Abstract
Two effects arise for the modelling of block maxima from dependent time series: a reduced rate of convergence for the block maxima probability distribution towards the generalised extreme value distribution, and an increase in uncertainty of the parameter estimates compared to independent or short range dependent records. These effects are exemplified with a simulation study using a white noise, a short-range and a long-range dependent process. The two issues raised turned out to be relatively unproblematic for short-range dependent processes. For long-range dependent processes, especially the increased parameter uncertainty poses a problem. Incautious use of standard procedures would lead to a severe underestimation of the parameter uncertainty which implies a misconception of accuracy for derived quantities, such as return levels which are frequently used for risk assessment and dimensioning of hydraulic structures.
Similar content being viewed by others
References
S. Hallegatte, J. Appl. Meteorol. Climatol. 46, 1956 (2007)
Munich Re Group, Tech. Rep., Munich Re Group (2009)
J. Beirlant, Y. Goegebeur, H. Segers, J. Teugels, Statistics of Extremes: Theory and Applications (Wiley, 2004)
S.G. Coles, An Introduction to Statistical Modelling of Extreme Values (Springer, London, 2001)
P. Embrechts, C. Klüppelberger, T. Mikosch, Modelling Extremal Events for Insurance and Finance (Springer, Berlin, 1997)
J. Beran, Statistics for Long-Memory Processes (Chapman & Hall, 1994)
P.M. Robinson (ed.), Time Series with Long Memory (Oxford University Press, 2003)
T.A. Cohn, H.F. Lins, Geophys. Res. Lett. 32, L23402 (2005)
K. Fraedrich, R. Blender, Phys. Rev. Lett. 90, 108501 (2003)
R.L. Smith, Long-range Dependence and Global Warming, edited by V. Barnett, P. Turkman, Statistics for the Environment (John Wiley and Sons Ltd., 1993), pp. 141–161
D.I. Vyushin, V.E. Fioletov, T.G. Shepherd, J. Geophys. Res. 112, D14307 (2007)
A. Montanari, R. Rosso, M.S. Taqqu, Water Resour. Res. 36, 1249 (2000)
M. Mudelsee, Water Resour. Res. 43, W01202 (2007)
M. Santhanam, H. Kantz, Physica A 345, 713 (2005)
A. Bunde, J.F. Eichner, S. Havlin, J.W. Kantelhart, Physica A 330, 1 (2003)
E.G. Altmann, H. Kantz, Phys. Rev. E 71, 056106 (2005)
R.A. Fisher, L.H.C. Tippett, Proc. Cambridge Phil. Soc. 24, 180 (1928)
R.W. Katz, M.B. Parlange, C. Tebaldi, Clim. Change 60, 189 (2003)
D. Maraun, H.W. Rust, T.J. Osborn, Int. J. Climatol, doi: 10.1002/joc.1811 (2009) (in press)
H.W. Rust, D. Maraun, T.J. Osborn, Eur. Phys. J. Special Topics 174, 99 (2009)
J.R.M. Hosking, J.R. Wallis, E.F. Wood, Technometrics 27, 251 (1985)
R.L. Smith, Biometrika 72, 67 (1985)
H.W. Rust, O. Mestre, V.K.C. Venema, J. Geophys. Res. 113, D19110 (2008)
M.R. Leadbetter, G. Lindgren, H. Rootzén, Extremes and Related Properties of Random Sequences and Processes (Springer, New York, 1983)
P.J. Brockwell, R.A. Davis, Time Series: Theory and Methods (Springer, Berlin, 1991)
A. Montanari, R. Rosso, M.S. Taqqu, Water Resour. Res. 33, 1035 (1997)
A.C. Davison, D.B. Hinkley, Bootstrap Methods and their Application (Cambridge University Press, Cambridge, 1997)
H.W. Rust, M. Kallache, H.-J. Schellnhuber, J. Kropp, Confidence Intervals for Flood Return Level Estimates using a Bootstrap Approach, J. Kropp, H.-J. Schellnhuber, In Extremis (Springer, Berlin, forthcoming)
M. Kallache, H.W. Rust, J. Kropp, Nonlin. Proc. Geophys. 2, 201 (2005)
D. Maraun, H.W. Rust, J. Timmer, Nonlin. Proc. Geophys. 11, 495 (2004)
R Development Core Team, R: A Language and Environment for Statistical Computing (R Foundation for Statistical Computing, Vienna, 2004)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Rust, H. The effect of long-range dependence on modelling extremes with the generalised extreme value distribution. Eur. Phys. J. Spec. Top. 174, 91–97 (2009). https://doi.org/10.1140/epjst/e2009-01092-8
Published:
Issue Date:
DOI: https://doi.org/10.1140/epjst/e2009-01092-8