Title: General comparison results for factor models
Authors: Jonathan Ansari - University of Freiburg (Germany) [presenting]
Ludger Ruschendorf - University of Freiburg (Germany)
Abstract: In the setting of a factor model, the components of a random vector $X=(X_1,..., X_d)$ depend on a common factor variable $Z$ and some individual influences. Typically, the marginal distributions, as well as the copula of $X_i$ and $Z$, can be (partially) estimated from the data. However, the conditional distribution of $X$ given $Z$ is often not specified. Extending recently investigated ordering results for a multivariate version of *-products of bivariate copulas to the supermodular ordering, we provide some general comparison results for factor models in dependence on the specifications considering specifically the strong notion of the supermodular and directionally convex order. The proofs are based on standard classical ordering theory and rearrangement results, as well as on mass transfer theory. As a consequence of the ordering results, we derive best and worst case scenarios in relevant classes of factor models allowing, in particular, interesting applications to deriving sharp bounds in financial and insurance risk models. Further, we provide a new construction method for comprehensive, multi-parameter families of positively dependent, multivariate distributions that are increasing in the parameters w.r.t. the supermodular or directionally convex order.