  We consider the problem of estimating the number of false null hypotheses
among a very large number of independently tested hypotheses, focusing on the
situation in which the proportion of false null hypotheses is very small. We
propose a family of methods for establishing lower $100(1-\alpha)%$ confidence
bounds for this proportion, based on the empirical distribution of the
$p$-values of the tests. Methods in this family are then compared in terms of
ability to consistently estimate the proportion by letting $\alpha \to 0$ as
the number of hypothesis tests increases and the proportion decreases. This
work is motivated by a signal detection problem that occurs in astronomy.
