Abstract
We extend the notion of stochastic order to the pairwise comparison of fuzzy random variables. We consider expected utility, stochastic dominance and statistical preference, which are related to the comparisons of the expectations, distribution functions and medians of the underlying variables, and discuss how to generalize these notions to the fuzzy case, when an epistemic interpretation is given to the fuzzy random variables. In passing, we investigate to which extent the earlier extensions of stochastic dominance and expected utility to the comparison of sets of random variables can be useful as fuzzy rankings.
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Wang and Kerre assume that the fuzzy ranking produces a complete order, which is not always the case for our imprecise stochastic orders; this is why we have included (A0) in our discussion. In addition, axioms (A4), (A5) and (A6) correspond to (A4\(^{\prime }\)), (A6) and (A7) in that paper, their (A5) not being too interesting in our context.
Our result is more general because we establish the equality between these two lower previsions on gambles, and not only events, and for this we need to establish the complete monotonicity of these previsions.
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Acknowledgments
This is an extended version, with proofs and additional results, of earlier work presented at the SMPS’2014 and IFSA-EUSFLAT’2015 conferences. This work has benefited from discussions with Didier Dubois and Sébastien Destercke. We also acknowledge the financial support by Project TIN2014-59543-P.
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Montes, I., Miranda, E. & Montes, S. Imprecise stochastic orders and fuzzy rankings. Fuzzy Optim Decis Making 16, 297–327 (2017). https://doi.org/10.1007/s10700-016-9251-y
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DOI: https://doi.org/10.1007/s10700-016-9251-y