Multiple tests are designed to test a whole collection of null hypotheses simultaneously. Their quality is often judged by the false discovery rate (FDR), i.e. the expectation of the quotient of the number of false rejections divided by the number of all rejections. The widely cited Benjamini and Hochberg (BH) step up multiple test controls the FDR under various regularity assumptions. In this note we present a rapid approach to the BH step up and step down tests. Also, sharp FDR inequalities are discussed for dependent p-values and examples and counter-examples are considered. In particular, the Bonferroni bound is sharp under dependence for the control of the family-wise error rate.
Benditkis, J., Heesen, P., & Janssen, A. (2018). The false discovery rate (FDR) of multiple tests in a class room lecture. Statistics and Probability Letters, 134, 29–35. https://doi.org/10.1016/j.spl.2017.09.017