• Ahmad, M. I., , C. D. Sinclair, , and A. Werritty, 1988: Log-logistic flood frequency analysis. J. Hydrol., 98 , 215224.

  • Alila, Y., 1999: A hierarchical approach for the regionalization of precipitation annual maxima in Canada. J. Geophys. Res., 104 , 3164531655.

    • Search Google Scholar
    • Export Citation
  • Archer, G., , and J-M. Giovannoni, 1998: Statistical analysis with bootstrap diagnostics of atmospheric pollutants predicted in the APSIS experiment. Water Air Soil Pollut., 106 , 4381.

    • Search Google Scholar
    • Export Citation
  • Arnbjergnielsen, K., , P. Harremoes, , and H. Spliid, 1994: Nonparametric statistics on extreme rainfall. Nord. Hydrol., 25 , 267278.

  • Canty, A. J., , A. C. Davison, , and D. V. Hinkley, 1996: Reliable confidence intervals. Discussion of bootstrap confidence intervals by T. J. DiCiccio and B. Efron. Stat. Sci., 11 , 214219.

    • Search Google Scholar
    • Export Citation
  • Carpenter, J. R., 1999: Test-inversion bootstrap confidence intervals. J. Roy. Stat. Soc. Ser. A, B61 , 159172.

  • Carpenter, J. R., , and J. Bithell, 2000: Bootstrap confidence intervals: When, which, what? A practical guide for medical statisticians. Stat. Med., 19 , 11411164.

    • Search Google Scholar
    • Export Citation
  • Christensen, J. H., , and O. B. Christensen, 2003: Severe summertime flooding in Europe. Nature, 421 , 805806.

  • Coles, S., 2001: An Introduction to Statistical Modeling of Extreme Values. Springer Verlag, 208 pp.

  • Cooley, R. L., 1997: Confidence intervals for groundwater models using linearization, likelihood, and bootstrap methods. Ground Water, 35 , 869880.

    • Search Google Scholar
    • Export Citation
  • Davison, A. C., , and D. V. Hinkley, 1997: Bootstrap Methods and Their Application. Cambridge University Press, 592 pp.

  • Davison, A. C., , D. V. Hinkley, , and G. A. Young, 2003: Recent developments in bootstrap methodology. Stat. Sci., 18 , 141157.

  • DiCiccio, T. J., , and B. Efron, 1996: Bootstrap confidence intervals. Stat. Sci., 11 , 189228.

  • Dixon, R., 2002: Bootstrap resampling. The Encyclopedia of Environmetrics, A. H. El-Shaarawi and W. W. Piegorsch, Eds., John Wiley and Sons, 212–219.

    • Search Google Scholar
    • Export Citation
  • Dunn, P. K., 2001: Bootstrap confidence intervals for predicted rainfall quantiles. Int. J. Climatol., 21 , 8994.

  • Efron, B., 1979: Bootstrap methods: Another look at the jackknife. Ann. Stat., 7 , 126.

  • Efron, B., 1981: Censored data and the bootstrap. J. Amer. Stat. Assoc., 76 , 312319.

  • Efron, B., 1987: Better bootstrap confidence intervals. J. Amer. Stat. Assoc., 82 , 171200.

  • Efron, B., , and R. J. Tibshirani, 1993: An Introduction to the Bootstrap. Chapman and Hall, 436 pp.

  • Ferro, C. A. T., , A. Hannachi, , and D. B. Stephenson, 2005: Simple nonparametric techniques for exploring changing probability distributions of weather. J. Climate, 18 , 43444354.

    • Search Google Scholar
    • Export Citation
  • Fitzgerald, D. L., 2005: Analysis of extreme rainfall using the log-logistic distribution. Stochastic Environ. Res. Risk Assess., 19 , 249257.

    • Search Google Scholar
    • Export Citation
  • Fowler, H. J., , and C. G. Kilsby, 2003: A regional frequency analysis of United Kingdom extreme rainfall from 1961 to 2000. Int. J. Climatol., 23 , 13131334.

    • Search Google Scholar
    • Export Citation
  • Frei, C., , R. Schöll, , S. Fukutome, , J. Schmidli, , and P. L. Vidale, 2006: Future change of precipitation extremes in Europe: Intercomparison of scenarios from regional climate models. J. Geophys. Res., 111 .D06105, doi:10.1029/2005JD005965.

    • Search Google Scholar
    • Export Citation
  • Gao, X., , J. S. Pal, , and F. Giorgi, 2006: Projected changes in mean and extreme precipitation over the Mediterranean region from a high resolution double nested RCM simulation. Geophys. Res. Lett., 33 .L03706, doi:10.1029/2005GL024954.

    • Search Google Scholar
    • Export Citation
  • Glaves, R., , and P. Waylen, 1997: Regional flood frequency analysis in southern Ontario using L-moments. Can. Geogr., 41 , 178193.

  • Gnedenko, B., 1943: Sur la distribution limite du terme maximum d’une serie aleatoire. Ann. Math., 44 , 423453.

  • Groisman, P. Y., , R. W. Knight, , D. R. Easterling, , T. R. Karl, , G. C. Hegerl, , and V. A. N. Razuvaev, 2005: Trends in intense precipitation in the climate record. J. Climate, 18 , 13261350.

    • Search Google Scholar
    • Export Citation
  • Gumbel, E. J., 1958: Statistics of Extremes. Columbia University Press, 375 pp.

  • Hall, M. J., , H. F. P. van den Boogaard, , R. C. Fernando, , and A. E. Mynett, 2004: The construction of confidence intervals for frequency analysis using resampling techniques. Hydrol. Earth Syst. Sci., 8 , 235246.

    • Search Google Scholar
    • Export Citation
  • Hall, P., 1992: The Bootstrap and Edgeworth Expansion. Springer-Verlag, 372 pp.

  • Hosking, J. R. M., 1990: L-moments: Analysis and estimation of distributions using linear combinations of order statistics. J. Roy. Stat. Soc. Ser. A, B52 , 105124.

    • Search Google Scholar
    • Export Citation
  • Hosking, J. R. M., , and J. R. Wallis, 1997: Regional Frequency Analysis: An Approach Based on L-moments. Cambridge University Press, 224 pp.

    • Search Google Scholar
    • Export Citation
  • Hosking, J. R. M., , J. R. Wallis, , and E. F. Wood, 1985: Estimation of the generalized extreme-value distribution by the method of probability-weighted moments. Technometrics, 27 , 251261.

    • Search Google Scholar
    • Export Citation
  • Jones, R., , J. Murphy, , D. Hassell, , and R. Taylor, 2001: Ensemble mean changes in a simulation of the European climate of 2071–2100 using the new Hadley Centre regional modelling system HadAM3H/HadRM3H. Met Office Hadley Centre, 19 pp. [Available online at http://prudence.dmi.dk/public/publications/hadley_200208.pdf.].

  • Jones, R. G., , M. Noguer, , D. C. Hassell, , D. Hudson, , S. S. Wilson, , G. J. Jenkins, , and J. F. B. Mitchell, 2004: Generating high resolution climate change scenarios using PRECIS. Met Office Hadley Centre, 35 pp.

  • Katz, R. W., , M. B. Parlange, , and P. Naveau, 2002: Statistics of extremes in hydrology. Adv. Water Res., 25 , 12871304.

  • Khaliq, M. N., , A. St-Hilaire, , T. B. M. J. Ouarda, , and B. Bobee, 2005: Frequency analysis and temporal pattern of occurrences of southern Quebec heatwaves. Int. J. Climatol., 25 , 485504.

    • Search Google Scholar
    • Export Citation
  • Kharin, V. V., , and F. W. Zwiers, 2000: Changes in the extremes in an ensemble of transient climate simulations with a coupled atmosphere–ocean GCM. J. Climate, 13 , 37603788.

    • Search Google Scholar
    • Export Citation
  • Kharin, V. V., , and F. W. Zwiers, 2005: Estimating extremes in transient climate change simulations. J. Climate, 18 , 11561173.

  • Kiktev, D., , D. M. H. Sexton, , L. Alexander, , and C. K. Folland, 2003: Comparison of modeled and observed trends in indices of daily climate extremes. J. Climate, 16 , 35603571.

    • Search Google Scholar
    • Export Citation
  • Kjeldsen, T. R., , and D. Rosbjerg, 2002: Comparison of regional index flood estimation procedures based on the extreme value type I distribution. Stochastic Environ. Res. Risk Assess., 16 , 358373.

    • Search Google Scholar
    • Export Citation
  • Koutsoyiannis, D., 2004a: Statistics of extremes and estimation of extreme rainfall: I. Theoretical investigation. Hydrol. Sci. J., 49 , 575590.

    • Search Google Scholar
    • Export Citation
  • Koutsoyiannis, D., 2004b: Statistics of extremes and estimation of extreme rainfall: II. Empirical investigation of long rainfall records. Hydrol. Sci. J., 49 , 591610.

    • Search Google Scholar
    • Export Citation
  • Kyselý, J., 2002: Comparison of extremes in GCM-simulated, downscaled and observed central-European temperature series. Climate Res., 20 , 211222.

    • Search Google Scholar
    • Export Citation
  • Kyselý, J., , and J. Picek, 2007: Regional growth curves and improved design value estimates of extreme precipitation events in the Czech Republic. Climate Res., 33 , 243255.

    • Search Google Scholar
    • Export Citation
  • Lana, X., , and A. Burgueño, 1999: Comments on an extreme winter minimum temperature study in Catalonia, north-east Spain. Int. J. Climatol., 19 , 803809.

    • Search Google Scholar
    • Export Citation
  • Lee, S. M. S., , and G. A. Young, 1995: Asymptotic iterated bootstrap confidence intervals. Ann. Stat., 23 , 13011330.

  • Lee, S. M. S., , and G. A. Young, 2003: Prepivoting by weighted bootstrap iteration. Biometrika, 90 , 393410.

  • Lee, S. M. S., , and G. A. Young, 2005: Parametric bootstrapping with nuisance parameters. Stat. Probab. Lett., 71 , 143153.

  • Manteiga, W. G., , and J. M. P. Sánchez, 1994: The bootstrap—A review. Comp. Stat., 9 , 165205.

  • Martins, E. S., , and J. R. Stedinger, 2000: Generalized maximum-likelihood generalized extreme-value quantile estimators for hydrologic data. Water Resour. Res., 36 , 737744.

    • Search Google Scholar
    • Export Citation
  • Moberg, A., , and P. D. Jones, 2004: Regional climate model simulations of daily maximum and minimum near-surface temperatures across Europe compared with observed station data 1961–1990. Climate Dyn., 23 , 695715.

    • Search Google Scholar
    • Export Citation
  • Morrison, J. E., , and J. A. Smith, 2002: Stochastic modeling of flood peaks using the generalized extreme value distribution. Water Resour. Res., 38 .1305, doi:10.1029/2001WR000502.

    • Search Google Scholar
    • Export Citation
  • Paeth, H., , and A. Hense, 2005: Mean versus extreme climate in the Mediterranean region and its sensitivity to future global warming conditions. Meteor. Z., 14 , 329347.

    • Search Google Scholar
    • Export Citation
  • Pal, J. S., , F. Giorgi, , and X. Bi, 2004: Consistency of recent European summer precipitation trends and extremes with future regional climate projections. Geophys. Res. Lett., 31 .L13202, doi:10.1029/2004GL019836.

    • Search Google Scholar
    • Export Citation
  • Palmer, T. N., , and J. Räisänen, 2002: Quantifying the risk of extreme seasonal precipitation events in a changing climate. Nature, 415 , 512514.

    • Search Google Scholar
    • Export Citation
  • Pandey, M. D., , P. H. A. J. M. van Gelder, , and J. K. Vrijling, 2003: Bootstrap simulations for evaluating the uncertainty associated with peaks-over-threshold estimates of extreme wind velocity. Environmetrics, 14 , 2743.

    • Search Google Scholar
    • Export Citation
  • Pandey, M. D., , P. H. A. J. M. van Gelder, , and J. K. Vrijling, 2004: Dutch case studies of the estimation of extreme quantiles and associated uncertainty by bootstrap simulations. Environmetrics, 15 , 687699.

    • Search Google Scholar
    • Export Citation
  • Park, J-S., , H-S. Jung, , R-S. Kim, , and J-H. Oh, 2001: Modelling summer extreme rainfall over the Korean Peninsula using Wakeby distribution. Int. J. Climatol., 21 , 13711384.

    • Search Google Scholar
    • Export Citation
  • Semmler, T., , and D. Jacob, 2004: Modeling extreme precipitation events—A climate change simulation for Europe. Global Planet. Change, 44 , 119127.

    • Search Google Scholar
    • Export Citation
  • Shao, J., , and D. Tu, 1995: The Jackknife and Bootstrap. Springer, 540 pp.

  • Tebaldi, C., , K. Hayhoe, , J. M. Arblaster, , and G. A. Meehl, 2006: Going to the extremes. Climatic Change, 79 , 185211.

  • Ulrych, T. J., , D. R. Velis, , A. D. Woodbury, , and M. D. Sacchi, 2000: L-moments and C-moments. Stochastic Environ. Res. Risk Assess., 14 , 5068.

    • Search Google Scholar
    • Export Citation
  • van den Brink, H. W., , G. P. Koennen, , and J. D. Opsteegh, 2003: The reliability of extreme surge levels, estimated from observational records of order hundred years. J. Coastal Res., 19 , 376388.

    • Search Google Scholar
    • Export Citation
  • van den Brink, H. W., , G. P. Koennen, , and J. D. Opsteegh, 2004: Statistics of synoptic-scale wind speeds in ensemble simulations of current and future climate. J. Climate, 17 , 45644574.

    • Search Google Scholar
    • Export Citation
  • van den Brink, H. W., , G. P. Koennen, , and J. D. Opsteegh, 2005: Uncertainties in extreme surge level estimates from observational records. Philos. Trans. Roy. Soc. London, A363 , 13771386.

    • Search Google Scholar
    • Export Citation
  • Voss, R., , W. May, , and E. Roeckner, 2002: Enhanced resolution modelling study on anthropogenic climate change: Changes in extremes of the hydrological cycle. Int. J. Climatol., 22 , 755777.

    • Search Google Scholar
    • Export Citation
  • Weglarczyk, S., , W. G. Strupczewski, , and V. P. Singh, 2002: A note on the applicability of log-Gumbel and log-logistic probability distributions in hydrological analyses: II. Assumed pdf. Hydrol. Sci. J., 47 , 123137.

    • Search Google Scholar
    • Export Citation
  • Wilks, D. S., 1997: Resampling hypothesis tests for autocorrelated fields. J. Climate, 10 , 6582.

  • Zhang, X., , G. Hegerl, , F. W. Zwiers, , and J. Kenyo, 2005: Avoiding inhomogeneity in percentile-based indices of temperature extremes. J. Climate, 18 , 16411651.

    • Search Google Scholar
    • Export Citation
  • Zwiers, F. W., 1990: The effect of serial correlation on statistical inferences made with resampling procedures. J. Climate, 3 , 14521461.

    • Search Google Scholar
    • Export Citation
  • Zwiers, F. W., , and W. H. Ross, 1991: An alternative approach to the extreme value analysis of rainfall data. Atmos.–Ocean, 29 , 437461.

    • Search Google Scholar
    • Export Citation
  • Zwiers, F. W., , and V. V. Kharin, 1998: Changes in the extremes of the climate simulated by CCC GCM2 under CO2 doubling. J. Climate, 11 , 22002222.

    • Search Google Scholar
    • Export Citation
All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 262 262 45
PDF Downloads 214 214 25

A Cautionary Note on the Use of Nonparametric Bootstrap for Estimating Uncertainties in Extreme-Value Models

View More View Less
  • 1 Institute of Atmospheric Physics, Academy of Sciences of the Czech Republic, Prague, Czech Republic
© Get Permissions
Restricted access

Abstract

The parametric and nonparametric approaches to the bootstrap are compared as to their performance in estimating uncertainties in extreme-value models. Simulation experiments make use of several combinations of true and fitted probability distributions utilized in climatological and hydrological applications. The results demonstrate that for small to moderate sample sizes the nonparametric bootstrap should be interpreted with caution because it leads to confidence intervals that are too narrow and underestimate the real uncertainties involved in the frequency models. Although the parametric bootstrap yields confidence intervals that are slightly too liberal as well, it improves the uncertainty estimates in most examined cases, even under conditions in which an incorrect parametric model is adopted for the data. Differences among three examined types of bootstrap confidence intervals (percentile, bootstrap t, and bias corrected and accelerated) are usually smaller in comparison with those between the parametric and nonparametric versions of bootstrap. It is concluded that the parametric bootstrap should be preferred whenever inferences are based on small to moderate sample sizes (n ≤ 60) and a suitable model for the data is known or can be assumed, including applications to confidence intervals related to extremes in global and regional climate model projections.

Corresponding author address: Jan Kyselý, Institute of Atmospheric Physics, Academy of Sciences of the Czech Republic, Boční II 1401, 141 31 Prague, Czech Republic. Email: kysely@ufa.cas.cz

Abstract

The parametric and nonparametric approaches to the bootstrap are compared as to their performance in estimating uncertainties in extreme-value models. Simulation experiments make use of several combinations of true and fitted probability distributions utilized in climatological and hydrological applications. The results demonstrate that for small to moderate sample sizes the nonparametric bootstrap should be interpreted with caution because it leads to confidence intervals that are too narrow and underestimate the real uncertainties involved in the frequency models. Although the parametric bootstrap yields confidence intervals that are slightly too liberal as well, it improves the uncertainty estimates in most examined cases, even under conditions in which an incorrect parametric model is adopted for the data. Differences among three examined types of bootstrap confidence intervals (percentile, bootstrap t, and bias corrected and accelerated) are usually smaller in comparison with those between the parametric and nonparametric versions of bootstrap. It is concluded that the parametric bootstrap should be preferred whenever inferences are based on small to moderate sample sizes (n ≤ 60) and a suitable model for the data is known or can be assumed, including applications to confidence intervals related to extremes in global and regional climate model projections.

Corresponding author address: Jan Kyselý, Institute of Atmospheric Physics, Academy of Sciences of the Czech Republic, Boční II 1401, 141 31 Prague, Czech Republic. Email: kysely@ufa.cas.cz

Save