Significance Levels for Multiple Tests
Sergiu Hart and Benjamin Weiss
Abstract
Let X1, ..., Xn
be n random variables, with cumulative distribution functions
F1, ..., Fn.
Define ξi := Fi(Xi)
for all i, and let ξ(1) ≤ ...
≤ ξ(n) be the order
statistics of the (ξi)i.
Let α1 ≤ ... ≤ αn
be n numbers in the interval [0,1]. We show that the probability
of the event R := {ξ(i)
≤ αi
for all 1 ≤ i ≤ n}
is at most mini{n αi / 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 αi / 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.
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Statistics and Probability Letters
35 (1997), 1, 43-48