Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 33 / 4 / e / Solution

This is the generalized Pareto distribution for the excess $X-u$, with scale $\sigma >0$ and shape $\xi$. Its support requires $1+\xi (x-u)/\sigma >0$. The Pickands-Balkema-de Haan theorem says that, for distributions in an appropriate extreme-value domain of attraction, the conditional distribution of excesses above a sufficiently high threshold approaches a generalized Pareto form. This makes it the natural peaks-over-threshold method model; it is an asymptotic justification, not an assertion that every threshold is sufficiently high.

For $\xi >0$, the tail decays as a power and has no finite upper endpoint, giving a heavy tail. For $\xi =0$, the limiting distribution is exponential, with survival probability $\exp \{-(x-u)/\sigma \}$. For $\xi <0$, there is a finite upper endpoint $u-\sigma /\xi$, giving a bounded tail. Thus the sign of the shape parameter distinguishes heavy, exponential-type and bounded tails.