Asymptotic minimaxity of False Discovery Rate thresholding for sparse exponential data

Abstract

We apply FDR thresholding to a non-Gaussian vector whose coordinates X_i, i=1,..., n, are independent exponential with individual means ${\mu }_i$. The vector $\mu =({\mu }_i)$ is thought to be sparse, with most coordinates 1 but a small fraction significantly larger than 1; roughly, most coordinates are simply `noise,' but a small fraction contain `signal.' We measure risk by per-coordinate mean-squared error in recovering $\log ({\mu }_i)$, and study minimax estimation over parameter spaces defined by constraints on the per-coordinate p-norm of $\log ({\mu }_i)$: $\frac{1}{n}{\sum }_{i=1}^n{\log }^p({\mu }_i)\le {\eta }^p$. We show for large n and small $\eta$ that FDR thresholding can be nearly Minimax. The FDR control parameter 0<q<1 plays an important role: when $q\le 1/2$, the FDR estimator is nearly minimax, while choosing a fixed q>1/2 prevents near minimaxity. These conclusions mirror those found in the Gaussian case in Abramovich et al. [Ann. Statist. 34 (2006) 584--653]. The techniques developed here seem applicable to a wide range of other distributional assumptions, other loss measures and non-i.i.d. dependency structures.

Published at http://dx.doi.org/10.1214/009053606000000920 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Cited by in corpus (15)

• Innovated higher criticism for detecting sparse signals in correlated noise
• On the false discovery rate and an asymptotically optimal rejection curve
• Asymptotic Bayes-optimality under sparsity of some multiple testing procedures
• Properties of higher criticism under strong dependence
• Feature selection by Higher Criticism thresholding: optimal phase diagram
• Optimal rates of convergence for estimating the null density and proportion of nonnull effects in large-scale multiple testing
• A Selective Overview of Variable Selection in High Dimensional Feature Space (Invited Review Article)
• On false discovery rate thresholding for classification under sparsity
• Estimation of False Discovery Proportion with Unknown Dependence
• Distilled Sensing: Adaptive Sampling for Sparse Detection and Estimation
• Quick Search for Rare Events
• Search for astrophysical high energy neutrino point sources with a False Discovery Rate controlling procedure
• Asymptotically optimal sequential FDR and pFDR control with (or without) prior information on the number of signals
• On the Necessity of Irrelevant Variables
• Bayesian Multiple Testing Under Sparsity for Polynomial-Tailed Distributions