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

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.

[1]  Angel R. Martinez,et al.  Computational Statistics Handbook with MATLAB , 2001 .

[2]  Catherine Nicolis,et al.  Extreme value distributions in chaotic dynamics , 1995 .

[3]  E. Gumbel.  The Return Period of Flood Flows , 1941 .

[4]  Valerio Lucarini,et al.  A statistical mechanical approach for the computation of the climatic response to general forcings , 2010, 1008.0340.

[5]  Jose D. Salas,et al.  Regional Frequency Analysis of Extreme Precipitation in Northeastern Colorado and Fort Collins Flood of 1997 , 2002 .

[6]  A. C. Freitas,et al.  The extremal index, hitting time statistics and periodicity , 2010, 1008.1350.

[7]  S. Vannitsem,et al.  Statistical properties of the temperature maxima in an intermediate order Quasi-Geostrophic model , 2007 .

[8]  Richard L. Smith.  Extreme Value Analysis of Environmental Time Series: An Application to Trend Detection in Ground-Level Ozone , 1989 .

[9]  Valerio Lucarini,et al.  Extreme Value Statistics of the Total Energy in an Intermediate-Complexity Model of the Midlatitude Atmospheric Jet. Part II: Trend Detection and Assessment , 2006 .

[10]  Janet E. Heffernan,et al.  Dependence Measures for Extreme Value Analyses , 1999 .

[11]  G. Nicolis,et al.  Extreme events in deterministic dynamical systems. , 2006, Physical review letters.

[12]  Ana Cristina Moreira Freitas,et al.  Extreme values for Benedicks–Carleson quadratic maps , 2007, Ergodic Theory and Dynamical Systems.

[13]  Marcelo Cruz.  Modeling, Measuring and Hedging Operational Risk , 2002 .

[14]  H Kantz,et al.  Influence of the event magnitude on the predictability of an extreme event. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[15]  C. Cornell.  Engineering seismic risk analysis , 1968 .

[16]  D. Sornette,et al.  Rank‐ordering statistics of extreme events: Application to the distribution of large earthquakes , 1995, cond-mat/9510035.

[17]  Marc B. Parlange,et al.  STATISTICS OF EXTREMES: MODELING ECOLOGICAL DISTURBANCES , 2005 .

[18]  Chinmaya Gupta,et al.  Extreme-value distributions for some classes of non-uniformly partially hyperbolic dynamical systems , 2008, Ergodic Theory and Dynamical Systems.

[19]  A. Török,et al.  Extreme value distributions for non-uniformly hyperbolic dynamical systems ∗ , 2008 .

[20]  B. M. Hill,et al.  A Simple General Approach to Inference About the Tail of a Distribution , 1975 .

[21]  L. Young.  Recurrence times and rates of mixing , 1999 .

[22]  Z. Coelho,et al.  Limit laws of entrance times for homeomorphisms of the circle , 1996 .

[23]  L. Young,et al.  STATISTICAL PROPERTIES OF DYNAMICAL SYSTEMS WITH SOME HYPERBOLICITY , 1998 .

[24]  V. I. Arnolʹd,et al.  Ergodic problems of classical mechanics , 1968 .

[25]  Universal fluctuations and extreme-value statistics , 2001, cond-mat/0108007.

[26]  F. Longin,et al.  From value at risk to stress testing : The extreme value approach Franc ß ois , 2000 .

[27]  M. Gilli,et al.  An Application of Extreme Value Theory for Measuring Financial Risk , 2006 .

[28]  Holger Kantz,et al.  Dynamical Interpretation of Extreme Events: Predictability and Predictions , 2006 .

[29]  S. Resnick,et al.  Extreme Value Theory as a Risk Management Tool , 1999 .

[30]  Anthony C. Davison,et al.  Modelling Excesses over High Thresholds, with an Application , 1984 .

[31]  P. Burton.  Seismic risk in southern Europe through to India examined using Gumbel's third distribution of extreme values , 1979 .

[32]  C. Ferro,et al.  Robust extremes in chaotic deterministic systems. , 2009, Chaos.

[33]  E. Bertin,et al.  GLOBAL FLUCTUATIONS IN PHYSICAL SYSTEMS: A SUBTLE INTERPLAY BETWEEN SUM AND EXTREME VALUE STATISTICS , 2008, 0807.1649.

[34]  Giorgio Turchetti,et al.  Statistics of Poincar recurrences for a class of smooth circle maps , 2003, nlin/0312001.

[35]  Claudia Klüppelberg,et al.  Extreme Value Theory in Finance , 2008 .

[36]  Valerio Lucarini,et al.  Extreme Value Statistics of the Total Energy in an Intermediate Complexity Model of the Mid-latitude Atmospheric Jet. Part I: Stationary case , 2006 .

[37]  R. Fisher,et al.  Limiting forms of the frequency distribution of the largest or smallest member of a sample , 1928, Mathematical Proceedings of the Cambridge Philosophical Society.

[38]  J. Hüsler.  Extremes and related properties of random sequences and processes , 1984 .

[39]  Vladimir Kossobokov,et al.  Extreme events: dynamics, statistics and prediction , 2011 .

[40]  B. Gnedenko.  Sur La Distribution Limite Du Terme Maximum D'Une Serie Aleatoire , 1943 .

[41]  Eric Bertin,et al.  Global fluctuations and Gumbel statistics. , 2005, Physical review letters.

[42]  H. Lilliefors.  On the Kolmogorov-Smirnov Test for Normality with Mean and Variance Unknown , 1967 .

[43]  R. Katz.  Extreme value theory for precipitation: sensitivity analysis for climate change , 1999 .

[44]  J. Stedinger,et al.  Generalized maximum‐likelihood generalized extreme‐value quantile estimators for hydrologic data , 2000 .

[45]  G Turchetti,et al.  Statistics of Poincaré recurrences for maps with integrable and ergodic components. , 2004, Chaos.

[46]  Richard L. Smith.  Threshold Methods for Sample Extremes , 1984 .

[47]  Pierre Collet,et al.  Statistics of closest return for some non-uniformly hyperbolic systems , 1999, Ergodic Theory and Dynamical Systems.

[48]  Jorge Milhazes Freitas,et al.  On the link between dependence and independence in extreme value theory for dynamical systems , 2008 .

[49]  Ana Cristina Moreira Freitas,et al.  Extreme Value Laws in Dynamical Systems for Non-smooth Observations , 2010, 1006.3276.

[50]  A. Sper,et al.  Accessing extremes of mid latitudinal wave activity : methodology and application , 2022 .

[51]  Richard L. Smith,et al.  Models for exceedances over high thresholds , 1990 .

[52]  George Haiman,et al.  Extreme values of the tent map process , 2003 .

[53]  E. Zelenhasić,et al.  A Stochastic Model for Flood Analysis , 1970 .

[54]  Boris Hasselblatt,et al.  A First Course in Dynamics: with a Panorama of Recent Developments , 2003 .

[55]  J. Pickands.  Statistical Inference Using Extreme Order Statistics , 1975 .

[56]  P. Friederichs,et al.  Statistical Downscaling of Extreme Precipitation Events Using Censored Quantile Regression , 2007 .

[57]  James Pickands,et al.  Moment Convergence of Sample Extremes , 1968 .

[58]  R. Katz,et al.  Extreme events in a changing climate: Variability is more important than averages , 1992 .

[59]  Holger Kantz,et al.  Reactions to extreme events: Moving threshold model , 2006 .

[60]  Ana Cristina Moreira Freitas,et al.  Hitting time statistics and extreme value theory , 2008, 0804.2887.

[61]  J. Teugels,et al.  Statistics of Extremes , 2004 .