Uncertainty quantification for the family-wise error rate in multivariate copula models

We derive confidence regions for the realized family-wise error rate (FWER) of certain multiple tests which are empirically calibrated at a given (global) level of significance. To this end, we regard the FWER as a derived parameter of a multivariate parametric copula model. It turns out that the resulting confidence regions are typically very much concentrated around the target FWER level, while generic multiple tests with fixed thresholds are in general not FWER-exhausting. Since FWER level exhaustion and optimization of power are equivalent for the classes of multiple test problems studied in this paper, the aforementioned findings militate strongly in favor of estimating the dependency structure (i.e., copula) and incorporating it in a multivariate multiple test procedure. We illustrate our theoretical results by considering two particular classes of multiple test problems of practical relevance in detail, namely multiple tests for components of a mean vector and multiple support tests.