Exceedance rate, exceedance probability, and the duality of GEV and GPD for extreme hazard analysis

Abstract

This paper points out that equating the rate of exceedance over threshold to the probability of exceedance in the generalized Pareto distribution, as is often applied in practice, leads to erroneous model parameter estimation, under- or overestimation of hazard, and impairs the duality between the generalized Pareto (GPD) and the generalized extreme-value (GEV) distributions. The problem stems from the fundamental difference in the domain of definition: the rate of exceedance $\in \left( {0,\infty } \right)$ and the probability of exceedance $\in \left( {0,1} \right)$. The erroneous parameter estimation is a result of practice in model parameter estimation that uses the concept of ‘return period’ (the inverse of exceedance probability) for both the GEV and the GPD. By using the concept of ‘average recurrence interval’ (the inverse of exceedance rate) of extremes in stochastic processes, we illustrate that the erroneous hazard estimation of the GPD is resolved. The use of average recurrence interval along with the duality allows the use of either the GEV or GPD for extreme hazard analysis, regardless of whether the data are collected via block maxima or peaks over a threshold. Some recommendations with regard to the practice of distribution parameter estimation are given. We demonstrate the duality of the two distributions and the impact of using average recurrence interval instead of return period by analysis of wind gust data collected by an automatic weather station at Woomera, South Australia, Australia.