Data-based prediction under uncertainty: CDF-quantile distributions and info-gap robustness

Abstract

Data underlie understanding of processes and prediction of the future. However, things change; data from one population at one time may have uncertain relevance for modeling or prediction in another population or at another time. Data-based prediction in a changing world requires two complementary capabilities: versatile modeling, integrated with management of uncertainty. We develop a response to this challenge. We focus on statistical models of bounded random variables, associated with additional non-probabilistic uncertainties. We employ CDF-quantile distributions to model the probabilistic aspects of these phenomena. Non-probabilistic uncertainties in parameter values and in the shapes of probability distributions are modeled and managed with the method of robust satisficing from info-gap theory. The robustness to uncertainty is evaluated for alternative estimators. We show that putatively optimal estimators may be less robust than sub-optimal estimators, suggesting preference for a sub-optimal estimator in some circumstances. These two attributes –statistical accuracy and info-gap robustness –trade off against one another. The info-gap robustness function quantifies this trade off. Generic propositions specify the robustness functions and their trade offs, and characterize a class of situations in which preference for sub-optimal estimators can occur. Three examples are discussed.