Variability and stability of the false discovery proportion (Q2414483)

The ``false discovery rate'' is the expectation of the ``false discovery proportion'' (FDP) given by the ratio of the number of false rejections and all rejections. In the paper under review a closer look is provided at the FDP for adaptive linear step-up multiple tests, which extend the Benjamini and Hochberg test by estimating the unknown amount of the true null hypotheses. The authors present exact finite sample formulas for higher moments of the FDP and, in particular, for its variance. These results are applied to the variability and the stability analysis of FDP. The sufficient and necessary conditions for this stability are obtained. Next, the authors apply the results to study concrete adaptive step-up tests mainly based on the convex combinations of generalized Storey estimators with data dependent weights. It is shown that stability cannot be achieved in general. Under mild assumptions the adaptive tests based on the estimators mentioned above are superior compared to the classical Benjamini and Hochberg test. The least favorable configurations, which serve as useful technical tools, are also discussed.