## AbstractA multiple test problem is characterized by $m > 1$ hypotheses that shall be tested simultaneously under the scope of one statistical experiment. First theoretical investigations of multiple test problems reach back to the 1950s and were primarily concerned with biometrical and agricultural applications. Aim was the development of test procedures controlling the family-wise error rate (FWER), meaning that the probability for rejecting at least one true hypothesis is bounded by a pre-defined significance level $\alpha$. However, due to rapid technical progress in many scientific fields, the cardinality of systems of hypotheses under consideration can nowadays become almost arbitrarily large. Typical examples are genetic association studies (500,000 or one million SNPs) or signal detection problems in cosmology (the visible sky is divided into more than $10^6$ pixels). In such (explorative) analyses, the FWER is agreed to be a too conservative error measure and can lead to many false negatives. In 1995, Benjamini and Hochberg proposed controlling the false discovery rate (FDR) as an alternative criterion for such cases. The FDR is defined as the expected proportion of false rejections (i.\ e., type I errors) among all rejections. The linear step-up test is meanwhile a standard tool in many statistical software packages. Under certain independence assumptions, it controls the FDR at level $m_0 \alpha / m$, where $m_0$ denotes the number of true null hypotheses. We present recent work on powerful FDR-controlling test procedures which adapt to the (unknown) quantity $m_0$ in order to exhaust the FDR level better in case of $m_0 < m$. Besides explicitly adaptive approaches utilizing a pre-estimation of $m_0$, special focus will be laid on the asymptotically optimal rejection curve (AORC). Under independence assumptions, the AORC induces asymptotically ($m \to \infty$) optimal critical values for step-up-down (SUD) tests maximizing power. We discuss asymptotic results as well as modified AORC-based SUD tests for exact FDR control for any finite $m$.
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