In a recent paper (4), Efron pointed out that an important issue in large-scale
multiple hypothesis testing is that the null distribution may be unknown and need
to be estimated. Consider a Gaussian mixture model, where the null distribution
is known to be normal but both null parameters-the mean and the variance-are
unknown. We address the problem with a method based on Fourier transformation.
The Fourier approach was first studied by Jin and Cai (9), which focuses
on the scenario where any non-null effect has either the same or a larger variance
than that of the null effects. In this paper, we review the main ideas in (9), and
propose a generalized Fourier approach to tackle the problem under another scenario:
any non-null effect has a larger mean than that of the null effects, but no
constraint is imposed on the variance. This approach and that in (9) complement
with each other: each approach is successful in a wide class of situations where
the other fails. Also, we extend the Fourier approach to estimate the proportion
of non-null effects. The proposed procedures perform well both in theory and on
simulated data.