Multivariate multiple test procedures based on nonparametric copula estimation

Abstract

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.

Generating VaR Scenarios under Solvency II with Product Beta Distributions

“…It is applicable in arbitrary dimensions and generally superior to kernel density or classical and recent copula approaches, with respect to complexity, easy implementation (even in usual spreadsheet programs), and larger scenario VaR estimates. We have tested the procedure described in this paper with the 19-dimensional dataset discussed by Neumann et al (2018) and came to similar conclusions. A crucial point here is the estimation of the marginal distributions which, of course, influences the results to a certain extend, as does the value of m. However, in any case, the original data are exactly reproduced, and the selection of the steering parameters should depend on the purpose of the application.…”

Optimizing effective numbers of tests by vine copula modeling

“…Model 2.1 is a standard multiple testing model in the context of studies with M endpoints, which are all measured for the same n observational units; see, among many others, [7], [15], and [12]. Under Model 2.1, we make the following general assumptions.…”

“…Dependence modeling by means of copula functions has recently received a lot of attention in multiple testing, see [6], [2], [14], [13], [15], [3], [12], and Sections 2.2.4 and 4.4 of [5]. For example, in [6] it has explicitly been shown that the copula approach leads to the most general construction method for single-step multiple tests under known univariate marginal null distributions of test statistics or p-values, respectively.…”

New copulas based on general partitions-of-unity (part III) — the continuous case

“…In order to show the powerfulness of continuous PUC approaches in higher dimensions we conclude the applied section with a discussion of the 19-dimensional data set presented in Neumann et al [5], listed in Tab. For simplicity, we will consider only the Gamma copula in this section.…”

“…the d-dimensional Lebesgue measure. In this case, is a singular mixture of product densities given by 1 , , d a a  ( ) c u (5) ( ) ( 1 1 ( ) :…”