  This paper compares the higher criticism statistic (Donoho and Jin [Ann.
Statist. 32 (2004) 962-994]), a modification of the higher criticism statistic
also suggested by Donoho and Jin, and two statistics of the Berk-Jones [Z.
Wahrsch. Verw. Gebiete 47 (1979) 47-59] type. New approximations to the
significance levels of the statistics are derived, and their accuracy is
studied by simulations. By numerical examples it is shown that over a broad
range of sample sizes the Berk-Jones statistics have a better power function
than the higher criticism statistics to detect sparse mixtures. The
applications suggested by Meinshausen and Rice [Ann. Statist. 34 (2006)
373-393], to find lower confidence bounds for the number of false hypotheses,
and by Jeng, Cai and Li [Biometrika 100 (2013) 157-172], to detect copy number
variants, are also studied.
