Group for Research in Decision Analysis

# Testing for sparse normal means: is there a signal?

## Jon A. Wellner

Donoho and Jin (2004), following work of Ingster (1999), studied the problem of testing for a 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 a real number s with values between -1 and 2) which all achieve the Donoho-Jin-Ingster detection boundary. I will also briefly review recent work on estimating the proportion of non-zero means.