The immediate need for effective massive data mining gives rise to a recent new field in statistics: large-scale multiple simultaneous testing or multiple comparisons. In such settings, one tests thousands or even millions of hypotheses simultaneously:

H1,H2,...,Hn,

where associated with each hypothesis is a summary test statistic

X1,X2,...,Xn.

A problem of particular interest is to estimate the proportion of non-null effects, i.e., the pro- portion of hypotheses that are untrue.

In this talk, we report some recent progress on estimating the proportion. We model each Xj as normally distributed with individual mean μj and individual variance σj2, where the parameters satisfy that (μj , σj ) = (0, 1) if Hj is true, and (μj , σj ) ̸= (0, 1) otherwise. We show that, under natural identifiability conditions, universal oracle equivalence of the proportion can be constructed, which equals to the true proportion for any n and any set of parameters. The oracle naturally yields real estimators, which are uniformly consistent for the proportion over a wide class of situations.

This talk is based on collaborated works with (alphabetically) Tony Cai, David Donoho, Mark Low, Jie Peng, and Pei Wang.