Significance Levels for Multiple Tests
Sergiu Hart and Benjamin Weiss

Let X1, ..., Xn be n random variables, with cumulative distribution functions F1, ..., Fn. Define i := Fi(Xi) for all i, and let (1) ... (n) be the order statistics of the (i)i. Let 1 ... n be n numbers in the interval [0,1]. We show that the probability of the event R := {(i) i for all 1 i n} is at most mini{ni/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 mini{ni/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.

(Check "Notes on using HTML to present math" by Martin J. Osborne, to see how this abstract was produced; thanx, Martin!).

Statistics and Probability Letters 35 (1997), 43-48.
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