Jensen's and Cantelli's Inequalities with Imprecise Previsions
| Autor: | Pelessoni, Renato, Vicig, Paolo |
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| Rok vydání: | 2022 |
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| Druh dokumentu: | Working Paper |
| DOI: | 10.1016/j.fss.2022.06.021 |
| Popis: | We investigate how basic probability inequalities can be extended to an imprecise framework, where (precise) probabilities and expectations are replaced by imprecise probabilities and lower/upper previsions. We focus on inequalities giving information on a single bounded random variable $X$, considering either convex/concave functions of $X$ (Jensen's inequalities) or one-sided bounds such as $(X\geq c)$ or $(X\leq c)$ (Markov's and Cantelli's inequalities). As for the consistency of the relevant imprecise uncertainty measures, our analysis considers coherence as well as weaker requirements, notably $2$-coherence, which proves to be often sufficient. Jensen-like inequalities are introduced, as well as a generalisation of a recent improvement to Jensen's inequality. Some of their applications are proposed: extensions of Lyapunov's inequality and inferential problems. After discussing upper and lower Markov's inequalities, Cantelli-like inequalities are proven with different degrees of consistency for the related lower/upper previsions. In the case of coherent imprecise previsions, the corresponding Cantelli's inequalities make use of Walley's lower and upper variances, generally ensuring better bounds. Comment: Published in Fuzzy Sets and Systems - https://dx.doi.org/10.1016/j.fss.2022.06.021 |
| Databáze: | arXiv |
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