Introduction to Imprecise Probabilities

7.5 Frequentist statistics with imprecise probabilities

Now we discuss frequentist methods for statistical inference with imprecise probabilities. In this setting an imprecise sampling model, with the parameter understood as a fixed, non-probabilistic quantity, is considered. Methods currently available have almost exclusively been developed in the context of robust statistics, and thus presentation here also focuses on this area, highlighting those results from robust statistics where the relation to, and the (potential) impact on, imprecise probabilities is most striking.

Influenced by robust statistics, the methodology is typically relying on the sensitivity analysis point of view in the sense of Section 7.2.1. Moreover, aiming at robustness, imprecision is mainly used as a device to safeguard against potentially devastating consequences of slight deviations from an underlying traditional model.⁶¹ Thus in robust statistics the models considered are usually neighbourhood models, but many of the concepts, results and methods developed there are also of high relevance for general imprecise probability models. This in particular applies to hypothesis testing.

We start with describing the background, giving two prototypical examples of non-robustness of traditional statistical procedures. Then, in the second subsection, we review the comparatively widely developed theory of statistical hypothesis testing based on the so-called Huber-Strassen theorem. In the third subsection we ...