A Characterization of the Multivariate Pareto Distribution

Abstract

For a random vector $\mathbf{X}$ on $\mathbf{X} > \mathbf{b}$ whose mean exists, the mean residual lifetime $E(\mathbf{X} - \mathbf{c}\mid \mathbf{X} > \mathbf{c})$ is an affine function of $\mathbf{c}$ on $\mathbf{c} > \mathbf{b}$ if and only if $\mathbf{X}$ can be partitioned into independent random vectors which have shifted multivariate Pareto or exponential distributions. An interpretation in terms of income-distribution is suggested for the Pareto case. It is also shown that every multivariate distribution whose mean exists is determined by its mean residual lifetime.