Large-scale inference with block structure

The detection of weak and rare effects in large amounts data arises a number modern analysis problems. Known results show that this situation the potential statistical inference is severely limited by large-scale multiple testing inherent these Here, we fundamentally more powerful possible when there some structure signal can be exploited, for example, if clustered many small blocks, as case relevant applications. We derive boundary such where allow both blocks block length to grow polynomially with sample size. univariate multivariate settings well problem detecting clusters network. These recover special cases sparse (Ann. Statist. 32 (2004) 962–994) no signal, scan (Statist. Sinica 23 (2013) 409–428) comprises single interval. develop methodology allows optimal adaptive general setting, thus exploiting it present without incurring penalty structure. advantage considerable, means need increase at rate logn ensure detection, while presence even decrease polynomial rate.