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August 2021 On extended admissible procedures and their nonstandard Bayes risk
Haosui Duanmu, Daniel M. Roy
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Ann. Statist. 49(4): 2053-2078 (August 2021). DOI: 10.1214/20-AOS2026

Abstract

For finite parameter spaces, among decision procedures with finite risk functions, a decision procedure is extended admissible if and only if it is Bayes. Various relaxations of this classical equivalence have been established for infinite parameter spaces, but these extensions are each subject to technical conditions that limit their applicability, especially to modern (semi and nonparametric) statistical problems. Using results in mathematical logic and nonstandard analysis, we extend this equivalence to arbitrary statistical decision problems: informally, we show that, among decision procedures with finite risk functions, a decision procedure is extended admissible if and only if it has infinitesimal excess Bayes risk. In contrast to existing results, our equivalence holds in complete generality, that is, without regularity conditions or restrictions on the model or loss function. We also derive a nonstandard analogue of Blyth’s method that yields sufficient conditions for admissibility, and apply the nonstandard theory to derive a purely standard theorem: when risk functions are continuous on a compact Hausdorff parameter space, a procedure is extended admissible if and only if it is Bayes.

Funding Statement

This research was made possible through an NSERC Discovery Grant, Connaught Award, and U.S. Air Force Office of Scientific Research Grant #FA9550-15-1-0074.

Acknowledgments

The authors owe a debt of gratitude to William Weiss for detailed suggestions. We thank Gintarė Džiugaitė, Cameron Freer, and H. Jerome Keisler for early discussions and insights, and thank Nate Ackerman, Robert Anderson, Michael Evans, Jeffrey Negrea, Arno Pauly, and Aaron Smith for feedback on drafts and helpful discussions. We thank the anonymous referees for suggesting reformulations of Theorems 2.2, 5.18 and 6.6, and for the idea of pursuing the results in Section 7 and Appendix J. Finally, the authors would like to thank Peter Hoff for his course notes, which served as our first introduction to the topic. This work was done in part while the authors were visiting the Simons Institute for the Theory of Computing at UC Berkeley.

Citation

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Haosui Duanmu. Daniel M. Roy. "On extended admissible procedures and their nonstandard Bayes risk." Ann. Statist. 49 (4) 2053 - 2078, August 2021. https://doi.org/10.1214/20-AOS2026

Information

Received: 1 June 2017; Revised: 1 June 2020; Published: August 2021
First available in Project Euclid: 29 September 2021

Digital Object Identifier: 10.1214/20-AOS2026

Subjects:
Primary: 62A01 , 62C07 , 62C10
Secondary: 28E05

Keywords: Complete class theorems , decision theory , nonstandard analysis

Rights: Copyright © 2021 Institute of Mathematical Statistics

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Vol.49 • No. 4 • August 2021
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