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
2024-12-13
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Literature Cited

  1. Anderson CW. 1971. Contributions to the Asymptotic Theory of Extreme Values PhD Thesis, University of London, London, UK [Google Scholar]
  2. Anderson CW, Coles SG, Hüsler J. 1997. Maxima of Poisson-like variables and related triangular arrays. Ann. Appl. Probab. 7:953–71 [Google Scholar]
  3. Ballani F, Schlather M. 2011. A construction principle for multivariate extreme value distributions. Biometrika 98:633–45 [Google Scholar]
  4. Beirlant J, Dierckx G, Goegebeur Y, Matthys G. 1999. Tail index estimation and an exponential regression model. Extremes 2:177–200 [Google Scholar]
  5. Beirlant J, Goegebeur Y, Teugels J, Segers J. 2004. Statistics of Extremes: Theory and Applications New York: Wiley [Google Scholar]
  6. Blanchet J, Davison AC. 2011. Spatial modelling of extreme snow depth. Ann. Appl. Stat. 5:1699–725 [Google Scholar]
  7. Boldi M-O, Davison AC. 2007. A mixture model for multivariate extremes. J. R. Stat. Soc. B 69:217–29 [Google Scholar]
  8. Bortot P, Coles SG. 2003. Extremes of Markov chains with tail-switching potential. J. R. Stat. Soc. B 65:851–67 [Google Scholar]
  9. Bortot P, Gaetan C. 2014. A latent process model for temporal extremes. Scand. J. Stat. 41:606–21 [Google Scholar]
  10. Bortot P, Tawn JA. 1998. Models for the extremes of Markov chains. Biometrika 85:851–67 [Google Scholar]
  11. Brown BM, Resnick SI. 1977. Extreme values of independent stochastic processes. J. Appl. Probab. 14:732–39 [Google Scholar]
  12. Buishand TA, de Haan L, Zhou C. 2008. On spatial extremes: with application to a rainfall problem. Ann. Appl. Stat. 2:624–42 [Google Scholar]
  13. Butler A, Heffernan JE, Tawn JA, Flather RA. 2007. Trend estimation in extremes of synthetic North Sea surges. Appl. Stat. 56:395–414 [Google Scholar]
  14. Capéraà P, Fougères AL, Genest C. 1997. A nonparametric estimation procedure for bivariate extreme value copulas. Biometrika 84:567–77 [Google Scholar]
  15. Chandler RE, Bate S. 2007. Inference for clustered data using the independence log-likelihood. Biometrika 94:167–83 [Google Scholar]
  16. Choulakian V, Stephens MA. 2001. Goodness-of-fit tests for the generalized Pareto distribution. Technometrics 43:478–84 [Google Scholar]
  17. Coles SG. 2001. An Introduction to Statistical Modeling of Extreme Values New York: Springer [Google Scholar]
  18. Coles SG, Heffernan J, Tawn JA. 1999. Dependence measures for extreme value analyses. Extremes 2:339–65 [Google Scholar]
  19. Coles SG, Tawn JA. 1991. Modelling extreme multivariate events. J. R. Stat. Soc. B 53:377–92 [Google Scholar]
  20. Coles SG, Tawn JA. 1996. Modelling extremes of the areal rainfall process. J. R. Stat. Soc. B 58:329–47 [Google Scholar]
  21. Cooley D, Davis RA, Naveau P. 2010. The pairwise beta distribution: a flexible parametric multivariate model for extremes. J. Multivar. Anal. 101:2103–17 [Google Scholar]
  22. Cooley D, Nychka D, Naveau P. 2007. Bayesian spatial modeling of extreme precipitation return levels. J. Am. Stat. Assoc. 102:824–40 [Google Scholar]
  23. Danielsson J, de Haan L, Peng L, de Vries CG. 2001. Using a bootstrap method to choose the sample fraction in tail index estimation. J. Multivar. Anal. 76:226–48 [Google Scholar]
  24. Das B, Resnick SI. 2011. Conditioning on an extreme component: model consistency with regular variation on cones. Bernoulli 17:1226–52 [Google Scholar]
  25. Davison AC, Gholamrezaee MM. 2012. Geostatistics of extremes. Proc. R. Soc. Lond. A 468:581–608 [Google Scholar]
  26. Davison AC, Huser R, Thibaud E. 2013. Geostatistics of dependent and asymptotically independent extremes. Math. Geosci. 45:511–29 [Google Scholar]
  27. Davison AC, Padoan SA, Ribatet M. 2012. Statistical modelling of spatial extremes. Stat. Sci. 27:161–86; discussion187–201 [Google Scholar]
  28. Davison AC, Smith RL. 1990. Models for exceedances over high thresholds. J. R. Stat. Soc. B 52:393–425; discussion425–42 [Google Scholar]
  29. de Carvalho MB, Davison AC. 2014. Spectral density ratio models for multivariate extremes. J. Am. Stat. Assoc. 109:764–76 [Google Scholar]
  30. de Haan L. 1984. A spectral representation for max-stable processes. Ann. Probab. 12:1194–204 [Google Scholar]
  31. de Haan L, Ferreira A. 2006. Extreme Value Theory: An Introduction New York: Springer [Google Scholar]
  32. Dieker AB, Mikosch T. 2014. Exact simulation of Brown–Resnick random fields. arXiv:1406.5624v1 [math.PR]
  33. Dombry C, Ribatet M. 2014. Functional regular variations, Pareto processes and peaks over threshold. Stat. Interface. In press [Google Scholar]
  34. Dupuis DJ. 1998. Exceedances over high thresholds: a guide to threshold selection. Extremes 1:251–61 [Google Scholar]
  35. Dupuis DJ. 2005. Ozone concentrations: a robust analysis of multivariate extremes. Technometrics 47:191–201 [Google Scholar]
  36. Dupuis DJ, Field CA. 1998. Robust estimation of extremes. Can. J. Stat. 26:199–215 [Google Scholar]
  37. Dupuis DJ, Morgenthaler S. 2002. Robust weighted likelihood estimators with an application to bivariate extreme value problems. Can. J. Stat. 30:17–36 [Google Scholar]
  38. Eastoe EF, Tawn JA. 2009. Modelling non-stationary extremes with application to surface level ozone. Appl. Stat. 58:25–45 [Google Scholar]
  39. Eastoe EF, Tawn JA. 2012. Modelling the distribution for the cluster maxima of exceedances of subasymptotic thresholds. Biometrika 99:43–55 [Google Scholar]
  40. Einmahl JHJ, Li J, Liu RY. 2009. Thresholding events of extreme in simultaneous monitoring of multiple risks. J. Am. Stat. Assoc. 104:487982–92 [Google Scholar]
  41. Einmahl JHJ, Segers J. 2009. Maximum empirical likelihood estimation of the spectral measure of an extreme-value distribution. Ann. Stat. 37:2953–89 [Google Scholar]
  42. Embrechts P, Klüppelberg C, Mikosch T. 1997. Modelling Extremal Events for Insurance and Finance Berlin: Springer [Google Scholar]
  43. Engelke S, Malinowski A, Kabluchko Z, Schlather M. 2015. Estimation of Hüsler–Reiss distributions and Brown–Resnick processes. J. R. Stat. Soc. B 77:239–65 [Google Scholar]
  44. Fawcett L, Walshaw D. 2007. Improved estimation for temporally clustered extremes. Environmetrics 18:173–88 [Google Scholar]
  45. Fawcett L, Walshaw D. 2012. Estimating return levels from serially dependent extremes. Environmetrics 23:272–83 [Google Scholar]
  46. Ferreira A, de Haan L, Peng L. 2003. On optimising the estimation of high quantiles of a probability distribution. Statistics 37:401–34 [Google Scholar]
  47. Ferreira A, de Haan L, Zhou C. 2012. Exceedance probability of the integral of a stochastic process. J. Multivar. Anal. 105:241–57 [Google Scholar]
  48. Ferreira A, de Haan L. 2014. The generalized Pareto process; with a view towards application and simulation. Bernoulli 20:1717–37 [Google Scholar]
  49. Fisher RA, Tippett LHC. 1928. Limiting forms of the frequency distributions of the largest or smallest member of a sample. Math. Proc. Camb. Philos. Soc. 24:180–90 [Google Scholar]
  50. Fréchet M. 1927. Sur la loi de probabilité de l'écart maximum. Ann. Soc. Pol. Math. 6:93–116 [Google Scholar]
  51. Galambos J. 1978. The Asymptotic Theory of Extreme Order Statistics New York: Wiley [Google Scholar]
  52. Genton MG, Ma Y, Sang H. 2011. On the likelihood function of Gaussian max-stable processes. Biometrika 98:481–88 [Google Scholar]
  53. Gnedenko BV. 1943. Sur la distribution limite du terme maximum d'une série aléatoire. Ann. Math. 44:423–53 [Google Scholar]
  54. Gomes MI, de Haan L. 1999. Approximation by penultimate extreme value distributions. Extremes 2:71–85 [Google Scholar]
  55. Gomes MI, Pestana D. 2007. A sturdy reduced-bias extreme quantile (VaR) estimator. J. Am. Stat. Assoc. 102:280–92 [Google Scholar]
  56. Guillotte S, Perron F, Segers J. 2011. Nonparametric Bayesian inference on bivariate extremes. J. R. Stat. Soc. B 73:377–406 [Google Scholar]
  57. Guillou A, Hall PG. 2001. A diagnostic for selecting the threshold in extreme-value analysis. J. R. Stat. Soc. B 63:293–305 [Google Scholar]
  58. Gumbel EJ. 1958. Statistics of Extremes New York: Columbia Univ. Press [Google Scholar]
  59. Hall PG, Welsh AH. 1985. Adaptive estimates of parameters of regular variation. Ann. Stat. 13:331–41 [Google Scholar]
  60. Heffernan JE, Resnick SI. 2005. Hidden regular variation and the rank transform. Adv. Appl. Probab. 37:393–412 [Google Scholar]
  61. Heffernan JE, Resnick SI. 2007. Limit laws for random vectors with an extreme component. Ann. Appl. Probab. 17:537–71 [Google Scholar]
  62. Heffernan JE, Tawn JA. 2001. Extreme value analysis of a large designed experiment: a case study in bulk carrier safety. Extremes 4:359–78 [Google Scholar]
  63. Heffernan JE, Tawn JA. 2003. An extreme value analysis for the investigation into the sinking of the M. V. Derbyshire. Appl. Stat. 52:337–54 [Google Scholar]
  64. Heffernan JE, Tawn JA. 2004. A conditional approach for multivariate extreme values. J. R. Stat. Soc. B 66:497–530; discussion530–46 [Google Scholar]
  65. Hill BM. 1975. A simple general approach to inference about the tail of a distribution. Ann. Stat. 3:1163–74 [Google Scholar]
  66. Hosking JRM, Wallis JR. 1987. Parameter and quantile estimation for the generalized Pareto distribution. Technometrics 29:339–49 [Google Scholar]
  67. Hosking JRM, Wallis JR, Wood EF. 1985. Estimation of the generalized extreme-value distribution by the method of probability-weighted moments. Technometrics 27:251–61 [Google Scholar]
  68. Hsing T. 1987. On the characterization of certain point processes. Stoch. Process. Appl. 26:297–316 [Google Scholar]
  69. Hsing T, Hüsler J, Leadbetter MR. 1988. On the exceedance point process for a stationary sequence. Probab. Theory Rel. Fields 78:97–112 [Google Scholar]
  70. Huser R. 2013. Statistical Modeling and Inference for Spatio-Temporal Extremes PhD Thesis, Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland [Google Scholar]
  71. Huser R, Davison AC. 2013. Composite likelihood estimation for the Brown–Resnick process. Biometrika 100:511–18 [Google Scholar]
  72. Huser R, Davison AC. 2014. Space-time modelling of extreme events. J. R. Stat. Soc. B 76:439–61 [Google Scholar]
  73. Hüsler J, Reiss R. 1989. Maxima of normal random vectors: between independence and complete dependence. Stat. Probab. Lett. 7:283–86 [Google Scholar]
  74. Kabluchko Z, Schlather M, de Haan L. 2009. Stationary max-stable fields associated to negative definite functions. Ann. Probab. 37:2042–65 [Google Scholar]
  75. Kaufmann E. 2000. Penultimate approximations in extreme value theory. Extremes 3:39–55 [Google Scholar]
  76. Keef C, Papastatopoulos I, Tawn JA. 2013. Estimation of the conditional distribution of a multivariate variable given that one of its components is large: additional constraints for the Heffernan and Tawn model. J. Multivar. Anal. 115:396–404 [Google Scholar]
  77. Keef C, Tawn J, Svensson C. 2009. Spatial risk assessment for extreme river flows. J. R. Stat. Soc. C 58:601–18 [Google Scholar]
  78. Kotz S, Nadarajah S. 2000. Extreme Value Distributions: Theory and Applications London: Imperial College Press [Google Scholar]
  79. Leadbetter MR. 1991. On a basis for `Peaks over Threshold' modelling. Stat. Probab. Lett. 12:357–62 [Google Scholar]
  80. Leadbetter MR, Lindgren G, Rootzén H. 1983. Extremes and Related Properties of Random Sequences and Processes New York: Springer [Google Scholar]
  81. Ledford AW, Tawn JA. 1996. Statistics for near independence in multivariate extreme values. Biometrika 83:169–87 [Google Scholar]
  82. Ledford AW, Tawn JA. 1997. Modelling dependence within joint tail regions. J. R. Stat. Soc. B 59:475–99 [Google Scholar]
  83. Ledford AW, Tawn JA. 2003. Diagnostics for dependence within time series extremes. J. R. Stat. Soc. B 65:521–43 [Google Scholar]
  84. Liu Y, Tawn JA. 2014. Self-consistent estimation of conditional multivariate extreme distributions. J. Multivar. Anal 127:19–35 [Google Scholar]
  85. Martins E, Stedinger J. 2000. Generalized maximum-likelihood generalized extreme-value quantile estimators for hydrologic data. Water Resour. Res. 36:737–44 [Google Scholar]
  86. Nadarajah S, Mitov K. 2002. Asymptotics of maxima of discrete random variables. Extremes 5:287–94 [Google Scholar]
  87. Nikoloulopoulos AK, Joe H, Li H. 2009. Extreme value properties of multivariate t copulas.. Extremes 12:129–48 [Google Scholar]
  88. Northrop PJ, Jonathan P. 2011. Threshold modelling of spatially dependent non-stationary extremes with application to hurricane-induced wave heights. Environmetrics 22:799–809; discussion810–16 [Google Scholar]
  89. Oesting M, Kabluchlo Z, Schlather M. 2011. Simulation of Brown–Resnick processes. Extremes 15:89–107 [Google Scholar]
  90. Oesting M, Schlather M, Zhou C. 2013. On the normalized spectral representation of max-stable processes on a compact set. arXiv:1310.1813 [math.PR]
  91. Opitz T. 2013. Extremal t processes: elliptical domain of attraction and a spectral representation. J. Multivar. Anal. 122:409–13 [Google Scholar]
  92. Padoan SA, Ribatet M, Sisson SA. 2010. Likelihood-based inference for max-stable processes. J. Am. Stat. Assoc. 105:263–77 [Google Scholar]
  93. Pickands J. 1981. Multivariate extreme value distributions. Bull. Int. Stat. Inst 49:859–78 [Google Scholar]
  94. R Core Team. 2014. R: A Language and Environment for Statistical Computing Vienna: R Found. Stat. Comput http://www.R-project.org [Google Scholar]
  95. Ramos A, Ledford AW. 2009. A new class of models for bivariate joint extremes. J. R. Stat. Soc. B 71:219–41 [Google Scholar]
  96. Ramos A, Ledford AW. 2011. An alternative point process framework for modeling multivariate extreme values. Commun. Stat. Theory Methods 40:2205–24 [Google Scholar]
  97. Reich BJ, Shaby BA. 2012. A hierarchical max-stable spatial model for extreme precipitation. Ann. Appl. Stat. 6:1430–51 [Google Scholar]
  98. Reich BJ, Shaby BA, Cooley D. 2013. A hierarchical model for serially dependent extremes: a study of heat waves in the western US. J. Agric. Biol. Environ. Stat. 19:119–35 [Google Scholar]
  99. Resnick SI. 1987. Extreme Values, Regular Variation, and Point Processes New York: Springer [Google Scholar]
  100. Resnick SI. 2002. Hidden regular variation, second order regular variation and asymptotic independence. Extremes 5:303–36 [Google Scholar]
  101. Resnick SI. 2006. Heavy-Tail Phenomena: Probabilistic and Statistical Modeling New York: Springer [Google Scholar]
  102. Rootzén H, Tajvidi N. 2006. Multivariate generalized Pareto distributions. Bernoulli 12:917–30 [Google Scholar]
  103. Sabourin A, Naveau P. 2014. Bayesian Dirichlet mixture model for multivariate extremes: a re-parametrization. Comput. Stat. Data Anal. 71:542–67 [Google Scholar]
  104. Sang H, Gelfand AE. 2009. Hierarchical modeling for extreme values observed over space and time. Environ. Ecol. Stat. 16:407–26 [Google Scholar]
  105. Sang H, Gelfand AE. 2010. Continuous spatial process models for spatial extreme values. J. Agric. Biol. Environ. Stat. 15:49–65 [Google Scholar]
  106. Schlather M. 2002. Models for stationary max-stable random fields. Extremes 5:33–44 [Google Scholar]
  107. Shaby BA, Reich BJ. 2012. Bayesian spatial extreme value analysis to assess the changing risk of concurrent high temperatures across large portions of European cropland. Environmetrics 23:638–48 [Google Scholar]
  108. Shang H, Yan J, Zhang X. 2011. El Nino–Southern Oscillation influence on winter maximum daily precipitation in California in a spatial model. Water Resour. Res. 47:W11507 [Google Scholar]
  109. Smith RL. 1985. Maximum likelihood estimation in a class of non-regular cases. Biometrika 72:67–92 [Google Scholar]
  110. Smith RL. 1990. Max-stable processes and spatial extremes. Work. Pap., Dept. Math., Univ. Surrey, Guildford, UK. http://www.stat.unc.edu/postscript/rs/spatex.pdf
  111. Smith RL, Tawn JA, Coles SG. 1997. Markov chain models for threshold exceedances. Biometrika 84:249–68 [Google Scholar]
  112. Stephenson AG, Tawn JA. 2005. Exploiting occurrence times in likelihood inference for componentwise maxima. Biometrika 92:213–27 [Google Scholar]
  113. Süveges M, Davison AC. 2010. Model misspecification in peaks over threshold analysis. Ann. Appl. Stat. 4:203–21 [Google Scholar]
  114. Taleb NT. 2007. The Black Swan New York: Random House [Google Scholar]
  115. Thibaud E, Mutzner R, Davison AC. 2013. Threshold modeling of extreme spatial rainfall. Water Resour. Res. 49:4633–44 [Google Scholar]
  116. Thibaud E, Opitz T. 2013. Efficient inference and simulation for elliptical Pareto processes. arXiv:1401.0168 [stat.ME]
  117. von Mises R. 1936. La distribution de la plus grande de n valeurs. Rev. Math. l'Union Interbalk. (Athens) 1:141–60 [Google Scholar]
  118. Varin C. 2008. On composite marginal likelihoods. Adv. Stat. Anal. 92:11–28 [Google Scholar]
  119. Wadsworth JL, Tawn JA. 2012a. Likelihood-based procedures for threshold diagnostics and uncertainty in extreme value modelling. J. R. Stat. Soc. B 74:543–67 [Google Scholar]
  120. Wadsworth JL, Tawn JA. 2012b. Dependence modelling for spatial extremes. Biometrika 99:253–72 [Google Scholar]
  121. Wadsworth JL, Tawn JA. 2013. A new representation for multivariate tail probabilities. Bernoulli 19:2689–714 [Google Scholar]
  122. Wadsworth JL, Tawn JA. 2014. Efficient inference for spatial extreme value processes associated to log-Gaussian random functions. Biometrika 101:1–15 [Google Scholar]
  123. Weissman I. 1978. Estimation of parameters and large quantiles based on the k largest observations. J. Am. Stat. Assoc. 73:812–15 [Google Scholar]
  124. Weller GB, Cooley D. 2014. A sum characterization of hidden regular variation with likelihood inference via expectation-maximization. Biometrika 101:17–36 [Google Scholar]
  125. Westra S, Sisson SA. 2011. Detection of non-stationarity in precipitation extremes using a max-stable process model. J. Hydrol. 406:119–28 [Google Scholar]
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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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