Statistics of Extremes

Abstract

Statistics of extremes concerns inference for rare events. Often the events have never yet been observed, and their probabilities must therefore be estimated by extrapolation of tail models fitted to available data. Because data concerning the event of interest may be very limited, efficient methods of inference play an important role. This article reviews this domain, emphasizing current research topics. We first sketch the classical theory of extremes for maxima and threshold exceedances of stationary series. We then review multivariate theory, distinguishing asymptotic independence and dependence models, followed by a description of models for spatial and spatiotemporal extreme events. Finally, we discuss inference and describe two applications. Animations illustrate some of the main ideas. 9.1

Non-Stationary Dependence Structures for Spatial Extremes

“…For more details about univariate and multivariate extremes, see Beirlant et al (2004) and Davison and Huser (2015), and for an account of spatial extremes, see the review papers 5 by , Cooley et al (2012) and Davison et al (2013). See also the book by de Haan and Ferreira (2006), which explains the technicalities in depth.…”

A risk‐based forecast of extreme mortality events in small cetaceans: Using stranding data to inform conservation practice

“…The effect of extreme events on population dynamics is a growing area of conservation research (Barbraud et al., ; Chambert et al., ; Gutschick & BassiriRad, ). We predicted the effects of potential extreme ASMEs on small cetacean strandings along French coasts using EVT (Coles, ; Davison & Huser, ) and computed their associated risks, which enabled us to compare the predicted and observed figures across months and years for each species. The three species: harbor porpoises, common dolphins, and striped dolphins, each provided a different picture.…”

“…Our approach evaluates the impacts of extreme ASMEs by considering extreme stranding values (Davison & Huser, ). In our case, extreme ASMEs are defined as a large number of strandings over a 3‐day period.…”

Full likelihood inference for max‐stable data

“…Multivariate max‐stable models can be constructed similarly by substituting the processes W j ( s ) in Equation by analogous random vectors W j = ( W j 1 ,…, W jD ) ⊤ . From Equation , we deduce that the joint distribution of Z ( s ) at a finite collection of sites may be expressed as where the exponent function satisfies homogeneity and marginal constraints (see, e.g., Davison & Huser, ). As an illustration, Figure shows two independent realizations from the same Smith () model on defined by taking W j ( s ) = φ( s − U j ; σ 2 ), , in Equation , where φ(·; σ 2 ) is the normal density with zero mean and variance σ 2 , and the U j s are points from a unit rate Poisson point process on the real line.…”

“…This broad but constrained class of models may, at least theoretically, be used to extrapolate into the joint tail, hence providing a justified framework for risk assessment of extreme events. The probabilistic justification for using these models is that the max‐stable property arises in limiting models for suitably renormalized maxima of independent and identically distributed processes; see, for example, Davison, Padoan, and Ribatet (), Davison and Huser (), and Davison, Huser, and Thibaud ().…”