute value. The consideration of these hypotheses is motivated by the observation that in the high-dimensional regime, it is rare, and perhaps impossible, to have a null hypothesis that can be exactly modeled by assuming that all pairwise associations are precisely equal to zero. The formulation of the null hypothesis as a composite hypothesis makes the problem of constructing tests non-standard and in this paper we provide a solution for a broad class of dependence measures, which can be estimated by $U$-statistics. In particular we develop an asymptotic and a bootstrap level $\alpha$-test for the new hypotheses in the high-dimensional regime. We also prove that the new tests are minimax-optimal and investigate their finite sample properties by means of a small simulation study and a data example.

翻译：本文从不同视角审视高维向量分量间相互独立性的检验问题。我们不再检验分量间所有成对关联（例如所有成对Kendall's $\tau$）是否为零，而是关注原假设：所有成对关联的绝对值均不超过特定阈值。该假设的提出源于以下观察：在高维场景中，假设所有成对关联精确等于零的零假设既罕见且可能无法准确建模。将零假设设定为复合假设使检验构建问题具有非标准性，本文针对可通过$U$-统计量估计的广泛依赖度量类提出解决方案。具体而言，我们为高维场景下的新假设构建了渐近检验与自助法水平$\alpha$检验。同时证明新检验达到极小化最优性，并通过仿真实验与数据实例研究其有限样本性质。