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Adapting to unknown sparsity by controlling the false discovery rate. (English) Zbl 1092.62005
Summary: 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 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 $$q=\lim q_n\in [0,1/2]$$ and also $$q_n> \gamma/\log(n)$$, 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 sizes). We exhibit a close connection with FDR-controlling procedures under stringent control of the false discovery rate.

##### MSC:
 62C20 Minimax procedures in statistical decision theory 62J15 Paired and multiple comparisons; multiple testing 62G05 Nonparametric estimation 62G32 Statistics of extreme values; tail inference
EBayesThresh
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