Significance Levels for Multiple Tests

Sergiu Hart & 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 %n%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 mini %n%i/i%. This result is used in analyzing the significance level of multiple hypotheses testing. In particular, it implies that the R?ger tests dominate all tests with rejection regions of type R as above.

March, 1996
Published in: 
Statistics and Probability Letters 35 (1997), 43-48