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Unifying neighbourhood and distortion models: part I – new results on old models

ABSTRACT Neighbourhoods of precise probabilities are instrumental to perform robustness analysis, as they rely on very few parameters. Many such models, sometimes referred to as distortion models, have been proposed in the literature, such as the pari mutuel model, the linear vacuous mixtures or the constant odds ratio model. This paper is the first part of a two paper series where we study the sets of probabilities induced by such models, regarding them as neighbourhoods defined over specific metrics or premetrics. We also compare them in terms of a number of properties: precision, number of extreme points, n-monotonicity, behaviour under conditioning, etc. This first part tackles this study on some of the most popular distortion models in the literature, while the second part studies less known neighbourhood models and summarises our findings.