Some Remarks on the Supermodular Order

On the distortion of a copula and its margins

“…According to Theorem 3.3 and Theorem 3.4 in Müller and Scarsini (2000), we can finally conclude X ≤ sm Y. Additional equivalent characterizations of stop-loss order can be found in Denuit et al (2005).…”

“…≥ 0 for any 1 ≤ i < j ≤ n, then it follows from Theorem 2.2 in Müller and Scarsini (2000) that C X is supermodular. In addition, the composition C X (F * 1 (·), .…”

“…According to Theorem 3.4 in Müller and Scarsini (2000), since X ≤ sm Y, then X and Y have the same margins. The existence of the copulas immediately follows from Sklar (1959).…”

“…, x n denote the individual claims coming from n policies, then the supermodularity of the function has the implication of a worse loss to the insurer, given that an increase of a single claim increases the values of some of the other claims. More about supermodular functions and their properties can be found in Bäuerle (1997) and Müller and Scarsini (2000).…”

When Are On–off Sources Sis?: Conditions and Applications

“…Additional information on the sm ordering can be found in [3,4,11,12,16,20]. In Sections 7 and 8 we shall need the fact that the sm ordering is closed under convolution.…”