Skip to main content

Abstract

In this paper, I review a variety of approaches to the estimation of extremal properties of a probability distribution or stochastic process. Statistical methods based on exceedances over a high threshold have gained in popularity in recent years, as compared with the much older methods based directly on the extreme value distributions. Nevertheless, there remain some critical questions about their application.

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

Access this chapter

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  • Anderson, C.W. and Dancy, G. (1992), The severity of extreme events. Preprint, Sheffield University.

    Google Scholar 

  • Bloomfield, P., Royle, A. and Yang, Q. (1993), Accounting for meteorological effects in measuring urban ozone levels and trends. Technical Report, National Institute of Statistical Sciences, P.O. Box 14162, Research Triangle Park, N.C. 27709-4162.

    Google Scholar 

  • Coles, S.G. and Tawn, J.A. (1991), Modelling extreme multivariate events. J. R. Statist. Soc. B 53, 377–392.

    MATH  MathSciNet  Google Scholar 

  • Coles, S.G. and Tawn, J.A. (1994), Statistical methods for multivariate extremes: An application to structural design (with discussion). Applied Statistics, to appear.

    Google Scholar 

  • Coles, S.G., Tawn, J.A. and Smith, R.L. (1993), A seasonal Markov model for extremely low temperatures. Submitted to Environmetrics.

    Google Scholar 

  • Cox, D.R. and Lewis, P.A.W. (1966), The Statistical Analysis of Series of Events. Methuen, London.

    MATH  Google Scholar 

  • Davison, A.C. (1984), Modelling excesses over high thresholds, with an application. In Statistical Extremes and Applications, J. Tiago de Oliveira (ed.), Reidel, Dordrecht, 461–482.

    Google Scholar 

  • Davison, A.C. and Smith, R.L. (1990), Models for exceedances over high thresholds (with discussion). J.R. Statist. Soc. B 52, 393–442.

    MATH  MathSciNet  Google Scholar 

  • Dekkers, A.L.M. and de Haan, L. (1989), On the estimation of the extreme-value index and large quantile estimation. Ann. Statist. 17, 1795–1832.

    Article  MATH  MathSciNet  Google Scholar 

  • Dekkers, A.L.M., Einmahl, J.H.J, and de Haan, L. (1989), A moment estimator for the index of an extreme-value distribution. Ann. Statist. 17, 1833–1855.

    Article  MATH  MathSciNet  Google Scholar 

  • Galambos, J. (1987), The Asymptotic Theory of Extreme Order Statistics (2nd. edn.). Krieger, Melbourne, FI. (First edn. published 1978 by John Wiley, New York.)

    Google Scholar 

  • Gumbel, E.J. (1960), Distributions des valeurs extremes en plusiers dimensions. Publ. Inst. Statist. Univ. Paris 9, 171–173.

    MATH  MathSciNet  Google Scholar 

  • Gumbel, E.J. and Mustafi, C.K. (1967), Some analytic properties of bivariate extremal distributions. J. Amer. Statist. Assoc. 62, 569–588.

    Article  MATH  MathSciNet  Google Scholar 

  • Haan, L. de and Resnick, S.I. (1977), Limit theory for multivariate sample extremes. Z. Wahrscheinlichkeitstheorie v. Geb. 40, 317–337.

    Article  MATH  Google Scholar 

  • Haan, L. de and Rootzen, H. (1993), Journal of Statistical Planning and Inference.

    Google Scholar 

  • Hosking, J.R.M., and Wailis, J.R. (1987), Parameter and quantile estimation for the generalized Pareto distribution. Technometrics 29, 339–349.

    Article  MATH  MathSciNet  Google Scholar 

  • Hosking, J.R.M., Wailis, J.R. and Wood, E.F. (1985), Estimation of the generalised extreme-value distribution by the method of probability- weighted moments. Technometrics 27, 251–261.

    Article  MathSciNet  Google Scholar 

  • Hsing, T. (1987), On the characterization of certain point processes. Stoch. Proc. Appl. 26, 297–316.

    Article  MATH  MathSciNet  Google Scholar 

  • Hsing, T., Hüsler, J. and Leadbetter, M.R. (1988), On the exceedance point process for a stationary sequence. Probability Theory and Related Fields 78, 97–112.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  • Hüsler, J. and Reiss, R.-D. (1989), Maxima of normal random vectors: Between independence and complete dependence. Statistics and Probability Letters 7, 283–286.

    Article  MATH  MathSciNet  Google Scholar 

  • Joe, H. (1990), Families of minstable multivariate exponential and multivariate extreme value distributions. Statistics and Probability Letters 9, 75–81.

    Article  MATH  MathSciNet  Google Scholar 

  • Joe, H., Smith, R.L. and Weissman, I. (1992), Bivariate threshold methods for extremes. J.R. Statist. Soc. B., 54, 171–183.

    MATH  MathSciNet  Google Scholar 

  • Leadbetter, M.R. (1983), Extremes and local dependence in stationary sequences. Z. Wahrsch. v. Geb. 65, 291–306.

    Article  MATH  MathSciNet  Google Scholar 

  • Leadbetter, M.R. (1991), On a basis for ‘peaks over thresholds’ modeling. Statistics and Probability Letters 12, 357–362.

    Article  MATH  MathSciNet  Google Scholar 

  • Leadbetter, M.R., Lindgren, G. and Rootzén, H. (1983), Extremes and Related Properties of Random Sequences and Series. Springer Verlag, New York.

    Book  Google Scholar 

  • Leadbetter, M.R., Weissman, I., de Haan, L. and Rootzen, H. (1989), On clustering of high levels in statistically stationary series. Proceedings of the Fourth International Meeting on Statistical Climatology, ed. John Sansom. New Zealand Meteorological Service, P.O. Box 722, Wellington, New Zealand.

    Google Scholar 

  • Marshall, A.W. and Olkin, I. (1967), A multivariate extremal distribution. J. Amer. Statist. Assoc. 62, 30–44.

    Article  MATH  MathSciNet  Google Scholar 

  • Nandagopalan, S. (1990), Multivariate extremes and the estimation of the extremal index. Ph.D. dissertation available as Technical Report Number 315, Center for Stochastic Processes, Department of Statistics, University of North Carolina.

    Google Scholar 

  • Pickands, J. (1971), The two-dimensional Poisson process and extremal processes. J. Appl. Prob. 8, 745–756.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  • Resnick, S. (1975), Weak convergence to extremal processes. Ann. Probab. 3, 951–960.

    Article  MATH  MathSciNet  Google Scholar 

  • Resnick, S. (1987), Extreme Values, Point Processes and Regular Variation. Springer Verlag, New York.

    Google Scholar 

  • Smith, R.L. (1984), Threshold methods for sample extremes. In Statistical Extremes and Applications, J. Tiago de Oliveira (ed.), Reidel Dordrecht, 621–638.

    Google Scholar 

  • Smith, R.L. (1985), Maximum likelihood estimation in a class of nonregular cases. Biometrika, 72, 67–90.

    Article  MATH  MathSciNet  Google Scholar 

  • Smith, R.L. (1987), Estimating tails of probability distributions. Ann. Statist., 15, 1174–1207.

    Article  MATH  MathSciNet  Google Scholar 

  • Smith, R.L. (1989), Extreme value analysis of environmental time series: An application to trend detection in ground-level ozone (with discussion). Statistical Science 4, 367–393.

    Article  MATH  MathSciNet  Google Scholar 

  • Smith, R.L. (1990), Extreme value theory. Chapter 14 of Handbook of Applicable Mathematics, W. Ledermann (ed.). John Wiley, Chichester, pp. 437–472.

    Google Scholar 

  • Smith, R.L. (1991), Max-stable processes and spatial extremes, preprint under revision.

    Google Scholar 

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

    Article  MATH  Google Scholar 

  • Smith, R.L. and Huang, L.-S. (1993), Modeling high threshold exceedances of urban ozone. Technical Report, National Institute of Statistical Sciences, P.O. Box 14162, Research Triangle Park, N.C. 27709 - 4162.

    Google Scholar 

  • Smith, R.L., Tawn, J.A. and Coles, S.G. (1993), Markov chain models for threshold exceedances. Submitted to Biometrika.

    Google Scholar 

  • Smith, R.L. and Weissman, I. (1994), Estimating the extremal index. To appear, J.R. Statist. Soc. B.

    Google Scholar 

  • Tawn, J.A. (1988), Bivariate extreme value theory - models and estimation. Biometrika 75, 397–415.

    Article  MATH  MathSciNet  Google Scholar 

  • Tawn, J.A. (1990), Modelling multivariate extreme value distributions. Biometrika 77, 245–253.

    Article  MATH  Google Scholar 

  • Tiago de Oliveira, J. (1984b), Bivariate models for extremes. In Statistical Extremes and Applications, J. Tiago de Oliveira (ed.), 131–153, Reidel, Dordrecht.

    Google Scholar 

  • Yun, S. (1993), The extremal index of a higher-order stationary Markov chain. Manuscript in preparation.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1994 Kluwer Academic Publishers

About this chapter

Cite this chapter

Smith, R.L. (1994). Multivariate Threshold Methods. In: Galambos, J., Lechner, J., Simiu, E. (eds) Extreme Value Theory and Applications. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-3638-9_14

Download citation

  • DOI: https://doi.org/10.1007/978-1-4613-3638-9_14

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4613-3640-2

  • Online ISBN: 978-1-4613-3638-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics