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Comparing generalized Pareto models fitted to extreme observations: an application to the largest temperatures in Spain

Stochastic Environmental Research and Risk Assessment volume 28, pages 1221–1233 (2014)Cite this article

Abstract

In this paper, a subsampling-based testing procedure for the comparison of the exceedance distributions of stationary time series is introduced. The proposed testing procedure has a number of advantages including the fact that the assumption of stationary can be relaxed for some specific forms of non-stationary and also that the two time series are not required to be independently-generated. For this purpose, a test based on the Kolmogorov–Smirnov and the L 1-Wasserstein distances between generalized Pareto distributions is introduced and studied in some detail. The performance of the testing procedure is illustrated through a simulation study and with an empirical application to a set of data concerning daily maximum temperature in the 17 autonomous communities of Spain for the period 1990–2004. The autonomous communities were clustered according to the similarities of the fitted generalized Pareto models and then mapped. The cluster analysis reveals a clear distinction between the four northeast communities on the shores of the Bay of Biscay (which are the regions exhibiting milder temperatures) and the remaining regions. A second cluster corresponds to the southern Mediterranean area and the central region which corresponds to the communities with highest temperatures.

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References

  • Alonso AM, Maharaj EA (2006) Comparison of time series using subsampling. Comput Stat Data Anal 50:2589–2599

    Article  Google Scholar 

  • Balkema A, de Haan L. (1974) Residual life time at great age. Ann Probab 2:792–804

    Article  Google Scholar 

  • Brunet M, Jones PD, Sigro J, Saladi O, Aguilar E, Moberg A, Della-Marta PM, Lister D, Walther A, Lopez D (2007) Temporal and spatial temperature variability and change over Spain during 1850–2005. J Geophys Res 112:D12117. doi:10.1029/2006JD008249

    Article  Google Scholar 

  • Cameron D, Bevin K, Tawn J (2001) Modelling extreme rainfalls using a modified pulse Bartlett–Lewis stochastic rainfall model (with uncertainty). Adv Water Resour 24:203–211

    Article  Google Scholar 

  • Castillo E, Hadi AS (1997) Fitting the generalized Pareto distribution to data. J Am Stat Assoc 92:1609–1620

    Article  Google Scholar 

  • Castillo E, Hadi AS, Balakrishnan N, Sarabia JM (2004) Extreme value and related models with applications in engineering and science. Wiley, New Jersey, p 362

  • Chavez-Demoulin V, Embrechts P (2004) Smooth extremal models in finance and insurance. J Risk Insur 71:183–199

    Article  Google Scholar 

  • Coles SG (2001) An introduction to statistical modeling of extreme values. Springer, London, p 228

  • Davison AC, Smith RL (1990) Models for exceedances over high thresholds. J Roy Stat Soc B 52:393–442

    Google Scholar 

  • de Haan L, Ferreira A (2006) Extreme value theory: an introduction. Springer, New York, p 417

  • de Zea Bermudez P, Kotz S (2010a) Parameter estimation of the generalized Pareto distribution. Part I. J Stat Plan Inference 40:1353–1373

    Article  Google Scholar 

  • de Zea Bermudez P, Kotz S (2010b) Parameter estimation of the generalized Pareto distribution. Part II. J Stat Plan Inference 40:1374–1388

    Article  Google Scholar 

  • Draghicescu D, Ignaccolo R (2009) Modeling threshold exceedance probabilities of spatially correlated time series. Electron J Stat 3:149–164

    Article  Google Scholar 

  • DuMouchel WH (1983) Estimating the stable index α in order to measure tail thickness: a critique. Ann Stat 11:1019–1031

    Google Scholar 

  • Eastoe EF, Tawn JA (2012) Modelling the distribution of the cluster maxima of exceedances of subasymptotic thresholds. Biometrika 99:43–55

    Article  Google Scholar 

  • Fernández-Montes S, Rodrigo FS (2012) Trends in seasonal indices of daily temperature extremes in the Iberian Peninsula, 1929–2005. Int J Climatol 32:2320–2332

    Article  Google Scholar 

  • Furió D, Meneu V (2011) Analysis of extreme temperatures for four sites across Peninsular Spain. Theor Appl Climatol 104:83–99

    Article  Google Scholar 

  • Furrer EM, Katz RW (2008) Improving the simulation of extreme precipitation events by stochastic weather generators. Water Resour Res 44, W12439. doi:10.1029/2008WR007316

    Google Scholar 

  • García-Herrera R, Díaz J, Trigo RM, Hernández E (2005) Extreme summer temperatures in Iberia: health impacts and associated synoptic conditions. Ann Geophys 23:239–251

    Article  Google Scholar 

  • Gupta A, Liang B (2005) Do hedge funds have enough capital? A value at risk approach. J Financial Econ 77:219–253

    Article  Google Scholar 

  • Jonathan P, Ewans K (2013) Statistical modelling of extreme ocean environments for marine design: a review. Ocean Eng 62:91–109

    Article  Google Scholar 

  • Katz RW, Parlange MB, Naveau P (2002) Statistics of extremes in hydrology. Adv Water Resour 25:1287–1304

    Article  Google Scholar 

  • Kotz S, Nadarajah S (2000) Extreme value distributions: theory and applications. Imperial College Press, London, p 196

  • Lauridsen S (2000) Estimation of value at risk by extreme value methods. Extremes 3:107–144

    Article  Google Scholar 

  • Leadbetter MR, Lindgren G, Rootzén H (1983) Extremes and related properties of random sequences and processes. Springer, Berlin, p 336

  • Letretel C, Marcos M, Martín-Míguez B, Woppelmann G (2010) Sea level extremes in Marseille (NW Mediterranean) during 1885–2008. Cont Shelf Res 30:1267–1274

    Article  Google Scholar 

  • Mackay E, Challenor P, AbuBakr S (2011) A comparison of estimators for the generalised Pareto distribution. Ocean Eng 38:1338–1346

    Article  Google Scholar 

  • Mendes JM, de Zea Bermudez P, Pereira J, Turkman KF, Vasconcelos MJP (2009) Spatial extremes of wild fire sizes: Bayesian hierarchical models for extremes. Environ Ecol Stat 17:1–28

    Article  Google Scholar 

  • Méndez FJ, Menéndez M, Luceño A, Losada IJ (2007) Analyzing monthly extreme sea levels with a time-dependent GEV model. J Atmos Oceanic Technol 24:894–911

    Article  Google Scholar 

  • Menéndez M, Méndez FJ, Izaguirre C, Losada IJ (2009) The influence of seasonality on estimating return values of significant wave height. Coast Eng 56:211–219

    Article  Google Scholar 

  • Mínguez R, Tomás A, Méndez FJ, Medina R (2013) Mixed extreme wave climate model for reanalysis databases. Stoch Environ Res Risk Assess 27:757–768

    Article  Google Scholar 

  • Naveau P, Nogaj M, Ammann C, Yiou P, Cooley D, Jomelli V (2005) Statistical methods for the analysis of climate extremes. C R Geosci 337:1013–1022

    Article  Google Scholar 

  • Niu XF (1997) Extreme value for a class of non-stationary time series with applications. Ann Appl Probab 7:508–522

    Article  Google Scholar 

  • Nogaj M, Yiou P, Parey S, Malek F, Naveau P (2006) Amplitude and frequency of temperature extremes over the North Atlantic region. Geophys Res Lett 33:L10801. doi:10.1029/2005GL024251

  • Northrop PJ, Jonathan P (2011) Threshold modelling of spatially-dependent non-stationary extremes with application to hurricane- induced wave heights (with discussion). Environmetrics 22:799–809

    Article  Google Scholar 

  • Pickands J (1975) Statistical inference using extreme order statistics. Ann Stat 3:119–131

    Article  Google Scholar 

  • Politis DN, Romano JP, Wolf M (1999) Subsampling. Springer, New York, p 363

  • Ribatet M, Sauquet E, Grésillon J-M, Ouarda TBMJ (2007) A regional Bayesian POT model for flood frequency analysis. Stoch Environ Res Risk Assess 21:327–339

    Article  Google Scholar 

  • Scarrott C, MacDonald A (2012) A review of extreme value threshold estimation and uncertainty quantification. REVSTAT 10:33–60

    Google Scholar 

  • Scotto MG, Alonso AM, Barbosa SM (2010) Clustering time series of sea levels: extreme value approach. J Waterw Port Coast Ocean Eng 136:215–225

    Article  Google Scholar 

  • Scotto MG, Barbosa SM, Alonso AM (2011) Extreme value and cluster analysis of European daily temperature series. J Appl Stat 38:2793–2804

    Article  Google Scholar 

  • Tobías A, Scotto MG (2005) Prediction of extreme ozone levels in Barcelona, Spain. Environ Monit Assess 100(1–3):23–32

    Article  Google Scholar 

  • Vanem E (2011) Long-term time-dependent stochastic modelling of extreme waves. Stoch Environ Res Risk Assess 25:185–209

    Article  Google Scholar 

Download references

Acknowledgments

The authors would like to express their gratitude to the editor and the two referees. They offered extremely valuable perspectives on our work and effective suggestions for improvements. The second author would like to thank the support of the Department of Statistics of the University Carlos III, Madrid (Spain), where she stayed during her last 4 months of Sabbatical Leave. This stay was financially supported by the grant SFRH/BSAB/ 1138/2011, from Fundação para a Ciência e Tecnologia-FCT, Portugal. The research was also funded by FCT projects PEst-OE/MAT/UI0006/2011 and PTDC/MAT/118335/2010. The first author also acknowledges the support of CICYT (Spain) Grants SEJ2007-64500, ECO2011-25706 and ECO2012-38442. The third author was supported by FEDER funds through COMPETE–Operational Programme Factors of Competitiveness (“Programa Operacional Factores de Competitividade”) and by Portuguese funds through the Center for Research and Development in Mathematics and Applications (University of Aveiro) and the Portuguese Foundation for Science and Technology (“FCT–Fundação para a Ciência e a Tecnologia”), within project PEst-C/MAT/UI4106/2011 with COMPETE number FCOMP-01-0124-FEDER-022690.

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Authors and Affiliations

  1. Department of Statistics and INEACU, Universidad Carlos III de Madrid, Madrid, Spain

    Andrés M. Alonso

  2. Department of Statistics and Operations Research and CEAUL, Universidade de Lisboa, Lisbon, Portugal

    Patricia de Zea Bermudez

  3. Department of Mathematics, Universidade de Aveiro, Aveiro, Portugal

    Manuel G. Scotto

Authors
  1. Andrés M. Alonso
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  2. Patricia de Zea Bermudez
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  3. Manuel G. Scotto
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Correspondence to Patricia de Zea Bermudez.

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Alonso, A.M., de Zea Bermudez, P. & Scotto, M.G. Comparing generalized Pareto models fitted to extreme observations: an application to the largest temperatures in Spain. Stoch Environ Res Risk Assess 28, 1221–1233 (2014). https://doi.org/10.1007/s00477-013-0809-8

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  • DOI: https://doi.org/10.1007/s00477-013-0809-8

Keywords

  • Extreme value theory
  • Generalized Pareto distribution
  • Peaks over threshold
  • Subsampling
  • Stationarity and non-stationary time series