A two-step framework for over-threshold modelling of environmental extremes

Abstract

The evaluation of the probability of occurrence of extreme natural events is important for the protection of urban areas, industrial facilities and others. Traditionally, the extreme value theory (EVT) offers a valid theoretical framework on this topic. In an over-threshold modelling (OTM) approach, Pickands' theorem, (Pickands, 1975) states that, for a sample composed by independent and identically distributed (i.i.d.) values, the distribution of the data exceeding a given threshold converges through a generalized Pareto distribution (GPD). Following this theoretical result, the analysis of realizations of environmental variables exceeding a threshold spread widely in the literature. However, applying this theorem to an auto-correlated time series logically involves two successive and complementary steps: the first one is required to build a sample of i.i.d. values from the available information, as required by the EVT; the second to set the threshold for the optimal convergence toward the GPD. In the past, the same threshold was often employed both for sampling observations and for meeting the hypothesis of extreme value convergence. This confusion can lead to an erroneous understanding of methodologies and tools available in the literature. This paper aims at clarifying the conceptual framework involved in threshold selection, reviewing the available methods for the application of both steps and illustrating it with a double threshold approach.