What finite-additivity can add to decision theory

Mark J. Schervish (), Teddy Seidenfeld (), Rafael B. Stern () and Joseph B. Kadane ()
Additional contact information
Mark J. Schervish: Carnegie Mellon University
Teddy Seidenfeld: Carnegie Mellon University
Rafael B. Stern: Federal University of São Carlos
Joseph B. Kadane: Carnegie Mellon University

Statistical Methods & Applications, 2020, vol. 29, issue 2, No 2, 237-263

Abstract: Abstract We examine general decision problems with loss functions that are bounded below. We allow the loss function to assume the value $$\infty $$∞. No other assumptions are made about the action space, the types of data available, the types of non-randomized decision rules allowed, or the parameter space. By allowing prior distributions and the randomizations in randomized rules to be finitely-additive, we prove very general complete class and minimax theorems. Specifically, under the sole assumption that the loss function is bounded below, we show that every decision problem has a minimal complete class and all admissible rules are Bayes rules. We also show that every decision problem has a minimax rule and a least-favorable distribution and that every minimax rule is Bayes with respect to the least-favorable distribution. Some special care is required to deal properly with infinite-valued risk functions and integrals taking infinite values.

Keywords: Admissible rule; Bayes rule; Complete class; Least-favorable distribution; Minimax rule; Primary 62C07; Secondary 62C20 (search for similar items in EconPapers)
Date: 2020
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s10260-019-00486-6 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:stmapp:v:29:y:2020:i:2:d:10.1007_s10260-019-00486-6

Ordering information: This journal article can be ordered from
http://www.springer. ... cs/journal/10260/PS2

DOI: 10.1007/s10260-019-00486-6

Access Statistics for this article

Statistical Methods & Applications is currently edited by Tommaso Proietti

More articles in Statistical Methods & Applications from Springer, Società Italiana di Statistica
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().