Statistical Hypothesis Testing Under Interval Uncertainty: An Overview

An important part of statistical data analysis is hypothesis testing. For example, we know the probability distribution of the characteristics corresponding to a certain disease, we have the values of the characteristics describing a patient, and we must make a conclusion whether this patient has this disease. Traditional hypothesis testing techniques are based on the assumption that we know the exact values of the characteristic(s) x describing a patient. In practice, the value x̃ comes from measurements and is, thus, only known with uncertainty: x̃ 6= x. In many practical situations, we only know the upper bound ∆ on the (absolute value of the) measurement error ∆x def = x̃−x. In such situation, after the measurement, the only information that we have about the (unknown) value x of this characteristic is that x belongs to the interval [x̃−∆, x̃ + ∆]. In this paper, we overview different approaches on how to test a hypothesis under such interval uncertainty. This overview is based on a general approach to decision making under interval uncertainty, approach developed by the 2007 Nobelist L. Hurwicz.