• Arcones, M. A., and E. Giné, 1989: The bootstrap of the mean with arbitrary bootstrap sample. Ann. Inst. Henri Poincaré, 25, 457481.

    • Search Google Scholar
    • Export Citation
  • Arcones, M. A., and E. Giné, 1991: Additions and corrections to “The bootstrap of the mean with arbitrary bootstrap sample.” Ann. Inst. Henri Poincaré, 27, 583595.

    • Search Google Scholar
    • Export Citation
  • Athreya, K. B., 1987a: Bootstrap of the mean in the infinite variance case. Proc. First World Congress of the Bernoulli Society, Utrecht, Netherlands, Bernoulli Society, 9598.

  • Athreya, K. B., 1987b: Bootstrap of the mean in the infinite variance case. Ann. Stat., 15, 724731, https://doi.org/10.1214/aos/1176350371.

  • Bickel, P. J., and D. A. Freedman, 1981: Some asymptotic theory for the bootstrap. Ann. Stat., 9, 11961217, https://doi.org/10.1214/aos/1176345637.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Bickel, P. J., F. Götze, and W. R. van Zwet, 1997: Resampling fewer than n observations: Gains, losses, and remedies for losses. Stat. Sin., 7, 131.

    • Search Google Scholar
    • Export Citation
  • Büecher, A., and J. Segers, 2017: On the maximum likelihood estimator for the generalized extreme-value distribution. Extremes, 20, 839872, https://doi.org/10.1007/s10687-017-0292-6.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Canty, A., and B. Ripley, 2017: boot: Bootstrap R (S-Plus) Functions, version 1.3-20. R package, http://statwww.epfl.ch/davison/BMA/.

  • Cooley, D., 2013: Return periods and return levels under climate change. Extremes in a Changing Climate: Detection, Analysis and Uncertainty, Springer, 97114.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Davison, A., and D. Hinkley, 1997: Bootstrap Methods and Their Application. Cambridge University Press, 582 pp.

  • Deheuvels, P., D. M. Mason, and G. R. Shorack, 1993: Some results on the influence of extremes on the bootstrap. Ann. Inst. Henri Poincaré Probab. Stat., 29, 83103.

    • Search Google Scholar
    • Export Citation
  • Fawcett, L., and D. Walshaw, 2012: Estimating return levels from serially dependent extremes. Environmetrics, 23, 272283, https://doi.org/10.1002/env.2133.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Feigin, P., and S. I. Resnick, 1997: Linear programming estimators and bootstrapping for heavy-tailed phenomena. Adv. Appl. Probab., 29, 759805, https://doi.org/10.2307/1428085.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Fukuchi, J.-I., 1994: Bootstrapping extremes of random variables. Ph.D. thesis, Iowa State University, 101 pp.

  • Gilleland, E., 2017: distillery: Method Functions for Confidence Intervals and to Distill Information from an Object, version 1.0-4. R package, https://www.ral.ucar.edu/staff/ericg.

  • Gilleland, E., 2020: Bootstrap methods for statistical inference. Part I: Comparative forecast verification for continuous variables. J. Atmos. Oceanic Technol., 36, 21172134, https://doi.org/10.1175/JTECH-D-20-0069.1.

    • Search Google Scholar
    • Export Citation
  • Gilleland, E., and R. W. Katz, 2016: extRemes 2.0: An extreme value analysis package in R. J. Stat. Software, 72, 139, https://doi.org/10.18637/jss.v072.i08.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Gilleland, E., R. W. Katz, and P. Naveau, 2017: Quantifying the risk of extreme events under climate change. Chance, 30, 3036, https://doi.org/10.1080/09332480.2017.1406757.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Giné, E., and J. Zinn, 1989: Necessary conditions for the bootstrap of the mean. Ann. Stat., 17, 684691, https://doi.org/10.1214/aos/1176347134.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hall, P., 1990: Asymptotic properties of the bootstrap for heavy-tailed distributions. Ann. Probab., 18, 13421360, https://doi.org/10.1214/aop/1176990748.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Heffernan, J. E., and J. A. Tawn, 2004: A conditional approach for multivariate extreme values (with discussion). J. Roy. Stat. Soc., 66B, 497546, https://doi.org/10.1111/j.1467-9868.2004.02050.x.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Katz, R. W., 2013: Statistical methods for non-stationary extremes. Extremes in a Changing Climate: Detection, Analysis and Uncertainty, Springer, 1537.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Katz, R. W., M. B. Parlange, and P. Naveau, 2002: Statistics of extremes in hydrology. Adv. Water Resour., 25, 12871304, https://doi.org/10.1016/S0309-1708(02)00056-8.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kinateder, J. G., 1992: An invariance principle applicable to the bootstrap. Exploring the Limits of Bootstrap, Wiley Series in Probability and Mathematical Statistics, Wiley, 157181.

  • Knight, K., 1989: On the bootstrap of the sample mean in the infinite variance case. Ann. Stat., 17, 11681175, https://doi.org/10.1214/aos/1176347262.

    • Crossref
    • 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, https://doi.org/10.3354/cr020211.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lee, S., 1999: On a class of m out of n bootstrap confidence intervals. J. Roy. Stat., 61B, 901911, https://doi.org/10.1111/1467-9868.00209.

  • LePage, R., 1992: Bootstrapping signs. Exploring the Limits of Bootstrap, Wiley Series in Probability and Mathematical Statistics, Wiley, 215224.

  • R Core Team, 2017: R: A language and environment for statistical computing. R Foundation for Statistical Computing, https://www.R-project.org/.

  • Resnick, S. I., 2007: Heavy-Tail Phenomena: Probabilistic and Statistical Modeling. Springer Series in Operations Research and Financial Engineering, Springer, 404 pp.

  • Schendel, T., and R. Thongwichian, 2015: Flood frequency analysis: Confidence interval estimation by test inversion bootstrapping. Adv. Water Resour., 83, 19, https://doi.org/10.1016/j.advwatres.2015.05.004.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Schendel, T., and R. Thongwichian, 2017: Confidence intervals for return levels for the peaks-over-threshold approach. Adv. Water Resour., 99, 5359, https://doi.org/10.1016/j.advwatres.2016.11.011.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Shao, J., and T. Dongsheng, 1995: The Jackknife and the Bootstrap. Springer, 123 pp.

  • Smith, R. L., 1985: Maximum likelihood estimation in a class of nonregular cases. Biometrika, 72, 6790, https://doi.org/10.1093/biomet/72.1.67.

    • Crossref
    • Search Google Scholar
    • Export Citation
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Bootstrap Methods for Statistical Inference. Part II: Extreme-Value Analysis

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  • 1 National Center for Atmospheric Research, Boulder, Colorado
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Abstract

This paper is the sequel to a companion paper on bootstrap resampling that reviews bootstrap methodology for making statistical inferences for atmospheric science applications where the necessary assumptions are often not met for the most commonly used resampling procedures. In particular, this sequel addresses extreme-value analysis applications with discussion on the challenges for finding accurate bootstrap methods in this context. New bootstrap code from the R packages “distillery” and “extRemes” is introduced. It is further found that one approach for accurate confidence intervals in this setting is not well suited to the case when the random sample’s distribution is not stationary.

Supplemental information related to this paper is available at the Journals Online website: https://doi.org/10.1175/JTECH-D-20-0070.s1.

© 2020 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Eric Gilleland, ericg@ucar.edu

This article has a companion article which can be found at http://journals.ametsoc.org/doi/abs/10.1175/JTECH-D-20-0069.1.

Abstract

This paper is the sequel to a companion paper on bootstrap resampling that reviews bootstrap methodology for making statistical inferences for atmospheric science applications where the necessary assumptions are often not met for the most commonly used resampling procedures. In particular, this sequel addresses extreme-value analysis applications with discussion on the challenges for finding accurate bootstrap methods in this context. New bootstrap code from the R packages “distillery” and “extRemes” is introduced. It is further found that one approach for accurate confidence intervals in this setting is not well suited to the case when the random sample’s distribution is not stationary.

Supplemental information related to this paper is available at the Journals Online website: https://doi.org/10.1175/JTECH-D-20-0070.s1.

© 2020 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Eric Gilleland, ericg@ucar.edu

This article has a companion article which can be found at http://journals.ametsoc.org/doi/abs/10.1175/JTECH-D-20-0069.1.

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