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June 2017 On coverage and local radial rates of credible sets
Eduard Belitser
Ann. Statist. 45(3): 1124-1151 (June 2017). DOI: 10.1214/16-AOS1477

Abstract

In the mildly ill-posed inverse signal-in-white-noise model, we construct confidence sets as credible balls with respect to the empirical Bayes posterior resulting from a certain two-level hierarchical prior. The quality of the posterior is characterized by the contraction rate which we allow to be local, that is, depending on the parameter. The issue of optimality of the constructed confidence sets is addressed via a trade-off between its “size” (the local radial rate) and its coverage probability. We introduce excessive bias restriction (EBR), more general than self-similarity and polished tail condition recently studied in the literature. Under EBR, we establish the confidence optimality of our credible set with some local (oracle) radial rate. We also derive the oracle estimation inequality and the oracle posterior contraction rate. The obtained local results are more powerful than global: adaptive minimax results for a number of smoothness scales follow as consequence, in particular, the ones considered by Szabó et al. [Ann. Statist. 43 (2015) 1391–1428].

Citation

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Eduard Belitser. "On coverage and local radial rates of credible sets." Ann. Statist. 45 (3) 1124 - 1151, June 2017. https://doi.org/10.1214/16-AOS1477

Information

Received: 1 January 2016; Revised: 1 May 2016; Published: June 2017
First available in Project Euclid: 13 June 2017

zbMATH: 1371.62044
MathSciNet: MR3662450
Digital Object Identifier: 10.1214/16-AOS1477

Subjects:
Primary: 62C05, 62G15
Secondary: 62G99

Rights: Copyright © 2017 Institute of Mathematical Statistics

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Vol.45 • No. 3 • June 2017
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