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), 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 increasing function of the number of rejections. An example is $FR/R^\gamma$, the number of false rejections divided by the number of rejections to a power $0\leq \gamma\leq 1$. 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 FWER and FDR.