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Some optimality properties of FDR controlling rules under sparsity

Frommlet, Florian and Bogdan, Malgorzata LU (2013) In Electronic Journal of Statistics 7(1). p.1328-1368
Abstract

False Discovery Rate (FDR) and the Bayes risk are two different statistical measures, which can be used to evaluate and compare multiple testing procedures. Recent results show that under sparsity FDR controlling procedures, like the popular Benjamini-Hochberg (BH) procedure, perform also very well in terms of the Bayes risk. In particular asymptotic Bayes optimality under sparsity (ABOS) of BH was shown previously for location and scale models based on log-concave densities. This article extends previous work to a substantially larger set of distributions of effect sizes under the alternative, where the alternative distribution of true signals does not change with the number of tests m, while the sample size n slowly increases. ABOS of... (More)

False Discovery Rate (FDR) and the Bayes risk are two different statistical measures, which can be used to evaluate and compare multiple testing procedures. Recent results show that under sparsity FDR controlling procedures, like the popular Benjamini-Hochberg (BH) procedure, perform also very well in terms of the Bayes risk. In particular asymptotic Bayes optimality under sparsity (ABOS) of BH was shown previously for location and scale models based on log-concave densities. This article extends previous work to a substantially larger set of distributions of effect sizes under the alternative, where the alternative distribution of true signals does not change with the number of tests m, while the sample size n slowly increases. ABOS of BH and the corresponding step-down procedure based on FDR levels proportional to n-1/2 are proved. A simulation study shows that these asymptotic results are relevant already for relatively small values of m and n. Apart from showing asymptotic optimality of BH, our results on the optimal FDR level provide a natural extension of the well known results on the significance levels of Bayesian tests.

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publishing date
type
Contribution to journal
publication status
published
subject
keywords
Asymptotic optimality, Bayes risk, False discovery rate, Multiple testing, Two groups model
in
Electronic Journal of Statistics
volume
7
issue
1
pages
41 pages
publisher
Institute of Mathematical Statistics
external identifiers
  • scopus:84884955624
ISSN
1935-7524
DOI
10.1214/13-EJS808
language
English
LU publication?
no
id
dafb5714-fdbd-44d7-8b96-62981dec25a5
date added to LUP
2023-12-08 09:26:22
date last changed
2023-12-11 12:55:05
@article{dafb5714-fdbd-44d7-8b96-62981dec25a5,
  abstract     = {{<p>False Discovery Rate (FDR) and the Bayes risk are two different statistical measures, which can be used to evaluate and compare multiple testing procedures. Recent results show that under sparsity FDR controlling procedures, like the popular Benjamini-Hochberg (BH) procedure, perform also very well in terms of the Bayes risk. In particular asymptotic Bayes optimality under sparsity (ABOS) of BH was shown previously for location and scale models based on log-concave densities. This article extends previous work to a substantially larger set of distributions of effect sizes under the alternative, where the alternative distribution of true signals does not change with the number of tests m, while the sample size n slowly increases. ABOS of BH and the corresponding step-down procedure based on FDR levels proportional to n<sup>-1/2</sup> are proved. A simulation study shows that these asymptotic results are relevant already for relatively small values of m and n. Apart from showing asymptotic optimality of BH, our results on the optimal FDR level provide a natural extension of the well known results on the significance levels of Bayesian tests.</p>}},
  author       = {{Frommlet, Florian and Bogdan, Malgorzata}},
  issn         = {{1935-7524}},
  keywords     = {{Asymptotic optimality; Bayes risk; False discovery rate; Multiple testing; Two groups model}},
  language     = {{eng}},
  number       = {{1}},
  pages        = {{1328--1368}},
  publisher    = {{Institute of Mathematical Statistics}},
  series       = {{Electronic Journal of Statistics}},
  title        = {{Some optimality properties of FDR controlling rules under sparsity}},
  url          = {{http://dx.doi.org/10.1214/13-EJS808}},
  doi          = {{10.1214/13-EJS808}},
  volume       = {{7}},
  year         = {{2013}},
}