New distributions for modeling subjective lower and upper probabilities

Abstract

This paper presents an investigation of a relatively unstudied approach to modeling lower and upper subjective probabilities. It is based on the fact that every cumulative distribution function (CDF) with support (0,1) has a “dual” CDF that obeys the conjugacy relation between coherent lower and upper probabilities. A new 2-parameter family of “CDF-Quantile” distributions with support (0,1) is extended via a third parameter for the purpose of modeling lower–upper probabilities via this approach. The extension exploits certain properties of the CDF-Quantile family, and the fact that continuous CDFs on (0,1) random variables form an algebraic group that is closed under composition. This extension also yields methods for testing specific models of lower–upper probability assignments. Finally, the new models are applied to real data-sets, and compared with alternative approaches for their relative advantages and drawbacks.