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.

## References

*Trans. Amer. Math. Soc.*

**271**667–687. MR0654856 10.2307/1998904Anderson, R. M. (1982). Star-finite representations of measure spaces.

*Trans. Amer. Math. Soc.*

**271**667–687. MR0654856 10.2307/1998904

*Statistical Decision Theory and Bayesian Analysis*, 2nd ed.

*Springer Series in Statistics*. Springer, New York. MR0804611 10.1007/978-1-4757-4286-2Berger, J. O. (1985).

*Statistical Decision Theory and Bayesian Analysis*, 2nd ed.

*Springer Series in Statistics*. Springer, New York. MR0804611 10.1007/978-1-4757-4286-2

*Theory of Charges*:

*A Study of Finitely Additive Measures*.

*Pure and Applied Mathematics*

**109**. Academic Press [Harcourt Brace Jovanovich, Publishers], New York. MR0751777Bhaskara Rao, K. P. S. and Bhaskara Rao, M. (1983).

*Theory of Charges*:

*A Study of Finitely Additive Measures*.

*Pure and Applied Mathematics*

**109**. Academic Press [Harcourt Brace Jovanovich, Publishers], New York. MR0751777

*Ann. Math. Stat.*

**42**855–903. MR0286209 10.1214/aoms/1177693318Brown, L. D. (1971). Admissible estimators, recurrent diffusions, and insoluble boundary value problems.

*Ann. Math. Stat.*

**42**855–903. MR0286209 10.1214/aoms/1177693318

*Fundamentals of Statistical Exponential Families with Applications in Statistical Decision Theory*.

*Institute of Mathematical Statistics Lecture Notes—Monograph Series*

**9**. IMS, Hayward, CA. MR0882001Brown, L. D. (1986).

*Fundamentals of Statistical Exponential Families with Applications in Statistical Decision Theory*.

*Institute of Mathematical Statistics Lecture Notes—Monograph Series*

**9**. IMS, Hayward, CA. MR0882001

*Ann. Statist.*

**13**262–271. MR0773166 10.1214/aos/1176346591Cohen, M. P. and Kuo, L. (1985). The admissibility of the empirical distribution function.

*Ann. Statist.*

**13**262–271. MR0773166 10.1214/aos/1176346591

*Nonstandard Analysis*(L. O. Arkeryd, N. J. Cutland and C. W. Henson, eds.).

*NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci.*

**493**51–76. Kluwer Academic, Dordrecht. MR1603229 10.1007/978-94-011-5544-1_2Cutland, N. J. (1997). Nonstandard real analysis. In

*Nonstandard Analysis*(L. O. Arkeryd, N. J. Cutland and C. W. Henson, eds.).

*NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci.*

**493**51–76. Kluwer Academic, Dordrecht. MR1603229 10.1007/978-94-011-5544-1_2

*Developments in Nonstandard Mathematics*(N. J. Cutland, V. Neves, F. Oliveira and J. Sousa-Pinto, eds.).

*Pitman Research Notes in Mathematics Series*

**336**151–176. Longman, Harlow. MR1394215Cutland, N. J., Neves, V., Oliveira, F. and Sousa-Pinto, J. (1995). Loeb measure theory. In

*Developments in Nonstandard Mathematics*(N. J. Cutland, V. Neves, F. Oliveira and J. Sousa-Pinto, eds.).

*Pitman Research Notes in Mathematics Series*

**336**151–176. Longman, Harlow. MR1394215

*Comment. Math. Univ. Carolin.*

**59**467–485. MR3914713 10.14712/1213-7243.2015.270Duanmu, H. and Weiss, W. (2018). Finitely-additive, countably-additive and internal probability measures.

*Comment. Math. Univ. Carolin.*

**59**467–485. MR3914713 10.14712/1213-7243.2015.270

*Mathematical Statistics*:

*A Decision Theoretic Approach*.

*Probability and Mathematical Statistics*

**1**. Academic Press, New York. MR0215390Ferguson, T. S. (1967).

*Mathematical Statistics*:

*A Decision Theoretic Approach*.

*Probability and Mathematical Statistics*

**1**. Academic Press, New York. MR0215390

*Games Econom. Behav.*

**68**155–179. MR2577384 10.1016/j.geb.2009.03.013Halpern, J. Y. (2010). Lexicographic probability, conditional probability, and nonstandard probability.

*Games Econom. Behav.*

**68**155–179. MR2577384 10.1016/j.geb.2009.03.013

*Proc*. 4

*th Berkeley Sympos. Math. Statist. and Prob*.,

*Vol. I*361–379. Univ. California Press, Berkeley, CA. MR0133191James, W. and Stein, C. (1961). Estimation with quadratic loss. In

*Proc*. 4

*th Berkeley Sympos. Math. Statist. and Prob*.,

*Vol. I*361–379. Univ. California Press, Berkeley, CA. MR0133191

*Rethinking the Foundations of Statistics*211.Kadane, J. B., Schervish, M. J. and Seidenfeld, T. (1999). Statistical implications of finitely additive probability. In

*Rethinking the Foundations of Statistics*211.

*Ann. Math. Stat.*

**28**573–601. MR0092325 10.1214/aoms/1177706874Kiefer, J. (1957). Invariance, minimax sequential estimation, and continuous time processes.

*Ann. Math. Stat.*

**28**573–601. MR0092325 10.1214/aoms/1177706874

*Ann. Math. Stat.*

**26**69–81. MR0067443 10.1214/aoms/1177728594LeCam, L. (1955). An extension of Wald’s theory of statistical decision functions.

*Ann. Math. Stat.*

**26**69–81. MR0067443 10.1214/aoms/1177728594

*Applied Statistical Decision Theory*.

*Studies in Managerial Economics*. Division of Research, Graduate School of Business Administration, Harvard Univ., Boston, MA. MR0117844Raiffa, H. and Schlaifer, R. (1961).

*Applied Statistical Decision Theory*.

*Studies in Managerial Economics*. Division of Research, Graduate School of Business Administration, Harvard Univ., Boston, MA. MR0117844

*Ann. Math. Stat.*

**34**751–768. MR0150908 10.1214/aoms/1177704001Sacks, J. (1963). Generalized Bayes solutions in estimation problems.

*Ann. Math. Stat.*

**34**751–768. MR0150908 10.1214/aoms/1177704001

*Ann. Math. Stat.*

**26**518–522. MR0070929 10.1214/aoms/1177728497Stein, C. (1955). A necessary and sufficient condition for admissibility.

*Ann. Math. Stat.*

**26**518–522. MR0070929 10.1214/aoms/1177728497

*Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability*, 1954

*–*1955,

*Vol. I*197–206. Univ. California Press, Berkeley and Los Angeles. MR0084922Stein, C. (1956). Inadmissibility of the usual estimator for the mean of a multivariate normal distribution. In

*Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability*, 1954

*–*1955,

*Vol. I*197–206. Univ. California Press, Berkeley and Los Angeles. MR0084922

*Ann. Math. Stat.*

**38**818–822. MR0226780 10.1214/aoms/1177698876Stone, M. (1967). Generalized Bayes decision functions, admissibility and the exponential family.

*Ann. Math. Stat.*

**38**818–822. MR0226780 10.1214/aoms/1177698876

*Ann. Math. Stat.*

**10**299–326. MR0000932 10.1214/aoms/1177732144Wald, A. (1939). Contributions to the theory of statistical estimation and testing hypotheses.

*Ann. Math. Stat.*

**10**299–326. MR0000932 10.1214/aoms/1177732144

*Ann. Math. Stat.*

**18**549–555. MR0023499 10.1214/aoms/1177730345Wald, A. (1947). An essentially complete class of admissible decision functions.

*Ann. Math. Stat.*

**18**549–555. MR0023499 10.1214/aoms/1177730345

*Econometrica*

**15**279–313. MR0024113 10.2307/1905331Wald, A. (1947). Foundations of a general theory of sequential decision functions.

*Econometrica*

**15**279–313. MR0024113 10.2307/1905331

*Ann. Math. Stat.*

**20**165–205. MR0044802 10.1214/aoms/1177730030Wald, A. (1949). Statistical decision functions.

*Ann. Math. Stat.*

**20**165–205. MR0044802 10.1214/aoms/1177730030