Proportion of non‐zero normal means: universal oracle equivalences and uniformly consistent estimators

Summary. Since James and Stein's seminal work, the problem of estimating n normal means has received plenty of enthusiasm in the statistics community. Recently, driven by the fast expansion of the field of large‐scale multiple testing, there has been a resurgence of research interest in the n normal means problem. The new interest, however, is more or less concentrated on testing n normal means: to determine simultaneously which means are 0 and which are not. In this setting, the proportion of the non‐zero means plays a key role. Motivated by examples in genomics and astronomy, we are particularly interested in estimating the proportion of non‐zero means, i.e. given n independent normal random variables with individual means Xj∼N(μj,1), j=1,…,n, to estimate the proportion ɛn=(1/n) #{j:μj /= 0}. We propose a general approach to construct the universal oracle equivalence of the proportion. The construction is based on the underlying characteristic function. The oracle equivalence reduces the problem of estimating the proportion to the problem of estimating the oracle, which is relatively easier to handle. In fact, the oracle equivalence naturally yields a family of estimators for the proportion, which are consistent under mild conditions, uniformly across a wide class of parameters. The approach compares favourably with recent works by Meinshausen and Rice, and Genovese and Wasserman. In particular, the consistency is proved for an unprecedentedly broad class of situations; the class is almost the largest that can be hoped for without further constraints on the model. We also discuss various extensions of the approach, report results on simulation experiments and make connections between the approach and several recent procedures in large‐scale multiple testing, including the false discovery rate approach and the local false discovery rate approach.