## The Annals of Statistics

### Dominating countably many forecasts

#### Abstract

We investigate differences between a simple Dominance Principle applied to sums of fair prices for variables and dominance applied to sums of forecasts for variables scored by proper scoring rules. In particular, we consider differences when fair prices and forecasts correspond to finitely additive expectations and dominance is applied with infinitely many prices and/or forecasts.

#### Article information

Source
Ann. Statist., Volume 42, Number 2 (2014), 728-756.

Dates
First available in Project Euclid: 20 May 2014

Permanent link to this document
https://projecteuclid.org/euclid.aos/1400592176

Digital Object Identifier
doi:10.1214/14-AOS1203

Mathematical Reviews number (MathSciNet)
MR3210985

Zentralblatt MATH identifier
1295.62006

Subjects
Primary: 62A01: Foundations and philosophical topics
Secondary: 62C05: General considerations

#### Citation

Schervish, M. J.; Seidenfeld, Teddy; Kadane, J. B. Dominating countably many forecasts. Ann. Statist. 42 (2014), no. 2, 728--756. doi:10.1214/14-AOS1203. https://projecteuclid.org/euclid.aos/1400592176

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#### Supplemental materials

• Supplementary material: Infinite previsions and finitely additive expectations. The expectation of a random variable $X$ defined on $\Omega$ is usually defined as the integral of $X$ over the set $\Omega$ with respect to the underlying probability measure defined on subsets of $\Omega$. In the countably additive setting, such integrals can be defined (except for certain cases involving $\infty-\infty$) uniquely from a probability measure on $\Omega$. Dunford and Schwartz [(1958), Chapter III] give a detailed analysis of integration with respect to finitely additive measures that attempts to replicate the uniqueness of integrals. Their analysis requires additional assumptions if one wishes to integrate unbounded random variables. We choose the alternative of defining integrals as special types of linear functionals. This is the approach used in the study of the Daniell integral. [See Royden (1968), Chapter 13.] Then the measure of a set becomes the integral of its indicator function. De Finetti’s concept of prevision turns out to be a finitely additive generalization of the Daniell integral. (See Definition 10 in Appendix A.2.) We provide details on the finitely additive Daniell integral along with details about the meaning of infinite previsions and how to extend coherence$_{1}$ and coherence$_{3}$ to deal with random variables having infinite previsions. Infinite previsions invariably arise when dealing with general sets of unbounded random variables.