Adapting to Unknown Sparsity by controlling the False Discovery Rate
| dc.creator | Abramovich, Felix | |
| dc.creator | Benjamini, Yoav | |
| dc.creator | Donoho, David L. | |
| dc.creator | Johnstone, Iain M. | |
| dc.date | 2005-05-18 | |
| dc.date.accessioned | 2026-07-07T08:06:54Z | |
| dc.date.available | 2026-07-07T08:06:54Z | |
| dc.description | We attempt to recover an $n$-dimensional vector observed in white noise, where $n$ is large and the vector is known to be sparse, but the degree of sparsity is unknown. We consider three different ways of defining sparsity of a vector: using the fraction of nonzero terms; imposing power-law decay bounds on the ordered entries; and controlling the $\ell_p$ norm for $p$ small. We obtain a procedure which is asymptotically minimax for $\ell^r$ loss, simultaneously throughout a range of such sparsity classes. The optimal procedure is a data-adaptive thresholding scheme, driven by control of the {\it False Discovery Rate} (FDR). FDR control is a relatively recent innovation in simultaneous testing, ensuring that at most a certain fraction of the rejected null hypotheses will correspond to false rejections. In our treatment, the FDR control parameter $q_n$ also plays a determining role in asymptotic minimaxity. If $q = \lim q_n \in [0,1/2]$ and also $q_n > γ/\log(n)$ we get sharp asymptotic minimaxity, simultaneously, over a wide range of sparse parameter spaces and loss functions. On the other hand, $ q = \lim q_n \in (1/2,1]$, forces the risk to exceed the minimax risk by a factor growing with $q$. To our knowledge, this relation between ideas in simultaneous inference and asymptotic decision theory is new. Our work provides a new perspective on a class of model selection rules which has been introduced recently by several authors. These new rules impose complexity penalization of the form $2 \cdot \log({potential model size} / {actual model size})$. We exhibit a close connection with FDR-controlling procedures under stringent control of the false discovery rate. | |
| dc.description | This is a complete version of a paper to appear in Annals of Statitistics. The paper in AoS has certain proofs abbreviated that are given here in detail | |
| dc.identifier | https://arxiv.org/abs/math/0505374 | |
| dc.identifier | http://arxiv.org/abs/math/0505374 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/130767 | |
| dc.subject | Statistics Theory | |
| dc.subject | 62F10; 62G12 | |
| dc.title | Adapting to Unknown Sparsity by controlling the False Discovery Rate | |
| dc.type | text |