Skip to main content
SpringerLink
Log in

On extremal dependence: some contributions

  • Original Paper
  • Published:
TEST Aims and scope Submit manuscript

Abstract

The usual coefficients of tail dependence are based on exceedances of high values. These extremal events are useful and widely used in literature but an adverse situation may also occur with the upcrossing of a high level. In this context we define upcrossings-tail dependence coefficients and analyze all types of dependence coming out. We will prove that these coefficients are related to multivariate tail dependence coefficients already known in literature. We shall see that the upcrossings-tail dependence coefficients have the interesting feature of congregating both “temporal” and “spatial” dependence.

The coefficients of tail dependence can also be applied to stationary sequences and hence measure the tail dependence in time. Results concerning connections with the extremal index and the upcrossings index as well as with local dependence conditions will be stated. Several illustrative examples will be exploited and a small note on inference will be given by presenting estimators derived from the stated results and respective properties.

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

  • Alpuim MT (1989) An extremal markovian sequence. J Appl Probab 26:219–232

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Billingsley P (1968) Convergence of probability measures. Wiley, New York

    MATH  Google Scholar 

  • Canto e Castro L (1992) Sobre a teoria assintótica de extremos. PhD thesis, FCUL

  • Coles S, Heffernan J, Tawn JA (1999) Dependence measures for extreme value analyses. Extremes 2:339–365

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Draisma G (2001) Duration of extremes at sea. In: Parametric and semi-parametric methods in E.V.T. PhD thesis, Erasmus University, pp 137–143

  • Draisma G, Drees H, Ferreira A, de Haan L (2004) Bivariate tail estimation: dependence in asymptotic independence. Bernoulli 10:251–280

    Article  MathSciNet  MATH  Google Scholar 

  • Drees H (2003) Extreme quantile estimation for dependent data with applications to finance. Bernoulli 9:617–657

    Article  MathSciNet  MATH  Google Scholar 

  • Ferreira H (2006) The upcrossings index and the extremal index. J Appl Probab 43(4):927–937

    Article  MathSciNet  MATH  Google Scholar 

  • Ferreira H (2011a) Dependence between two multivariate extremes. Stat Probab Lett 81(5):586–591

    Article  MATH  Google Scholar 

  • Ferreira M (2011b, submitted) On tail dependence: a characterization for first order max-autoregressive processes. In: Notes and communications of CEAUL, 09/11

  • Ferreira M, Canto e Castro L (2008) Tail and dependence behaviour of levels that persist for a fixed period of time. Extremes 11:113–133

    Article  MathSciNet  MATH  Google Scholar 

  • Frahm G, Junker M, Schmidt R (2005) Estimating the tail-dependence coefficient: properties and pitfalls. Insur Math Econ 37:80–100

    Article  MathSciNet  MATH  Google Scholar 

  • Heffernan JE, Tawn JA, Zhang Z (2007) Asymptotically (in)dependent multivariate maxima of moving maxima processes. Extremes 10:57–82

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  • Leadbetter MR, Nandagopalan S (1989) In: Hüsler J, Reiss R-D (eds) Extreme value theory on exceedance point processes for stationary sequences under mild oscillation restrictions. Springer, Berlin, pp 69–80

    Google Scholar 

  • Ledford A, Tawn JA (1996) Statistics for near independence in multivariate extreme values. Biometrika 83:169–187

    Article  MathSciNet  MATH  Google Scholar 

  • Ledford A, Tawn JA (1997) Modelling Dependence within joint tail regions. J R Stat Soc B 59:475–499

    Article  MathSciNet  MATH  Google Scholar 

  • Peng L (1999) Estimation of the coefficient of the tail dependence in bivariate extremes. Stat Probab Lett 43:399–409

    Article  MATH  Google Scholar 

  • Schmidt R, Stadtmüller U (2006) Nonparametric estimation of tail dependence. Scand J Stat 33:307–335

    Article  MATH  Google Scholar 

  • Sibuya M (1960) Bivariate extreme statistics. Ann Inst Stat Math 11:195–210

    Article  MathSciNet  MATH  Google Scholar 

  • Sklar A (1959) Fonctions de répartition à n dimensions et leurs marges. Research report, University of Paris, Institute of Statistics, France

  • Tiago de Oliveira J (1962/63) Structure theory of bivariate extremes, extensions. Est. Mat., Estat. e Econ. 7, 165–195

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. DMAT-CMAT, U. Minho, Braga, Portugal

    Marta Ferreira

  2. Dep. Mat., U. Beira Interior, Covilhã, Portugal

    Helena Ferreira

Authors
  1. Marta Ferreira
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Helena Ferreira
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marta Ferreira.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ferreira, M., Ferreira, H. On extremal dependence: some contributions. TEST 21, 566–583 (2012). https://doi.org/10.1007/s11749-011-0261-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11749-011-0261-3

Keywords

Mathematics Subject Classification (2000)

Search

Navigation