Thorsten Dickhaus: Multiple hypotheses testing in multivariate copula models

Abstract: We are concerned with simultaneous testing of a family of m > 1 null hypotheses under a single statistical model. In this, we assume that the m individual tests are carried out by means of (marginal) p-values and that these p-values, regarded as random variables, are dependent. The traditional type I error measure in multiple testing is the family-wise error rate (FWER). The FWER denotes the probability of the occurrence of at least one type I error among the m individual tests. In the first part of the presentation, we express the rejection threshold of an FWER-controlling simultaneous test procedure (STP) in the sense of Gabriel (1969) in terms of the copula function of the vector of p-values, assuming that each of these p-values is marginally uniformly distributed on the unit interval under the corresponding null hypothesis. This offers the opportunity to exploit the rich literature on copula-based modeling of multivariate dependency structures for the construction of STPs in non-Gaussian situations. 

The second part deals with the estimation of an unknown p-value copula and the analysis of the impact of the estimation uncertainty on the FWER behaviour of the (empirically calibrated) STP. The presentation is based on Stange, Bodnar and Dickhaus (2015).​