Significance Levels for Multiple Tests
Sergiu Hart and Benjamin Weiss
Abstract
Let X_{1}, ..., X_{n}
be n random variables, with cumulative distribution functions
F_{1}, ..., 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 α_{1} ≤ ... ≤ 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_{i}{n α_{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_{i}{n α_{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.

Statistics and Probability Letters
35 (1997), 1, 4348