###
Significance Levels for Multiple Tests

by

Sergiu Hart and Benjamin Weiss

**Abstract**

Let
*X*_{1}*, ..., X*_{n}
be *n* random variables, with
cumulative distribution functions
*F*_{1}*, ..., F*_{n}.
Define _{i} :=
*F*_{i}(*X*_{i}) 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
*min*_{i}{*n*_{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*_{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.

(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.

**The paper is available for downloading:**