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, Volume 23, Issue 2, pp 388–408 | Cite as

Extremal properties of M4 processes

  • A. P. Martins
  • H. Ferreira
Original Paper

Abstract

The existence of data with different dependence structures motivates the development of models which can capture several types of dependence. In this paper we consider a stationary sequence of moving maxima vectors \(\{{\mathbf {X}}_n=(X_{n1},\ldots ,X_{nd})\}_{n\ge 1}\) having innovations \({\mathbf {Z}}_{l,n}=(Z_{l,n,1},\ldots ,Z_{l,n,d})\) with totally dependent margins for certain values of \(l,\) \(l\in I_1,\) and independent margins for the remaining values of \(l,\) \(l\in I_2.\) We obtain in this way a \(d\)-dimensional process \(\{{\mathbf {X}}_n\}_{n\ge 1}\) whose extremal dependence, measured by the tail dependence coefficients, lies between asymptotic independence and total dependence. The extremal properties of these M4 processes are studied and examined both theoretically and through simulation studies: we derive the multivariate extremal index, the tail dependence coefficients and co-movement indices.

Keywords

Multivariate extremes M4 processes Tail dependence  Extremal index Co-movement index 

Mathematics Subject Classification (2000)

60G70 

Notes

Acknowledgments

The authors are grateful to the reviewers for their insightful comments and suggestions. This research was partially supported by the research unit “Centro de Matemática” of the University of Beira Interior and the research project PEst-OE/MAT/UI0212/2014 through the Foundation for Science and Technology (FCT) co-financed by FEDER/COMPETE.

References

  1. Deheuvels P (1983) Point processes and multivariate extreme values. J Multivar Anal 13:257–272CrossRefzbMATHMathSciNetGoogle Scholar
  2. Deheuvels P (1985) Point processes and multivariate extreme values II. In: Krishnaiah PR (ed) Multivariate analysis VI. North Holland, Amsterdam, pp 145–164Google Scholar
  3. Falk M, Ticky D (2012a) Asymptotic conditional distribution of exceedance counts. Adv Appl Probab 44(1):270–291CrossRefzbMATHGoogle Scholar
  4. Falk M, Ticky D (2012b) Asymptotic conditional distribution of exceedance counts: fragility index with different margins. Ann Inst Stat Math 64(5):1071–1085CrossRefzbMATHGoogle Scholar
  5. Ferreira H (2012) Multivariate maxima of moving multivariate maxima. Stat Probab Lett 82:1489–1496CrossRefzbMATHGoogle Scholar
  6. Ferreira H, Ferreira M (2012) On extremal dependence of block vectors. Kybernetika 48(5):988–1006zbMATHMathSciNetGoogle Scholar
  7. Ferreira H, Ferreira M (2012b). Extremal behavior of pMAX processes (math.ST)Google Scholar
  8. Geluk JL, De Haan L, De Vries CG (2007) Week and strong financial fragility. Tinbergen Institute Discussion Paper, TI 2007–023/2Google Scholar
  9. Heffernan J, Tawn J, Zhang Z (2007) Asymptotically (in)dependent multivariate maxima of moving maxima processes. Extremes 10:57–82CrossRefzbMATHMathSciNetGoogle Scholar
  10. Li H (2009) Orthant tail dependence of multivariate extreme value distributions. J Multivar Anal 100(1):243–256CrossRefzbMATHGoogle Scholar
  11. Li H, Sun Y (2009) Tail dependece for heavy-tailed scale mixtures of multivariate distributions. J Appl Probab 46(4):925–937CrossRefzbMATHMathSciNetGoogle Scholar
  12. Liebscher E (2008) Construction of asymmetric multivariate copulas. J Multivar Anal 99:2234–2250CrossRefzbMATHMathSciNetGoogle Scholar
  13. Marshall AW, Olkin I (1983) Domains of attraction of multivariate extreme value distributions. Ann Probab 11:168–177CrossRefzbMATHMathSciNetGoogle Scholar
  14. Martins AP, Ferreira H (2005) The multivariate extremal index and the dependence structure of a multivariate extreme value distribution. Test 14(2):433–448CrossRefzbMATHMathSciNetGoogle Scholar
  15. Robert C (2008) Estimating the multivariate extremal index function. Bernoulli 14(4):1027–1064CrossRefzbMATHMathSciNetGoogle Scholar
  16. Sibuya M (1960) Bivariate extreme statistics. Ann Inst Stat Math 11:195–210CrossRefzbMATHMathSciNetGoogle Scholar
  17. Smith RL, Weissman I (1996) Characterization and estimation of the multivariate extremal index. Technical report, University of North Carolina at Chapel Hill, NC. http://www.stat.unc.edu/postscript/rs/extremal.pdf
  18. Tiago de Oliveira J (1962/3) Structure theory of bivariate extremes; extensions. Estudos de Mat Estat e Econ 7:165–195Google Scholar
  19. Zhang Z, Shinki K (2007) Extreme co-movement and extreme impacts in high frequency data in finance. J Bank Financ 31:1399–1415CrossRefGoogle Scholar
  20. Zhang Z (2009) On approximating max-stable processes and constructing extremal copula functions. Stat Inf Stoch Proc 12(1):89–114CrossRefzbMATHGoogle Scholar
  21. Zhang Z, Smith RL (2004) The behavior of multivariate maxima of moving maxima processes. J Appl Probab 41(4):1113–1123CrossRefzbMATHMathSciNetGoogle Scholar
  22. Zhang Z, Smith RL (2010) On the estimation and application of max-stable processes. J Stat Plan Inf 140(5):1135–1153CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Sociedad de Estadística e Investigación Operativa 2014

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Beira InteriorCovilhãPortugal

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