Significance levels for multiple tests

Let X₁, ... ,X_n be n random variables, with cumulative distribution functions F₁, ... ,F_n. Define ξ_i:= F_i(X_i) for all i, and let ξ⁽¹⁾ ≤ ⋯ ≤ ξ⁽ⁿ⁾ be the order statistics of the (ξ_i)_i. Let α₁ ≤ ⋯ ≤ α_n be n numbers in the interval [0,1]. We show that the probability of the event R := {ξ⁽ⁱ⁾ ≤ α_i for all 1 ≤ i ≤ n} is at most min_inα_i/i}. Moreover, this bound is exact: for any given n marginal distributions (F_i)_i, there exists a joint distribution with these marginals such that the probability of R is exactly min_inα_i/i}. This result is used in analyzing the significance level of multiple hypotheses testing. In particular, it implies that the Rüger tests dominate all tests with rejection regions of type R as above.