Statistics of Extremes

Annual Review of Statistics and Its Application

Volumn 2, Issue , 2015, Pages 203-235

  Davison, A C ^a   Huser, R ^b  

^a EPFL   (Switzerland)

^b KING ABDULLAH UNIVERSITY OF SCIENCE AND TECHNOLOGY   (Saudi Arabia)

Author keywords

Asymptotic dependence; Asymptotic independence; Extrapolation; Generalized extreme value distribution; Generalized Pareto distribution; Max stability; Pareto process; Peaks over thresholds; Poisson process

Indexed keywords

EID: 84928104089     PISSN: 23268298     EISSN: 2326831X     Source Type: Journal    
DOI: 10.1146/annurev-statistics-010814-020133     Document Type: Article

References (125)

1. Anderson CW. 1971. Contributions to the Asymptotic Theory of Extreme Values. PhD Thesis, University of London, London, UK
2. Anderson CW, Coles SG,Hüsler J. 1997. Maxima of Poisson-like variables and related triangular arrays. Ann. Appl. Probab. 7:953-71
3. Ballani F, SchlatherM. 2011. A construction principle for multivariate extreme value distributions. Biometrika 98:633-45
4. Beirlant J, Dierckx G, Goegebeur Y, Matthys G. 1999. Tail index estimation and an exponential regression model. Extremes 2:177-200
5. Beirlant J, Goegebeur Y, Teugels J, Segers J. 2004. Statistics of Extremes: Theory and Applications. New York: Wiley
6. Blanchet J, Davison AC. 2011. Spatial modelling of extreme snow depth. Ann. Appl. Stat. 5:1699-725
7. Boldi M-O, Davison AC. 2007. A mixture model for multivariate extremes. J. R. Stat. Soc. B 69:217-29
8. Bortot P, Coles SG. 2003. Extremes of Markov chains with tail-switching potential. J. R. Stat. Soc. B 65:851-67
9. Bortot P, Gaetan C. 2014. A latent process model for temporal extremes. Scand. J. Stat. 41:606-21
10. Bortot P, Tawn JA. 1998. Models for the extremes of Markov chains. Biometrika 85:851-67
11. Brown BM, Resnick SI. 1977. Extreme values of independent stochastic processes. J. Appl. Probab. 14:732-39
12. Buishand TA, de Haan L, Zhou C. 2008. On spatial extremes: with application to a rainfall problem. Ann. Appl. Stat. 2:624-42
13. Butler A, Heffernan JE, Tawn JA, Flather RA. 2007. Trend estimation in extremes of synthetic North Sea surges. Appl. Stat. 56:395-414
14. Capéraà P, Fougères AL, Genest C. 1997. A nonparametric estimation procedure for bivariate extreme value copulas. Biometrika 84:567-77
15. Chandler RE, Bate S. 2007. Inference for clustered data using the independence log-likelihood. Biometrika 94:167-83
16. Choulakian V, Stephens MA. 2001. Goodness-of-fit tests for the generalized Pareto distribution. Technometrics 43:478-84
17. Coles SG. 2001. An Introduction to Statistical Modeling of Extreme Values. New York: Springer
18. Coles SG, Heffernan J, Tawn JA. 1999. Dependence measures for extreme value analyses. Extremes 2:339-65
19. Coles SG, Tawn JA. 1991. Modelling extreme multivariate events. J. R. Stat. Soc. B 53:377-92
20. Coles SG, Tawn JA. 1996. Modelling extremes of the areal rainfall process. J. R. Stat. Soc. B 58:329-47
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
22. Cooley D, Nychka D, Naveau P. 2007. Bayesian spatial modeling of extreme precipitation return levels. J. Am. Stat. Assoc. 102:824-40
23. Danielsson J, deHaan 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
24. Das B, Resnick SI. 2011. Conditioning on an extreme component: model consistency with regular variation on cones. Bernoulli 17(1):226-52
25. Davison AC, Gholamrezaee MM. 2012. Geostatistics of extremes. Proc. R. Soc. Lond. A 468:581-608
26. Davison AC, Huser R,Thibaud E. 2013. Geostatistics of dependent and asymptotically independent extremes. Math. Geosci. 45:511-29
27. Davison AC, Padoan SA, Ribatet M. 2012. Statistical modelling of spatial extremes. Stat. Sci. 27:161-86; discussion pp. 187-201
28. Davison AC, Smith RL. 1990. Models for exceedances over high thresholds. J. R. Stat. Soc. B 52:393-425; discussion pp. 425-42
29. de Carvalho MB, Davison AC. 2014. Spectral density ratio models for multivariate extremes. J. Am. Stat. Assoc. 109:764-76
30. de Haan L. 1984. A spectral representation for max-stable processes. Ann. Probab. 12:1194-204
31. de Haan L, Ferreira A. 2006. Extreme Value Theory: An Introduction. New York: Springer
32. Dieker AB,Mikosch T. 2014. Exact simulation ofBrown-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
34. Dupuis DJ. 1998. Exceedances over high thresholds: a guide to threshold selection. Extremes 1:251-61
35. Dupuis DJ. 2005. Ozone concentrations: a robust analysis of multivariate extremes. Technometrics 47:191-201
36. Dupuis DJ, Field CA. 1998. Robust estimation of extremes. Can. J. Stat. 26:199-215
37. Dupuis DJ, Morgenthaler S. 2002. Robust weighted likelihood estimators with an application to bivariate extreme value problems. Can. J. Stat. 30:17-36
38. Eastoe EF, Tawn JA. 2009. Modelling non-stationary extremes with application to surface level ozone. Appl. Stat. 58:25-45
39. Eastoe EF, Tawn JA. 2012. Modelling the distribution for the cluster maxima of exceedances of subasymptotic thresholds. Biometrika 99:43-55
40. Einmahl JHJ, Li J, Liu RY. 2009. Thresholding events of extreme in simultaneous monitoring of multiple risks. J. Am. Stat. Assoc. 104(487):982-92
41. Einmahl JHJ, Segers J. 2009. Maximum empirical likelihood estimation of the spectralmeasure of an extremevalue distribution. Ann. Stat. 37:2953-89
42. Embrechts P, Klüppelberg C, Mikosch T. 1997. Modelling Extremal Events for Insurance and Finance. Berlin: Springer
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
44. Fawcett L,Walshaw D. 2007. Improved estimation for temporally clustered extremes. Environmetrics 18:173-88
45. Fawcett L, Walshaw D. 2012. Estimating return levels from serially dependent extremes. Environmetrics 23:272-83
46. Ferreira A, de Haan L, Peng L. 2003. On optimising the estimation of high quantiles of a probability distribution. Statistics 37:401-34
47. FerreiraA, deHaan L, Zhou C. 2012. Exceedance probability of the integral of a stochastic process. J. Multivar. Anal. 105:241-57
48. Ferreira A, de Haan L. 2014. The generalized Pareto process; with a view towards application and simulation. Bernoulli 20:1717-37
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
50. Fréchet M. 1927. Sur la loi de probabilité de l'écart maximum. Ann. Soc. Pol. Math. 6:93-116
51. Galambos J. 1978. The Asymptotic Theory of Extreme Order Statistics. New York: Wiley
52. Genton MG, Ma Y, Sang H. 2011. On the likelihood function of Gaussian max-stable processes. Biometrika 98:481-88
53. Gnedenko BV. 1943. Sur la distribution limite du terme maximum d'une série aléatoire. Ann. Math. 44:423-53
54. Gomes MI, de Haan L. 1999. Approximation by penultimate extreme value distributions. Extremes 2:71-85
55. Gomes MI, Pestana D. 2007. A sturdy reduced-bias extreme quantile (VaR) estimator. J. Am. Stat. Assoc. 102:280-92
56. Guillotte S, Perron F, Segers J. 2011. Nonparametric Bayesian inference on bivariate extremes. J. R. Stat. Soc. B 73:377-406
57. Guillou A, Hall PG. 2001. A diagnostic for selecting the threshold in extreme-value analysis. J. R. Stat. Soc. B 63:293-305
58. Gumbel EJ. 1958. Statistics of Extremes. New York: Columbia Univ. Press
59. Hall PG, Welsh AH. 1985. Adaptive estimates of parameters of regular variation. Ann. Stat. 13:331-41
60. Heffernan JE, Resnick SI. 2005. Hidden regular variation and the rank transform. Adv. Appl. Probab. 37:393-412
61. Heffernan JE, Resnick SI. 2007. Limit laws for random vectors with an extreme component. Ann. Appl. Probab. 17:537-71
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
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
64. Heffernan JE, Tawn JA. 2004. A conditional approach for multivariate extreme values. J. R. Stat. Soc. B 66:497-530; discussion pp. 530-46
65. Hill BM. 1975. A simple general approach to inference about the tail of a distribution. Ann. Stat. 3:1163-74
66. Hosking JRM, Wallis JR. 1987. Parameter and quantile estimation for the generalized Pareto distribution. Technometrics 29:339-49
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
68. Hsing T. 1987. On the characterization of certain point processes. Stoch. Process. Appl. 26:297-316
69. Hsing T, Hüsler J, Leadbetter MR. 1988. On the exceedance point process for a stationary sequence. Probab. Theory Rel. Fields 78:97-112
70. HuserR. 2013. StatisticalModeling and Inference for Spatio-Temporal Extremes. PhDThesis,Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland
71. Huser R, Davison AC. 2013. Composite likelihood estimation for the Brown-Resnick process. Biometrika 100:511-18
72. Huser R, Davison AC. 2014. Space-time modelling of extreme events. J. R. Stat. Soc. B 76:439-61
73. Hüsler J, Reiss R. 1989. Maxima of normal random vectors: between independence and complete dependence. Stat. Probab. Lett. 7:283-86
74. Kabluchko Z, Schlather M, de Haan L. 2009. Stationary max-stable fields associated to negative definite functions. Ann. Probab. 37:2042-65
75. Kaufmann E. 2000. Penultimate approximations in extreme value theory. Extremes 3:39-55
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
77. Keef C, Tawn J, Svensson C. 2009. Spatial risk assessment for extreme river flows. J. R. Stat. Soc. C 58:601-18
78. Kotz S, Nadarajah S. 2000. Extreme Value Distributions: Theory and Applications. London: Imperial College Press
79. Leadbetter MR. 1991. On a basis for 'Peaks over Threshold' modelling. Stat. Probab. Lett. 12:357-62
80. LeadbetterMR, Lindgren G, Rootzén H. 1983. Extremes and Related Properties of Random Sequences and Processes. New York: Springer
81. Ledford AW, Tawn JA. 1996. Statistics for near independence in multivariate extreme values. Biometrika 83:169-87
82. Ledford AW, Tawn JA. 1997. Modelling dependence within joint tail regions. J. R. Stat. Soc. B 59:475-99
83. Ledford AW, Tawn JA. 2003. Diagnostics for dependence within time series extremes. J. R. Stat. Soc. B 65:521-43
84. Liu Y,Tawn JA. 2014. Self-consistent estimation of conditional multivariate extreme distributions. J. Multivar. Anal. 127:19-35
85. Martins E, Stedinger J. 2000. Generalizedmaximum-likelihood generalized extreme-value quantile estimators for hydrologic data. Water Resour. Res. 36:737-44
86. Nadarajah S, Mitov K. 2002. Asymptotics of maxima of discrete random variables. Extremes 5:287-94
87. Nikoloulopoulos AK, Joe H, Li H. 2009. Extreme value properties ofmultivariate t copulas. Extremes 12:129-48
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; discussion pp. 810-16
89. Oesting M, Kabluchlo Z, Schlather M. 2011. Simulation of Brown-Resnick processes. Extremes 15:89-107
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
92. Padoan SA, Ribatet M, Sisson SA. 2010. Likelihood-based inference for max-stable processes. J. Am. Stat. Assoc. 105:263-77
93. Pickands J. 1981. Multivariate extreme value distributions. Bull. Int. Stat. Inst. 49:859-78
94. R Core Team. 2014. R: A Language and Environment for Statistical Computing. Vienna: R Found. Stat. Comput. http://www.R-project.org
95. Ramos A, Ledford AW. 2009. A new class of models for bivariate joint extremes. J. R. Stat. Soc. B 71:219-41
96. Ramos A, Ledford AW. 2011. An alternative point process framework for modeling multivariate extreme values. Commun. Stat. Theory Methods 40:2205-24
97. Reich BJ, Shaby BA. 2012. A hierarchical max-stable spatial model for extreme precipitation. Ann. Appl. Stat. 6:1430-51
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
99. Resnick SI. 1987. Extreme Values, Regular Variation, and Point Processes. New York: Springer
100. Resnick SI. 2002. Hidden regular variation, second order regular variation and asymptotic independence. Extremes 5:303-36
101. Resnick SI. 2006. Heavy-Tail Phenomena: Probabilistic and Statistical Modeling. New York: Springer
102. Rootzén H, Tajvidi N. 2006. Multivariate generalized Pareto distributions. Bernoulli 12:917-30
103. Sabourin A, Naveau P. 2014. Bayesian Dirichletmixturemodel for multivariate extremes: a re-parametrization. Comput. Stat. Data Anal. 71:542-67
104. Sang H, Gelfand AE. 2009. Hierarchical modeling for extreme values observed over space and time. Environ. Ecol. Stat. 16:407-26
105. Sang H, Gelfand AE. 2010. Continuous spatial processmodels for spatial extreme values. J. Agric. Biol. Environ. Stat. 15:49-65
106. Schlather M. 2002. Models for stationary max-stable random fields. Extremes 5:33-44
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
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
109. Smith RL. 1985. Maximum likelihood estimation in a class of non-regular cases. Biometrika 72:67-92
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
112. Stephenson AG, Tawn JA. 2005. Exploiting occurrence times in likelihood inference for componentwise maxima. Biometrika 92:213-27
113. Süveges M, Davison AC. 2010. Model misspecification in peaks over threshold analysis. Ann. Appl. Stat. 4:203-21
114. Taleb NT. 2007. The Black Swan. New York: Random House
115. Thibaud E, Mutzner R, Davison AC. 2013. Threshold modeling of extreme spatial rainfall. Water Resour. Res. 49:4633-44
116. Thibaud E, Opitz T. 2013. Efficient inference and simulation for elliptical Pareto processes. arXiv:1401.0168 [stat.ME]
117. vonMises R. 1936. La distribution de la plus grande de n valeurs. Rev. Math. l'Union Interbalk. (Athens) 1:141-60
118. Varin C. 2008. On composite marginal likelihoods. Adv. Stat. Anal. 92(1):1-28
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
120. Wadsworth JL, Tawn JA. 2012b. Dependence modelling for spatial extremes. Biometrika 99:253-72
121. Wadsworth JL, Tawn JA. 2013. A new representation formultivariate tail probabilities. Bernoulli 19:2689-714
122. Wadsworth JL, Tawn JA. 2014. Efficient inference for spatial extreme value processes associated to log-Gaussian random functions. Biometrika 101:1-15
123. Weissman I. 1978. Estimation of parameters and large quantiles based on the k largest observations. J. Am. Stat. Assoc. 73:812-15
124. Weller GB, Cooley D. 2014. A sum characterization of hidden regular variation with likelihood inference via expectation-maximization. Biometrika 101:17-36
125. Westra S, Sisson SA. 2011. Detection of non-stationarity in precipitation extremes using a max-stable process model. J. Hydrol. 406:119-28