Journal of Statistical Physics

, Volume 145, Issue 5, pp 1156–1180 | Cite as

Numerical Convergence of the Block-Maxima Approach to the Generalized Extreme Value Distribution

  • Davide Faranda
  • Valerio Lucarini
  • Giorgio Turchetti
  • Sandro Vaienti


In this paper we perform an analytical and numerical study of Extreme Value distributions in discrete dynamical systems. In this setting, recent works have shown how to get a statistics of extremes in agreement with the classical Extreme Value Theory. We pursue these investigations by giving analytical expressions of Extreme Value distribution parameters for maps that have an absolutely continuous invariant measure. We compare these analytical results with numerical experiments in which we study the convergence to limiting distributions using the so called block-maxima approach, pointing out in which cases we obtain robust estimation of parameters. In regular maps for which mixing properties do not hold, we show that the fitting procedure to the classical Extreme Value Distribution fails, as expected. However, we obtain an empirical distribution that can be explained starting from a different observable function for which Nicolis et al. (Phys. Rev. Lett. 97(21): 210602, 2006) have found analytical results.


Extreme values Dynamical systems EVT GEV Mixing Logistic map Chaos 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Altmann, E.G., Hallerberg, S., Kantz, H.: Reactions to extreme events: moving threshold model. Phys. A, Stat. Mech. Appl. 364, 435–444 (2006) CrossRefGoogle Scholar
  2. 2.
    Arnold, V.I., Avez, A.: Ergodic Problems of Classical Mechanics. Benjamin, New York (1968) Google Scholar
  3. 3.
    Balakrishnan, V., Nicolis, C., Nicolis, G.: Extreme value distributions in chaotic dynamics. J. Stat. Phys. 80(1), 307–336 (1995). ISSN 0022-4715 MATHADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Beirlant, J.: Statistics of Extremes: Theory and Applications. Wiley, New York (2004). ISBN 0471976474 MATHCrossRefGoogle Scholar
  5. 5.
    Bertin, E.: Global fluctuations and Gumbel statistics. Phys. Rev. Lett. 95(17), 170601 (2005). ISSN 1079-7114 ADSCrossRefGoogle Scholar
  6. 6.
    Brodin, E., Kluppelberg, C.: Extreme value theory in finance. In: Encyclopedia of Quantitative Risk Analysis and Assesment (2008). doi: 10.1002/9780470061596.risk0431. ISBN:0-470-03549-8, 978-0-470-03549-8 Google Scholar
  7. 7.
    Buric, N., Rampioni, A., Turchetti, G.: Statistics of Poincaré recurrences for a class of smooth circle maps. Chaos Solitons Fractals 23(5), 1829–1840 (2005) MATHADSMathSciNetGoogle Scholar
  8. 8.
    Burton, P.W.: Seismic risk in southern Europe through to India examined using Gumbel’s third distribution of extreme values. Geophys. J. R. Astron. Soc. 59(2), 249–280 (1979). ISSN 1365-246X CrossRefGoogle Scholar
  9. 9.
    Clusel, M., Bertin, E.: Global fluctuations in physical systems: a subtle interplay between sum and extreme value statistics. Int. J. Mod. Phys. B 22(20), 3311–3368 (2008). ISSN 0217-9792 MATHADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    Coelho, Z., De Faria, E.: Limit laws of entrance times for homeomorphisms of the circle. Isr. J. Math. 93(1), 93–112 (1996). ISSN 0021-2172 MATHCrossRefGoogle Scholar
  11. 11.
    Coles, S., Heffernan, J., Tawn, J.: Dependence measures for extreme value analyses. Extremes 2(4), 339–365 (1999). ISSN 1386-1999 MATHCrossRefGoogle Scholar
  12. 12.
    Collet, P.: Statistics of closest return for some non-uniformly hyperbolic systems. Ergod. Theory Dyn. Syst. 21(02), 401–420 (2001) MATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Cornell, C.A.: Engineering seismic risk analysis. Bull. Seismol. Soc. Am. 58(5), 1583 (1968). ISSN 0037-1106 Google Scholar
  14. 14.
    Cruz, M.G.: Modeling, Measuring and Hedging Operational Risk. Wiley, New York (2002). ISBN 0471515604 Google Scholar
  15. 15.
    Dahlstedt, K., Jensen, H.J.: Universal fluctuations and extreme-value statistics. J. Phys. A, Math. Gen. 34, 11193 (2001) MATHADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    Davison, A.C.: Modelling excesses over high thresholds, with an application. In: Statistical Extremes and Applications, pp. 461–482. Reidel, Dordrecht (1984) Google Scholar
  17. 17.
    Davison, A.C., Smith, R.L.: Models for exceedances over high thresholds. J. R. Stat. Soc., Ser. B, Methodol. 52(3), 393–442 (1990). ISSN 0035-9246 MATHMathSciNetGoogle Scholar
  18. 18.
    Embrechts, P., Resnick, S.I., Samorodnitsky, G.: Extreme value theory as a risk management tool. N. Am. Actuar. J. 3, 30–41 (1999). ISSN 1092-0277 MATHMathSciNetGoogle Scholar
  19. 19.
    Felici, M., Lucarini, V., Speranza, A., Vitolo, R.: Extreme value statistics of the total energy in an intermediate complexity model of the mid-latitude atmospheric jet. Part I: Stationary case. J. Atmos. Sci. 64, 2137–2158 (2007) ADSCrossRefGoogle Scholar
  20. 20.
    Felici, M., Lucarini, V., Speranza, A., Vitolo, R.: Extreme value statistics of the total energy in an intermediate complexity model of the mid-latitude atmospheric jet. Part II: Trend detection and assessment. J. Atmos. Sci. 64, 2159–2175 (2007) ADSCrossRefGoogle Scholar
  21. 21.
    Fisher, R.A., Tippett, L.H.C.: Limiting forms of the frequency distribution of the largest or smallest member of a sample. Proc. Camb. Philos. Soc. 24, 180 (1928) MATHADSCrossRefGoogle Scholar
  22. 22.
    Freitas, A.C.M., Freitas, J.M.: On the link between dependence and independence in extreme value theory for dynamical systems. Stat. Probab. Lett. 78(9), 1088–1093 (2008). ISSN 0167-7152 MATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Freitas, A.C.M., Freitas, J.M.: Extreme values for Benedicks-Carleson quadratic maps. Ergod. Theory Dyn. Syst. 28(04), 1117–1133 (2008). ISSN 0143-3857 MATHMathSciNetGoogle Scholar
  24. 24.
    Freitas, A.C.M., Freitas, J.M., Todd, M.: Hitting time statistics and extreme value theory. Probab. Theory Relat. Fields 147(3–4), 675–710 (2010) MATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Freitas, A.C.M., Freitas, J.M., Todd, M.: Extremal index, hitting time statistics and periodicity. Arxiv preprint, arXiv:1008.1350 (2010)
  26. 26.
    Freitas, A.C.M., Freitas, J.M., Todd, M., Gardas, B., Drichel, D., Flohr, M., Thompson, R.T., Cummer, S.A., Frauendiener, J., Doliwa, A., et al.: Extreme value laws in dynamical systems for non-smooth observations. Arxiv preprint, arXiv:1006.3276 (2010)
  27. 27.
    Ghil, M., et al.: Extreme events: dynamics, statistics and prediction. Nonlinear Process. Geophys. 18, 295–350 (2011) ADSCrossRefGoogle Scholar
  28. 28.
    Gilli, M., Këllezi, E.: An application of extreme value theory for measuring financial risk. Comput. Econ. 27(2), 207–228 (2006). ISSN 0927-7099 MATHCrossRefGoogle Scholar
  29. 29.
    Gnedenko, B.: Sur la distribution limite du terme maximum d’une série aléatoire. Ann. Math. 44(3), 423–453 (1943) MATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    Gumbel, E.J.: The return period of flood flows. Ann. Math. Stat. 12(2), 163–190 (1941). ISSN 0003-4851 MATHMathSciNetCrossRefGoogle Scholar
  31. 31.
    Gupta, C.: Extreme-value distributions for some classes of non-uniformly partially hyperbolic dynamical systems. Ergod. Theory Dyn. Syst. 30(03), 757–771 (2010) MATHCrossRefGoogle Scholar
  32. 32.
    Gupta, C., Holland, M., Nicol, M.: Extreme value theory for dispersing billiards and a class of hyperbolic maps with singularities. Preprint (2009) Google Scholar
  33. 33.
    Haiman, G.: Extreme values of the tent map process. Stat. Probab. Lett. 65(4), 451–456 (2003). ISSN 0167-7152 MATHMathSciNetCrossRefGoogle Scholar
  34. 34.
    Hallerberg, S., Kantz, H.: Influence of the event magnitude on the predictability of an extreme event. Phys. Rev. E 77(1), 11108 (2008). ISSN 1550-2376 ADSMathSciNetCrossRefGoogle Scholar
  35. 35.
    Hasselblatt, B., Katok, A.B.: A First Course in Dynamics: With a Panorama of Recent Developments. Cambridge University Press, Cambridge (2003) MATHGoogle Scholar
  36. 36.
    Hill, B.M.: A simple general approach to inference about the tail of a distribution. Ann. Stat. 3(5), 1163–1174 (1975). ISSN 0090-5364 MATHCrossRefGoogle Scholar
  37. 37.
    Holland, M., Nicol, M., Török, A.: Extreme value distributions for non-uniformly hyperbolic dynamical systems. Preprint (2008) Google Scholar
  38. 38.
    Hu, H., Rampioni, A., Rossi, L., Turchetti, G., Vaienti, S.: Statistics of Poincaré recurrences for maps with integrable and ergodic components. Chaos, Interdiscip. J. Nonlinear Sci. 14, 160 (2004) MATHMathSciNetCrossRefGoogle Scholar
  39. 39.
    Kantz, H., Altmann, E., Hallerberg, S., Holstein, D., Riegert, A.: Dynamical interpretation of extreme events: predictability and predictions. In: Extreme Events in Nature and Society, pp. 69–93. Springer, Berlin (2006) CrossRefGoogle Scholar
  40. 40.
    Katz, R.W.: Extreme value theory for precipitation: sensitivity analysis for climate change. Adv. Water Resour. 23(2), 133–139 (1999). ISSN 0309-1708 CrossRefGoogle Scholar
  41. 41.
    Katz, R.W., Brown, B.G.: Extreme events in a changing climate: variability is more important than averages. Clim. Change 21(3), 289–302 (1992). ISSN 0165-0009 CrossRefGoogle Scholar
  42. 42.
    Katz, R.W., Brush, G.S., Parlange, M.B.: Statistics of extremes: modeling ecological disturbances. Ecology 86(5), 1124–1134 (2005). ISSN 0012-9658 CrossRefGoogle Scholar
  43. 43.
    Leadbetter, M.R., Lindgren, G., Rootzen, H.: Extremes and Related Properties of Random Sequences and Processes. Springer, New York (1983) MATHCrossRefGoogle Scholar
  44. 44.
    Lilliefors, H.W.: On the Kolmogorov-Smirnov test for normality with mean and variance unknown. J. Am. Stat. Assoc. 62(318), 399–402 (1967). ISSN 0162-1459 CrossRefGoogle Scholar
  45. 45.
    Longin, F.M.: From value at risk to stress testing: the extreme value approach. J. Bank. Finance 24(7), 1097–1130 (2000). ISSN 0378-4266 CrossRefGoogle Scholar
  46. 46.
    Martinez, W.L., Martinez, A.R.: Computational Statistics Handbook with MATLAB. CRC Press, Boca Raton (2002) Google Scholar
  47. 47.
    Martins, E.S., Stedinger, J.R.: Generalized maximum-likelihood generalized extreme-value quantile estimators for hydrologic data. Water Resour. Res. 36(3), 737–744 (2000). ISSN 0043-1397 ADSCrossRefGoogle Scholar
  48. 48.
    Nicholls, N.: CLIVAR and IPCC interests in extreme events. In: Workshop Proceedings on Indices and Indicators for Climate Extremes, Asheville, NC. Sponsors, CLIVAR, GCOS and WMO (1997) Google Scholar
  49. 49.
    Nicolis, C., Balakrishnan, V., Nicolis, G.: Extreme events in deterministic dynamical systems. Phys. Rev. Lett. 97(21), 210602 (2006). ISSN 1079-7114 ADSCrossRefGoogle Scholar
  50. 50.
    Friederichs, P., Hense, A.: Statistical downscaling of extreme precipitation events using censored quantile regression. Mon. Weather Rev. 135(6), 2365–2378 (2007). ISSN 0027-0644 ADSCrossRefGoogle Scholar
  51. 51.
    Pickands III, J.: Moment convergence of sample extremes. Ann. Math. Stat. 39(3), 881–889 (1968) MATHMathSciNetCrossRefGoogle Scholar
  52. 52.
    Pickands III, J.: Statistical inference using extreme order statistics. Ann. Stat. 3(1), 119–131 (1975). ISSN 0090-5364 MATHMathSciNetCrossRefGoogle Scholar
  53. 53.
    Smith, R.L.: Threshold methods for sample extremes. Stat. Extremes Appl. 621, 638 (1984) Google Scholar
  54. 54.
    Smith, R.L.: Extreme value analysis of environmental time series: an application to trend detection in ground-level ozone. Stat. Sci. 4(4), 367–377 (1989). ISSN 0883-4237 MATHCrossRefGoogle Scholar
  55. 55.
    Sornette, D., Knopoff, L., Kagan, Y.Y., Vanneste, C.: Rank-ordering statistics of extreme events: application to the distribution of large earthquakes. J. Geophys. Res. 101(B6), 13883 (1996). ISSN 0148-0227 ADSCrossRefGoogle Scholar
  56. 56.
    Sveinsson, O.G.B., Boes, D.C.: Regional frequency analysis of extreme precipitation in Northeastern Colorado and Fort Collins flood of 1997. J. Hydrol. Eng. 7, 49 (2002) CrossRefGoogle Scholar
  57. 57.
    Todorovic, P., Zelenhasic, E.: A stochastic model for flood analysis. Water Resour. Res. 6(6), 1641–1648 (1970). ISSN 0043-1397 ADSCrossRefGoogle Scholar
  58. 58.
    Vannitsem, S.: Statistical properties of the temperature maxima in an intermediate order Quasi-Geostrophic model. Tellus A 59(1), 80–95 (2007). ISSN 1600-0870 ADSCrossRefGoogle Scholar
  59. 59.
    Vitolo, R., Holland, M.P., Ferro, C.A.T.: Robust extremes in chaotic deterministic systems. Chaos, Interdiscip. J. Nonlinear Sci. 19, 043127 (2009) CrossRefGoogle Scholar
  60. 60.
    Vitolo, R., Ruti, P.M., Dell’Aquila, A., Felici, M., Lucarini, V., Speranza, A.: Accessing extremes of mid-latitudinal wave activity: methodology and application. Tellus A 61(1), 35–49 (2009). ISSN 1600-0870 ADSCrossRefGoogle Scholar
  61. 61.
    Young, L.S.: Statistical properties of dynamical systems with some hyperbolicity. Ann. Math. 147(3), 585–650 (1998) MATHCrossRefGoogle Scholar
  62. 62.
    Young, L.S.: Recurrence times and rates of mixing. Isr. J. Math. 110(1), 153–188 (1999) MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Davide Faranda
    • 1
  • Valerio Lucarini
    • 1
    • 2
  • Giorgio Turchetti
    • 3
  • Sandro Vaienti
    • 4
  1. 1.Department of Mathematics and StatisticsUniversity of ReadingReadingUK
  2. 2.Department of Meteorology, Department of MathematicsUniversity of ReadingReadingUK
  3. 3.Department of Physics, INFN-BolognaUniversity of BolognaBolognaItaly
  4. 4.UMR-6207, Centre de Physique ThéoriqueCNRS, Universités d’Aix-Marseille I, II, Université du Sud Toulon-Var and FRUMAM (Fédération de Recherche des Unités de Mathématiques de Marseille)Marseille Cedex 09France

Personalised recommendations