False Discovery Rate and Asymptotics
After a short introduction into multiple testing and the concept of the false discovery rate (FDR), we are concerned with asymptotic FDR results with the number of hypotheses to be tested simultaneously tending to infinity. First, we investigate the asymptotic behavior of the popular linear step-up test by Benjamini and Hochberg in models with positive dependency between the marginal test statistics. As specific distributional examples, equi-correlated normal variates occuring in many-to-one comparisons and jointly studentized means leading to multivariate t-distributions will be treated. Connections to the theory of large deviations will be drawn for the case that these models tend to independence in a certain sense. In the second part, we present a new, asymptotically optimal (non-linear) rejection curve (AORC) for asymptotic exhaustion of the FDR level. Due to its non-linearity, a new methodology of proof for FDR control and for deriving upper FDR bounds for step-up-down (SUD) tests is required. Therefore, we will encapsulate essential distributional and procedural assumptions and provide a unified method for FDR calculations for SUD tests. Finally, ongoing work on modifications of AORC-based procedures for the finite case will be discussed. This is joint work with Helmut Finner, Veronika Gontscharuk, and Markus Roters.