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Humboldt-Universität zu Berlin - Mathematisch-Naturwissenschaftliche Fakultät - Research Unit 1735

Multiple testing under unspecified dependency structure

Although a variety of new multiple testing error criteria have been introduced and studied during the last decade in the literature, it is fair to say that the family-wise error rate (FWER) and the false discovery rate (FDR) are still the two most widely accepted and most commonly used type I error measures in (frequentist) multiple hypotheses testing. Classical procedures controlling these error rates are margin-based, meaning that only the marginal distributions of test statistics or p-values, respectively, are modeled explicitly. The adjustment for multiplicity is then performed by combining these summary statistics, for instance via their order statistics, and thresholding them according to probabilistic calculations or probability bounds, which make no or only qualitative assumptions on the dependence structure among the summary statistics.
In modern applications, however, summary statistics typically exhibit strong spatial, temporal, or spatio-temporal dependencies. Furthermore, often external structural information about these dependencies is available. One prominent example is the HapMap database in genetics (http://hapmap.ncbi.nlm.nih.gov), from which linkage disequilibrium matrices for several target populations can be downloaded. In cases where (i) such structural information is at hand or (ii) a calibration sample can be utilized to reliably estimate the dependency structure, or (iii) the dependency structure is a nuisance parameter which does not depend on the main parameter of statistical interest, multivariate multiple tests incorporating the dependencies can be constructed which typically outperform classical margin-based methods with respect to power.
Up to present, the vast majority of multivariate multiple test procedures rely on asymptotic normality (i. e., multivariate central limit theorems, see Hothorn et al. (2008)) or approximate the dependence structure by resampling. The former approach is restricted to elliptical dependencies and only asymptotically valid, and finite-sample properties of the latter are often hard to analyze. However, a third possibility to express dependency structures is constituted by the concept of copula functions and the famous theorem of Sklar (1959). This third possibility is the main topic of our project. Given the popularity of copula-based models in many areas of applied statistics, it is surprising that the first concrete reference on copula-based construction of multivariate multiple tests (controlling the FWER) seems to be Dickhaus and Gierl (2013).


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