Insurance Applications of Some New Dependence Models derived from Multivariate Collective Models

Abstract

Consider two different portfolios which have claims triggered by the same events. Their corresponding collective model over a fixed time period is given in terms of individual claim sizes $(X_i,Y_i), i\ge 1$ and a claim counting random variable $N$. In this paper we are concerned with the joint distribution function $F$ of the \ece{largest claim sizes} $(X_{N:N}, Y_{N:N})$. By allowing $N$ to depend on some parameter, say $\theta$, then $F=F(\theta)$ is for various choices of $N$ a tractable parametric family of bivariate distribution functions. We present three applications of the implied parametric models to some data from the literature and a new data set from a Swiss insurance company. Furthermore, we investigate both distributional and asymptotic properties of $(X_{N:N}, Y_{N:N})$.