A novel characterization of the generalized family wise error rate using empirical null distributions

We present a novel characterization of the generalized family wise error rate: kFWER. The interpretation allows researchers to view kFWER as a function of the test statistics rather than current methods based on p-values. Using this interpretation we present several theorems and methods (parametric and non-parametric) for estimating kFWER in various data settings. With this version of kFWER, researchers will have an estimate of kFWER in addition to knowing what tests are significant at the estimated kFWER. Additionally, we present methods that use empirical null distributions in place of parametric distributions in standard p-value kFWER controlling schemes. These advancements represent an improvement over common kFWER methods which are based on parametric assumptions and merely report the tests that are significant under a given value for kFWER.