Elements of asymptotic theory with outer probability measures
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Outer measures can be used for statistical inference in place of probability measures to bring flexibility in terms of model specification. The corresponding statistical procedures such as estimation or hypothesis testing need to be analysed in order to understand their behaviour, and motivate their use. In this article, we consider a simple class of outer measures based on the supremum of particular functions that we refer to as possibility functions. We then derive the asymptotic properties of the corresponding maximum likelihood estimators, likelihood ratio tests and Bayesian posterior uncertainties. These results are largely based on versions of both the law of large numbers and the central limit theorem that are adapted to possibility functions. Our motivation with outer measures is through the notion of uncertainty quantification, where verification of these procedures is of crucial importance. These introduced concepts naturally strengthen the link between the frequentist and Bayesian approaches.
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