  This paper revisits a meta-analysis method proposed by Pearson [Biometrika 26
(1934) 425--442] and first used by David [Biometrika 26 (1934) 1--11]. It was
thought to be inadmissible for over fifty years, dating back to a paper of
Birnbaum [J. Amer. Statist. Assoc. 49 (1954) 559--574]. It turns out that the
method Birnbaum analyzed is not the one that Pearson proposed. We show that
Pearson's proposal is admissible. Because it is admissible, it has better power
than the standard test of Fisher [Statistical Methods for Research Workers
(1932) Oliver and Boyd] at some alternatives, and worse power at others.
Pearson's method has the advantage when all or most of the nonzero parameters
share the same sign. Pearson's test has proved useful in a genomic setting,
screening for age-related genes. This paper also presents an FFT-based method
for getting hard upper and lower bounds on the CDF of a sum of nonnegative
random variables.
