1932

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.

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2015-04-10
2024-12-05
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    Illustration of the Extremal Types Theorem. For increasing values of , the left panels display the distribution of the maximum of independent uniform (), standard Gaussian (), unit exponential (), and 0.2-Pareto (), i.e., ()=1−−0.2, >1, random variables. The right panels display the distribution of −1() for appropriate sequences >0 and . In each case, is asymptotically degenerate, whereas −1() is not.

    Illustration of the point process of exceedances and the convergence to the generalized Pareto distribution (GPD). For increasing values of , the plots display the point process of rescaled times and rescaled variables, namely (/(+1), ()/), for data simulated from the uniform (), standard Gaussian (), unit exponential (), and 0.2-Pareto distributions. The side plots are histograms of the exceedances over the threshold (), i.e., −1() for which −1()>. The solid red curves are the corresponding asymptotic GPD densities.

    Illustration of extremal clustering for data simulated from ARMAX() processes with ≥0, i.e., =max(, ), =1, 2, …, where the are independent unit Fréchet random variables. The extremal index for this process is θ=max(1, )/(1+), so extreme events form clusters with mean size 2 when =1.

    Illustration of the estimation of low-probability events. The upper panels display asymptotically dependent () and asymptotically independent () data, along with a target extreme region . The numbers correspond to the true probability that a point lies in (), its naive empirical estimate (), its estimate under asymptotic dependence using Equation 12 () and its estimate under asymptotic independence using Equation 23 (). For and , extrapolation is based on the empirical estimate at the 0.95 level (), and uses an estimate of the coefficient of tail dependence proposed in an article by Ledford & Tawn (1996). The bottom panels show these probabilities as a function of the threshold (i.e., the -coordinate of the lower left corner of ).

    Illustration of the point process of exceedances in the bivariate framework for increasing . The upper left panel displays bivariate Student data with 2 degrees of freedom and standard Pareto marginals, rescaled by . These data are asymptotically dependent. The right-hand panels show the corresponding histograms of the pseudoradius and pseudoangle for points lying above the threshold defined by >10−2 (), with limiting densities superimposed in red. The bottom plots illustrate the Gaussian case, which is asymptotically independent; here the limiting distribution of places equal point masses of.5 at =0 and =1, and it has no mass elsewhere.

    Illustration of the construction and simulation of a max-stable process, here a unidimensional Smith model. A large (but in theory, infinite) number of random storms with random decreasing sizes are generated (), and the resulting max-stable process corresponds to the pointwise supermum ().

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