Monotonicity properties of multivariate distribution and survival functions — With an application to Lévy-frailty copulas

Abstract

The monotonicity properties of multivariate distribution functions are definitely more complicated than in the univariate case. We show that they fit perfectly well into the general theory of completely monotone and alternating functions on abelian semigroups. This allows us to prove a correspondence theorem which generalizes the classical version in two respects: the function in question may be defined on rather arbitrary product sets in R ¯ n , and it need not be grounded, i.e. disappear at the lower-left boundary. In 2009 a greatly interesting class of copulas was discovered by Mai and Scherer (cf. Mai and Scherer (2009) [4]), connecting in a very surprising way complete monotonicity with respect to the maximum operation on R + n and with respect to ordinary addition on N 0 . Based on the preceding results, we give another proof of this result.