Adapting to unknown sparsity by controlling the false discovery rate

Felix Abramovich, Yoav Benjamini, David L. Donoho, Iain M. Johnstone

Research output: Contribution to journalArticlepeer-review

Abstract

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 ℓ p norm for p small. We obtain a procedure which is asymptotically minimax for ℓ r loss, simultaneously throughout a range of such sparsity classes. The optimal procedure is a data-adaptive thresholding scheme, driven by control of the false discovery rate (FDR). FDR control is a relatively recent innovation in simultaneous testing, ensuring that at most a certain expected 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 ∈[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 ∈ (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 · log(potential model size/ actual model sizes). We exhibit a close connection with FDR-controlling procedures under stringent control of the false discovery rate.

Original languageEnglish
Pages (from-to)584-653
Number of pages70
JournalAnnals of Statistics
Volume34
Issue number2
DOIs
StatePublished - Apr 2006

Keywords

  • Minimax estimation
  • Model selection
  • Multiple comparisons
  • Smoothing parameter selection
  • Thresholding
  • Wavelet denoising

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