The Annals of Statistics

Symmetric Upper Probabilities

Larry Wasserman and Joseph B. Kadane

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As a first step toward developing statistical models based on upper and lower probabilities, we study upper probabilities and upper expectations on the unit interval that are symmetric, by which we mean invariant with respect to equimeasurability. These upper probabilities are generalizations of uniform probability measures. We give some characterizations of these upper probabilities. Specifically, we show that symmetry of the upper expectation functional is equivalent to the underlying set of densities being closed under majorization. We also show that a function is the upper distribution for a symmetric upper probability if and only if its lower graph is star-shaped with respect to the origin and to the point (1,1). We derive inner and outer approximations to symmetric classes of probabilities based on the upper probability. The class of symmetric upper expectations that are completely determined by their values on the indicator functions is characterized. We provide a geometric characterization of a hierarchy of upper probabilities including Fine's generalized upper probabilities and 2-alternating Choquet capacities. In particular, we establish a 1-1 correspondence between symmetric, 2-alternating capacities and nonincreasing density functions. We prove that undominated generalized upper probabilities do not exist in the symmetric case. Examples from robust statistics are considered. An example is given that shows that symmetry of upper probabilities does not imply symmetry of upper expectations. A corollary is that symmetry of the Choquet integral does not imply symmetry of the upper expectation functional.

Article information

Ann. Statist., Volume 20, Number 4 (1992), 1720-1736.

First available in Project Euclid: 12 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60A05: Axioms; other general questions
Secondary: 62A99: None of the above, but in this section

Choquet capacities doubly stochastic operators equimeasurability lower probabilities majorization 2-alternating capacities


Wasserman, Larry; Kadane, Joseph B. Symmetric Upper Probabilities. Ann. Statist. 20 (1992), no. 4, 1720--1736. doi:10.1214/aos/1176348887.

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