Open Access
February 2020 Needles and straw in a haystack: Robust confidence for possibly sparse sequences
Eduard Belitser, Nurzhan Nurushev
Bernoulli 26(1): 191-225 (February 2020). DOI: 10.3150/19-BEJ1122


In the general signal$+$noise (allowing non-normal, non-independent observations) model, we construct an empirical Bayes posterior which we then use for uncertainty quantification for the unknown, possibly sparse, signal. We introduce a novel excessive bias restriction (EBR) condition, which gives rise to a new slicing of the entire space that is suitable for uncertainty quantification. Under EBR and some mild exchangeable exponential moment condition on the noise, we establish the local (oracle) optimality of the proposed confidence ball. Without EBR, we propose another confidence ball of full coverage, but its radius contains an additional $\sigma n^{1/4}$-term. In passing, we also get the local optimal results for estimation, posterior contraction problems, and the problem of weak recovery of sparsity structure. Adaptive minimax results (also for the estimation and posterior contraction problems) over various sparsity classes follow from our local results.


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Eduard Belitser. Nurzhan Nurushev. "Needles and straw in a haystack: Robust confidence for possibly sparse sequences." Bernoulli 26 (1) 191 - 225, February 2020.


Received: 1 March 2018; Revised: 1 January 2019; Published: February 2020
First available in Project Euclid: 26 November 2019

zbMATH: 07140497
MathSciNet: MR4036032
Digital Object Identifier: 10.3150/19-BEJ1122

Keywords: confidence set , empirical Bayes posterior , local radial rate

Rights: Copyright © 2020 Bernoulli Society for Mathematical Statistics and Probability

Vol.26 • No. 1 • February 2020
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