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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.

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