Let X1, ... , Xn be n random variables, with cumulative distribution functions F1, ... , Fn. Define [xi]i: = Fi(Xi) for all i, and let [xi](1) [less-than-or-equals, slant] ... [less-than-or-equals, slant] [xi](n) be the order statistics of the ([xi]i)i. Let [alpha]1 [less-than-or-equals, slant] ... [less-than-or-equals, slant] [alpha]n be n numbers in the interval [0,1]. We show that the probability of the event R:= {[xi](i) [less-than-or-equals, slant] [alpha]i for all 1 [less-than-or-equals, slant] i [less-than-or-equals, slant] n} is at most minin[alpha]i/i}. Moreover, this bound is exact: for any given n marginal distributions (Fi)i, there exists a joint distribution with these marginals such that the probability of R is exactly minin[alpha]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.
