The joint distribution of the sum and maximum of dependent Pareto risks

Abstract

We develop a stochastic model for the sum $X$ and the maximum $Y$ of dependent, heavy-tail Pareto components. Our results include explicit forms of the probability density and cumulative distribution functions, marginal and conditional distributions, moments and related parameters, parameter estimation, and stochastic representations. We also derive mixed conditional tail expectations, $\mathrm{E}\left(X\right|Y>y)$ and $\mathrm{E}\left(Y\right|X>x)$, which provide measures of risk frequently used in finance and insurance. An extension incorporating a random number $N$ of components in the sum and the maximum, along with its basic properties, is included as well. Two data examples from finance illustrate modeling potential of these new multivariate distributions.