Randomized p-values for multiple testing of composite null hypotheses

We are considered with the problem of m simultaneous statistical test problems with composite null hypotheses. Usually, marginal p-values are computed under least favorable parameter configurations (LFCs), thus being over-conservative under non-LFCs. Our proposed randomized p-value leads to a tighter exhaustion of the marginal (local) significance level. In turn, it is stochastically larger than the LFC-based p-value under alternatives. While these distributional properties are typically nonsensical for m  =1, the exhaustion of the local significance level is extremely helpful for cases with m>1$m>1$ in connection with data-adaptive multiple tests as we will demonstrate by considering multiple one-sided tests for Gaussian means.