Quantile-Parameterized Distributions

Abstract

This paper introduces a new class of continuous probability distributions that are flexible enough to represent a wide range of uncertainties such as those that commonly arise in business, technology, and science. In many such cases, the nature of the uncertainty is more naturally characterized by quantiles than by parameters of familiar continuous probability distributions. In the practice of decision analysis, it is common to fit a hand-drawn curve to quantile outputs from probability elicitations on a continuous uncertain quantity and to then discretize the curve. The resulting discrete probability distribution is an approximation that cuts off the distribution's tails and eliminates intermediate values. Quantile-parameterized distributions address this problem by using quantiles themselves to parameterize a continuous probability distribution. We define quantile-parameterized distributions, illustrate their flexibility and range of applicability, and conclude with practical considerations when parameterizing distributions using inconsistent quantile assessments.