Computational Statistics

, Volume 32, Issue 4, pp 1481–1513 | Cite as

Multivariate moment based extreme value index estimators

  • Matias Heikkilä
  • Yves Dominicy
  • Pauliina Ilmonen
Original Paper
  • 128 Downloads

Abstract

Modeling extreme events is of paramount importance in various areas of science—biostatistics, climatology, finance, geology, and telecommunications, to name a few. Most of these application areas involve multivariate data. Estimation of the extreme value index plays a crucial role in modeling rare events. There is an affine invariant multivariate generalization of the well known Hill estimator—the separating Hill estimator. However, the Hill estimator is only suitable for heavy tailed distributions. As in the case of the separating multivariate Hill estimator, we consider estimation of the extreme value index under the assumptions of multivariate ellipticity and independent identically distributed observations. We provide affine invariant multivariate generalizations of the moment estimator and the mixed moment estimator. These estimators are suitable for both light and heavy tailed distributions. Asymptotic properties of the new extreme value index estimators are derived under multivariate elliptical distribution with known location and scatter. The effect of replacing true location and scatter by estimates is examined in a thorough simulation study. We also consider two data examples: one financial application and one meteorological application.

Keywords

Elliptical distribution Moment estimator Mixed moment estimator 

Notes

Acknowledgements

Matias Heikkil acknowledges financial support from the Magnus Ehrnrooth foundation. Yves Dominicy acknowledges financial support via a Mandat de Chargé de Recherches FNRS. The authors wish to thank the anonymous referees for their insightful comments that helped to improve the article greatly.

References

  1. Bollerslev T, Wooldridge J (1992) Quasi-maximum likelihood estimation and inference in dynamic models with time-varying covariances. Econom Rev 11(2):143–172CrossRefMATHMathSciNetGoogle Scholar
  2. Dekkers ALM, Einmahl JHJ, Haan LD (1989) A moment estimator for the index of an extreme-value distribution. Ann Stat 17(4):1833–1855CrossRefMATHMathSciNetGoogle Scholar
  3. Dominicy Y, Ogata H, Veredas D (2013) Inference for vast dimensional elliptical distributions. Comput Stat 28(4):1853–1880CrossRefMATHMathSciNetGoogle Scholar
  4. Dominicy Y, Ilmonen P, Veredas D (2015) Multivariate Hill estimators. Int Stat Rev. doi: 10.1111/insr.12120
  5. Fraga Alves MI, Gomes MI, de Haan L, Neves C (2009) Mixed moment estimator and location invariant alternatives. Extremes 12(2):149–185CrossRefMATHMathSciNetGoogle Scholar
  6. de Haan L, Ferreira A (2006) Extreme value theory: an introduction. (Springer Series in Operations Research and Financial Engineering). Springer, New YorkCrossRefGoogle Scholar
  7. Haeusler E, Teugels JL (1985) On asymptotic normality of Hill’s estimator for the exponent of regular variation. Ann Statistics 13(2):743–756CrossRefMATHMathSciNetGoogle Scholar
  8. Hall P (1982) On some simple estimates of an exponent of regular variation. J R Stat Soc Ser B (Methodol) 44(1):37–42MATHMathSciNetGoogle Scholar
  9. Hill BM (1975) A simple general approach to inference about the tail of a distribution. Ann Stat 3(5):1163–1174CrossRefMATHMathSciNetGoogle Scholar
  10. Hill JB (2015) Tail index estimation for a filtered dependent time series. Stat Sin 25(2):609–629MATHMathSciNetGoogle Scholar
  11. Hult H, Lindskog F (2002) Multivariate extremes, aggregation and dependence in elliptical distributions. Adv Appl Probab 34(3):587–608CrossRefMATHMathSciNetGoogle Scholar
  12. Mason DM (1982) Laws of large numbers for sums of extreme values. Ann Prob 10(3):754–764CrossRefMATHMathSciNetGoogle Scholar
  13. Mikosch T (1999) Regular Variation, Subexponentiality and Their Applications in Probability Theory. EURANDOM report, Eindhoven University of TechnologyGoogle Scholar
  14. Rousseeuw P (1985) Multivariate estimation with high breakdown point. In: Grossmann W, Pug G, Vincze I, Wertz W (eds) Math Stat Appl. Reidel, Dordrecht, pp 283–297CrossRefGoogle Scholar
  15. Rousseeuw PJ, Driessen KV (1999) A fast algorithm for the minimum covariance determinant estimator. Technometrics 41(3):212–223CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of Mathematics and Systems AnalysisAalto University School of ScienceEspooFinland
  2. 2.Solvay Brussels School of Economics and Management, ECARESUniversité libre de BruxellesBrusselsBelgium

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