Open Access
April 2006 Adapting to unknown sparsity by controlling the false discovery rate
Felix Abramovich, Yoav Benjamini, David L. Donoho, Iain M. Johnstone
Ann. Statist. 34(2): 584-653 (April 2006). DOI: 10.1214/009053606000000074


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 p norm for p small. We obtain a procedure which is asymptotically minimax for 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 qn also plays a determining role in asymptotic minimaxity. If q=lim qn∈[0,1/2] and also qn>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 qn∈(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.


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Felix Abramovich. Yoav Benjamini. David L. Donoho. Iain M. Johnstone. "Adapting to unknown sparsity by controlling the false discovery rate." Ann. Statist. 34 (2) 584 - 653, April 2006.


Published: April 2006
First available in Project Euclid: 27 June 2006

zbMATH: 1092.62005
MathSciNet: MR2281879
Digital Object Identifier: 10.1214/009053606000000074

Primary: 62C20
Secondary: 62G05 , 62G32

Keywords: minimax estimation , Model selection , Multiple comparisons , smoothing parameter selection , thresholding , wavelet denoising

Rights: Copyright © 2006 Institute of Mathematical Statistics

Vol.34 • No. 2 • April 2006
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