Symmetric Upper Probabilities

Abstract

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.