  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 $(X_{N:N}, Y_{N:N})$.
