Testing and estimation in a sparse normal means model, with connections to shape restricted inference

Donoho and Jin (2004), following work of Ingster (1999), studied the problem of testing for any signal in a sparse normal means model and showed that there is a “detection boundary” above which the signal can be detected and below which no test has any power. They showed that Tukey’s “higher criticism” statistic achieves the detection boundary. I will introduce a new family of test statistics based on phi-divergences indexed by s ∈ [−1, 2] which all achieve the Donoho-Jin- Ingster detection boundary. I will also review recent work on estimating the proportion of non-zero means and make some connections to shape-constrained estimation.