Neumann André, Bodnar Taras, Pfeifer Dietmar, Dickhaus Thorsten
Institute for Statistics, University of Bremen, Bibliothekstraße 1, D-28359, Bremen, Germany.
Department of Mathematics, Stockholm University, Roslagsvägen 101, SE-10691, Stockholm, Sweden.
Biom J. 2019 Jan;61(1):40-61. doi: 10.1002/bimj.201700205. Epub 2018 Jul 12.
Multivariate multiple test procedures have received growing attention recently. This is due to the fact that data generated by modern applications typically are high-dimensional, but possess pronounced dependencies due to the technical mechanisms involved in the experiments. Hence, it is possible and often necessary to exploit these dependencies in order to achieve reasonable power. In the present paper, we express dependency structures in the most general manner, namely, by means of copula functions. One class of nonparametric copula estimators is constituted by Bernstein copulae. We extend previous statistical results regarding bivariate Bernstein copulae to the multivariate case and study their impact on multiple tests. In particular, we utilize them to derive asymptotic confidence regions for the family-wise error rate (FWER) of multiple test procedures that are empirically calibrated by making use of Bernstein copulae approximations of the dependency structure among the test statistics. This extends a similar approach by Stange et al. (2015) in the parametric case. A simulation study quantifies the gain in FWER level exhaustion and, consequently, power that can be achieved by exploiting the dependencies, in comparison with common threshold calibrations like the Bonferroni or Šidák corrections. Finally, we demonstrate an application of the proposed methodology to real-life data from insurance.
多元多重检验程序最近受到了越来越多的关注。这是因为现代应用程序生成的数据通常是高维的，但由于实验中涉及的技术机制而具有明显的依赖性。因此，利用这些依赖性以获得合理的检验功效是可能的，而且通常是必要的。在本文中，我们以最一般的方式表达依赖性结构，即通过copula函数。一类非参数copula估计量由伯恩斯坦copula构成。我们将先前关于二元伯恩斯坦copula的统计结果扩展到多元情形，并研究它们对多重检验的影响。特别是，我们利用它们来推导多重检验程序的族错误率（FWER）的渐近置信区域，这些区域通过利用检验统计量之间依赖性结构的伯恩斯坦copula近似进行经验校准。这扩展了Stange等人（2015年）在参数情形下的类似方法。一项模拟研究量化了在FWER水平耗尽方面的增益，因此，与像邦费罗尼或西达克校正这样的常见阈值校准相比，通过利用依赖性可以实现检验功效。最后，我们展示了所提出方法在保险实际数据中的应用。
