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1. (i) It is well-known (e.g., Rüschendorf (2013)) that the 6scx-largest element of YF is comonotonic, and thus a comonotonic random vector has the largest DQρ α in this case. Note that such ρ does not include VaR. Indeed, as we have seen from Proposition 4, DQVaR α (X) = 1 for comonotonic X under mild conditions, which is not equal to its largest value n. (ii) In case n = 2, the 6scx-smallest element of YF is counter-comonotonic, and thus a comonotonic random vector has the smallest DQρ α.
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2. (iii) For n > 3, the 6scx-smallest elements of YF are generally hard to obtain. If each pair (Xi, Xj) is counter-monotonic for i 6= j, then X is a 6scx-smallest element of YF. Pairwise counter-monotonicity puts very strong restrictions on the marginal distributions. For instance, it rules out all continuous marginal distributions; see Puccetti and Wang (2015).
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3. (iv) If a joint mix, i.e., a random vector with a constant component-wise sum, exists in YF, then any joint mix is a 6scx-smallest element of YF by Jensen’s inequality. See Puccetti and Wang (2015) and Wang and Wang (2016) for results on the existence of joint mixes. In case a joint mix does not exist, the 6scx-smallest elements are obtained by Bernard et al. (2014) and Jakobsons et al. (2016) under some conditions on the marginal distributions such as monotonic densities.
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33. For some mapping on Xn , finding the best-case and worst-case values and structures over YF is known as a problem of risk aggregation under dependence uncertainty; see Bernard et al. (2014) and Embrechts et al. (2015). If ρ = (ρα)α∈I is a class of 6SSD-consistent risk measures such as ES, then, by Proposition EC.1, DQρ α is consistent with the sum-convex order on YF. This leads to the following observations on the corresponding dependence structures.
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