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We attempt to recover an -dimensional vector observed in white noise, where 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 norm for small. We obtain a procedure which is asymptotically minimax for loss, simultaneously throughout a range of such sparsity classes.

The optimal procedure is a data-adaptive thresholding scheme, driven by control of the (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 also plays a determining role in asymptotic minimaxity. If =lim ∈[0,1/2] and also >/log(), we get sharp asymptotic minimaxity, simultaneously, over a wide range of sparse parameter spaces and loss functions. On the other hand, =lim ∈(1/2,1] forces the risk to exceed the minimax risk by a factor growing with .

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.

Source Ann. Statist. , Volume 34, Number 2 (2006), 584-653.


Dates First available in Project Euclid: 27 June 2006

Permanent link to this document

Permanent link to this document

Digital Object Identifier doi:10.1214/009053606000000074

Digital Object Identifier


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Comments ( 10 )

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One of the most comprehensive reviews in the literature on this subject suggests that in healthy adults only around 180 mg of magnesium per day is required to maintain balance in healthy individuals (even when taking into account magnesium sweat losses). This analysis pooled magnesium data from 27 different tightly controlled balance studies (lasting more than 27 days) conducted in the USA. Cheap Footaction With Credit Card Online LS Top In Navy Marl With Double Layer Neck 6740 Tom Tailor yd2J05
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with an estimated average magnesium intake of just 246 mg/day (NHANES 2001–2002) and only 238 mg/day (NHANES 1999–2000). Magnesium intake for Mexican and African–American women of the same age group is even lower (185 mg/day and 169 mg/day, respectively, and 176 mg/day and 150 mg/day in those reporting no dietary supplement use). 77 Thus, the elderly population is especially at risk of magnesium deficiency due to low intake but also increased risk for chronic diseases that predispose to magnesium deficiency (ageing also reduces magnesium absorption from the diet, ie, achlorhydria). Table 1 covers themagnesium requirements from older balance studies.

View this table:
Table 1

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