Type I error rate control for testing many hypotheses: a survey with proofs

This paper presents a survey on some recent advances for the type I error rate control in multiple testing methodology. We consider the problem of controlling the k-family-wise error rate (kFWER, probability to make k false discoveries or more) and the false discovery proportion (FDP, proportion of false discoveries among the discoveries). The FDP is controlled either via its expectation, which is the so-called false discovery rate (FDR), or via its upper-tail distribution function. We aim at deriving general and unified results together with concise and simple mathematical proofs. Furthermore, while this paper is mainly meant to be a survey paper, some new contributions for controlling the kFWER and the upper-tail distribution function of the FDP are provided. In particular, we derive a new procedure based on the quantiles of the binomial distribution that controls the FDP under independence.