A comprehensive error rate for multiple testing

In multiple testing, a variety of control metrics have been introduced such as the Family-Wise Error Rate (FWER), the False Discovery Rate (FDR), the False Exceedence Rate (FER), etc. We found a way to embed these metrics into a continuous family of control metrics, all of which can be attained by applying a simple and general family of multiple testing procedures. The new general error rate (GER) limits the number of false positives relative to an arbitrary non-decreasing function of the number of rejections. An example is $V/R^\beta$, the number of false rejections divided by the number of rejections to a power $0\leq \beta\leq 1$. We investigated both the control of quantiles and of expectations and provide the corresponding multiple testing procedures. In the above example, the expectation of the criterion thus leads to a family of multiple testing procedures that bridges the gap between the FWER and the FDR.