Skip to main content
SpringerLink
Log in

Estimation of the extremal index using censored distributions

  • Published:
Extremes Aims and scope Submit manuscript

Abstract

The extremal index is an important parameter in the characterization of extreme values of a stationary sequence, since it measures short-range dependence at extreme values, and governs clustering of extremes. This paper presents a novel approach to estimation of the extremal index based on artificial censoring of inter-exceedance times. The censored estimator based on the maximum likelihood method is derived together with its variance, which is estimated from the expected Fisher information measure. In order to evaluate performance of the proposed estimator, a simulation study is carried out for various stationary processes satisfying the local dependence condition D(k)(un). An application to daily maximum temperatures at Uccle, Belgium, is also presented.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Ancona-Navarrete, M.A., Tawn, J.A.: A comparison of methods for estimating the extremal index. Extremes 3(1), 5–38 (2000)

    Article  MathSciNet  Google Scholar 

  • Barndorff-Nielsen, O., Cox, D.: Inference and Asymptotics. Chapman and Hall/CRC, London (1994)

    Book  Google Scholar 

  • Beirlant, J., Goegebeur, Y., Segers, J., Teugels, J., de Waal, D., Ferro, C.: Statistics of extremes: Theory and applications. Wiley (2004)

  • Cai, J.J.: A nonparametric estimator of the extremal arXiv:1911.06674 index. (2019). Accessed 10 December 2019 (2019)

  • Chernick, M.R., Hsing, T., McCormick, W.P.: Calculating the extremal index for a class of stationary sequences. Adv. Appl. Probab. 23, 835–850 (1991)

    Article  MathSciNet  Google Scholar 

  • Cohen, A.: Truncated and Censored Samples. Marcel Dekker, New York (1991) Helsel

    MATH  Google Scholar 

  • Fawcett, L., Walshaw, D.: Estimating return levels from serially dependent extremes. Environmetrics 23(3), 272–283 (2012)

    Article  MathSciNet  Google Scholar 

  • Ferreira, M.: Heuristic tools for the estimation of the extremal index: a comparison of methods. REVSTAT 16(1), 115–136 (2018a)

    MathSciNet  MATH  Google Scholar 

  • Ferreira, M.: Analysis of estimation methods for the extremal index. Electron. J. App. Stat. Anal. 11(1), 296–306 (2018b)

    MathSciNet  Google Scholar 

  • Ferreira, H., Ferreira, M.: Estimating the extremal index through local dependence. Annales de l’Institute Henri Poincare (B) Probability and Statistics 54 (2), 587–605 (2018)

    Article  MathSciNet  Google Scholar 

  • Ferro, C.A.T., Segers, J.: Inference for clusters of extreme values. J. R. Stat. Soc. B Met. 65(2), 545–556 (2003)

    Article  MathSciNet  Google Scholar 

  • Fukutome, S., Liniger, M.A., Süveges, M.: Automatic threshold and run parameter selection: a climatology for extreme hourly precipitation in Switzerland. Theor. Appl. Climatol. 120(3–4), 403–416 (2015)

    Article  Google Scholar 

  • Fukutome, S., Liniger, M.A., Süveges, M.: Correction to: Automatic threshold and run parameter selection: a climatology for extreme hourly precipitation in Switzerland. Theor. Appl. Climatol. 137(3–4), 3215 (2019)

    Article  Google Scholar 

  • Fusek, M., Michálek, J., Vávrová, M.: Evaluation of contamination data with non-detects using censored distributions. Fresenius Environ. Bull. 24, 4165–4172 (2015)

    Google Scholar 

  • Gomes, M.I.: On the estimation of parameters of rare events in environmental time series. In: Barnett, V., Turkman, K. (eds.) Statistics for the Environment 2: Water Related Issues, pp 225–241. Wiley, New York (1993)

  • Gomes, M.I., Hall, A., Miranda, M.C.: Subsampling techniques and the Jackknife methodology in the estimation of the extremal index. Comput. Stat. Data An. 52(4), 2022–2041 (2008)

    Article  MathSciNet  Google Scholar 

  • Helsel, D.R.: Fabricating data: How substituting values for nondetects can ruin results, and what can be done about it. Chemosphere 65, 2434–2439 (2006)

    Article  Google Scholar 

  • Holešovský, J.: Sensitivity assessment and comparison of maxima methods in the estimation of extremal index. In: MITAV 2017, Post-Conference Proceedings of Extended Versions of Selected Papers, pp. 110–120 (2017)

  • Holešovský, J., Fusek, M., Blachut, V., Michálek, J.: Comparison of precipitation extremes estimation using parametric and nonparametric methods. Hydrolog. Sci. J. 61(13), 2376–2386 (2016)

    Article  Google Scholar 

  • Holešovský, J., Fusek, M., Michálek, J.: Extreme value estimation for correlated observations. In: Proceedings of 20th International Conference on Soft Computing MENDEL 2014, pp. 359–364 (2014)

  • Hsing, T.: Estimating the parameters of rare events. Stoch. Proc. Appl. 37, 117–139 (1991)

    Article  MathSciNet  Google Scholar 

  • Joarder, A., Krishna, H., Kundu, D.: Inferences on Weibull parameters with conventional type-I censoring. Comput. Stat. Data Anal. 55, 1–11 (2011)

    Article  MathSciNet  Google Scholar 

  • Klein Tank, A.M.G.: Coauthors Daily dataset of 20th-century surface air temperature and precipitation series for the European Climate Assessment. Int. J. of Climatol. 22, 1441–1453 (2002)

    Article  Google Scholar 

  • Laurini, F., Tawn, J.A.: New estimators for the extremal index and other cluster characteristics. Extremes 6, 189–211 (2003)

    Article  MathSciNet  Google Scholar 

  • Leadbetter, M.R., Lindgren, G., Rootzén, H.: Extremes and related properties of random sequences and series. springer (1983)

  • Leadbetter, M.R., Nandagopalan, S.: On exceedance point processes for stationary sequences under mild oscillation restrictions. Lect. Notes Stat. 51, 69–80 (1989)

    Article  MathSciNet  Google Scholar 

  • Lehmann, E.L., Casella, G.: Theory of point estimation. Springer-Verlag (1998)

  • Northrop, P.J.: An efficient semiparametric maxima estimator of the extremal index. Extremes 18, 585–603 (2015)

    Article  MathSciNet  Google Scholar 

  • Robinson, M.E., Tawn, J.A.: Extremal analysis of processes sampled at different frequencies. J. R. Stat. Soc. B 62(1), 117–135 (2000)

    Article  MathSciNet  Google Scholar 

  • Segers, J.: Extreme events: dealing with dependence. EURANDOM report 2002–036 (2002)

  • Smith, R.L.: The extremal index for a Markov chain. J. Appl. Probab. 29, 37–45 (1992)

    Article  MathSciNet  Google Scholar 

  • Smith, R.L., Weissman, I.: Estimating the extremal index. J. R. Stat. Soc., Ser. B 56, 515–528 (1994)

    MathSciNet  MATH  Google Scholar 

  • Süveges, M.: Likelihood estimation of the extremal index. Extremes 10, 41–55 (2007)

    Article  MathSciNet  Google Scholar 

  • Süveges, M., Davison, A.C.: Model misspecification in peaks over threshold analysis. Ann. Appl. Stat. 4(1), 203–221 (2010)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The paper was supported by specific research project No. FAST-S-19-5878 at Brno University of Technology and by the project LO1408 AdMaS UP - Advanced Materials, Structures and Technologies, supported by Ministry of Education, Youth and Sports of the Czech Republic under the National Sustainability Programme I. The authors would like to thank the editor and the reviewers for a number of good suggestions which helped improve the manuscript.

Author information

Authors and Affiliations

  1. Faculty of Civil Engineering, Institute of Mathematics and Descriptive Geometry, Brno University of Technology, Žižkova 17, 602 00, Brno, Czech Republic

    Jan Holešovský

  2. Faculty of Electrical Engineering and Communication, Department of Mathematics, Brno University of Technology, Technická 2848/8, 61600, Brno, Czech Republic

    Michal Fusek

Authors
  1. Jan Holešovský
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michal Fusek
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jan Holešovský.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Holešovský, J., Fusek, M. Estimation of the extremal index using censored distributions. Extremes 23, 197–213 (2020). https://doi.org/10.1007/s10687-020-00374-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10687-020-00374-3

Keywords

AMS 2000 Subject Classifications

Search

Navigation