Abstract
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.
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Altmann, E.G., Hallerberg, S., Kantz, H.: Reactions to extreme events: moving threshold model. Phys. A, Stat. Mech. Appl. 364, 435–444 (2006)
Arnold, V.I., Avez, A.: Ergodic Problems of Classical Mechanics. Benjamin, New York (1968)
Balakrishnan, V., Nicolis, C., Nicolis, G.: Extreme value distributions in chaotic dynamics. J. Stat. Phys. 80(1), 307–336 (1995). ISSN 0022-4715
Beirlant, J.: Statistics of Extremes: Theory and Applications. Wiley, New York (2004). ISBN 0471976474
Bertin, E.: Global fluctuations and Gumbel statistics. Phys. Rev. Lett. 95(17), 170601 (2005). ISSN 1079-7114
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
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)
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
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
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
Coles, S., Heffernan, J., Tawn, J.: Dependence measures for extreme value analyses. Extremes 2(4), 339–365 (1999). ISSN 1386-1999
Collet, P.: Statistics of closest return for some non-uniformly hyperbolic systems. Ergod. Theory Dyn. Syst. 21(02), 401–420 (2001)
Cornell, C.A.: Engineering seismic risk analysis. Bull. Seismol. Soc. Am. 58(5), 1583 (1968). ISSN 0037-1106
Cruz, M.G.: Modeling, Measuring and Hedging Operational Risk. Wiley, New York (2002). ISBN 0471515604
Dahlstedt, K., Jensen, H.J.: Universal fluctuations and extreme-value statistics. J. Phys. A, Math. Gen. 34, 11193 (2001)
Davison, A.C.: Modelling excesses over high thresholds, with an application. In: Statistical Extremes and Applications, pp. 461–482. Reidel, Dordrecht (1984)
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
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
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)
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)
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)
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
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
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)
Freitas, A.C.M., Freitas, J.M., Todd, M.: Extremal index, hitting time statistics and periodicity. Arxiv preprint, arXiv:1008.1350 (2010)
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)
Ghil, M., et al.: Extreme events: dynamics, statistics and prediction. Nonlinear Process. Geophys. 18, 295–350 (2011)
Gilli, M., Këllezi, E.: An application of extreme value theory for measuring financial risk. Comput. Econ. 27(2), 207–228 (2006). ISSN 0927-7099
Gnedenko, B.: Sur la distribution limite du terme maximum d’une série aléatoire. Ann. Math. 44(3), 423–453 (1943)
Gumbel, E.J.: The return period of flood flows. Ann. Math. Stat. 12(2), 163–190 (1941). ISSN 0003-4851
Gupta, C.: Extreme-value distributions for some classes of non-uniformly partially hyperbolic dynamical systems. Ergod. Theory Dyn. Syst. 30(03), 757–771 (2010)
Gupta, C., Holland, M., Nicol, M.: Extreme value theory for dispersing billiards and a class of hyperbolic maps with singularities. Preprint (2009)
Haiman, G.: Extreme values of the tent map process. Stat. Probab. Lett. 65(4), 451–456 (2003). ISSN 0167-7152
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
Hasselblatt, B., Katok, A.B.: A First Course in Dynamics: With a Panorama of Recent Developments. Cambridge University Press, Cambridge (2003)
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
Holland, M., Nicol, M., Török, A.: Extreme value distributions for non-uniformly hyperbolic dynamical systems. Preprint (2008)
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)
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)
Katz, R.W.: Extreme value theory for precipitation: sensitivity analysis for climate change. Adv. Water Resour. 23(2), 133–139 (1999). ISSN 0309-1708
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
Katz, R.W., Brush, G.S., Parlange, M.B.: Statistics of extremes: modeling ecological disturbances. Ecology 86(5), 1124–1134 (2005). ISSN 0012-9658
Leadbetter, M.R., Lindgren, G., Rootzen, H.: Extremes and Related Properties of Random Sequences and Processes. Springer, New York (1983)
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
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
Martinez, W.L., Martinez, A.R.: Computational Statistics Handbook with MATLAB. CRC Press, Boca Raton (2002)
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
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)
Nicolis, C., Balakrishnan, V., Nicolis, G.: Extreme events in deterministic dynamical systems. Phys. Rev. Lett. 97(21), 210602 (2006). ISSN 1079-7114
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
Pickands III, J.: Moment convergence of sample extremes. Ann. Math. Stat. 39(3), 881–889 (1968)
Pickands III, J.: Statistical inference using extreme order statistics. Ann. Stat. 3(1), 119–131 (1975). ISSN 0090-5364
Smith, R.L.: Threshold methods for sample extremes. Stat. Extremes Appl. 621, 638 (1984)
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
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
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)
Todorovic, P., Zelenhasic, E.: A stochastic model for flood analysis. Water Resour. Res. 6(6), 1641–1648 (1970). ISSN 0043-1397
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
Vitolo, R., Holland, M.P., Ferro, C.A.T.: Robust extremes in chaotic deterministic systems. Chaos, Interdiscip. J. Nonlinear Sci. 19, 043127 (2009)
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
Young, L.S.: Statistical properties of dynamical systems with some hyperbolicity. Ann. Math. 147(3), 585–650 (1998)
Young, L.S.: Recurrence times and rates of mixing. Isr. J. Math. 110(1), 153–188 (1999)
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Faranda, D., Lucarini, V., Turchetti, G. et al. Numerical Convergence of the Block-Maxima Approach to the Generalized Extreme Value Distribution. J Stat Phys 145, 1156–1180 (2011). https://doi.org/10.1007/s10955-011-0234-7
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DOI: https://doi.org/10.1007/s10955-011-0234-7