Advertisement

TEST

, Volume 21, Issue 2, pp 330–354 | Cite as

Estimation of the third-order parameter in extreme value statistics

  • Yuri GoegebeurEmail author
  • Tertius de Wet
Original Paper

Abstract

We introduce a class of estimators for the third-order parameter in extreme value statistics when the distribution function underlying the data is heavy tailed. For appropriately chosen intermediate sequences of upper order statistics, consistency is established under the third-order tail condition and asymptotic normality under the fourth-order tail condition. Simulation experiments illustrate the finite sample behavior of some selected estimators.

Keywords

Pareto-type distribution Third-order parameter Fourth-order condition 

Mathematics Subject Classification (2000)

62G20 62G30 62G32 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Beirlant J, Dierckx G, Goegebeur Y, Matthys G (1999) Tail index estimation and an exponential regression model. Extremes 2:177–200 MathSciNetzbMATHCrossRefGoogle Scholar
  2. Beirlant J, Dierckx G, Guillou A, Stărică C (2002) On exponential representations of log-spacings of extreme order statistics. Extremes 5:157–180 MathSciNetzbMATHCrossRefGoogle Scholar
  3. Beirlant J, Goegebeur Y, Segers J, Teugels J (2004) Statistics of extremes—theory and applications. Wiley series in probability and statistics zbMATHCrossRefGoogle Scholar
  4. Beirlant J, Goegebeur Y, Verlaak R, Vynckier P (1998) Burr regression and portfolio segmentation. Insur Math Econ 23:231–250 zbMATHCrossRefGoogle Scholar
  5. Beirlant J, Vynckier P, Teugels J (1996) Tail index estimation, Pareto quantile plots, and regression diagnostics. J Am Stat Assoc 91:1659–1667 MathSciNetzbMATHGoogle Scholar
  6. Bingham NH, Goldie CM, Teugels JL (1987) Regular variation. Cambridge University Press, Cambridge zbMATHGoogle Scholar
  7. Caeiro F, Gomes MI (2006) A new class of estimators of a “scale” second order parameter. Extremes 9:193–211 MathSciNetzbMATHCrossRefGoogle Scholar
  8. Caeiro F, Gomes MI (2008) Minimum-variance reduced-bias tail index and high quantile estimation. REVSTAT Stat J 6:1–20 MathSciNetzbMATHGoogle Scholar
  9. Chernoff H, Gastwirth JL, Johns MV (1967) Asymptotic distribution of linear combinations of functions of order statistics with applications to estimation. Ann Math Stat 38:52–72 MathSciNetzbMATHCrossRefGoogle Scholar
  10. Ciuperca G, Mercadier C (2010) Semi-parametric estimation for heavy tailed distributions. Extremes 13:55–87 MathSciNetzbMATHCrossRefGoogle Scholar
  11. de Haan L, Ferreira A (2006) Extreme value theory: an introduction. Springer, Berlin zbMATHGoogle Scholar
  12. de Haan L, Stadtmüller U (1996) Generalized regular variation of second order. J Aust Math Soc A 61:381–395 zbMATHCrossRefGoogle Scholar
  13. Draisma G (2000) Parametric and semi-parametric methods in extreme value theory. Tinbergen Institute research series, vol 239. Erasmus Universiteit Rotterdam, Rotterdam Google Scholar
  14. Drees H, Kaufmann E (1998) Selecting the optimal sample fraction in univariate extreme value estimation. Stoch Process Appl 75:149–172 MathSciNetzbMATHCrossRefGoogle Scholar
  15. Feuerverger A, Hall P (1999) Estimating a tail exponent by modelling departure from a Pareto distribution. Ann Stat 27:760–781 MathSciNetzbMATHCrossRefGoogle Scholar
  16. Fraga Alves MI, de Haan L, Lin T (2006) Third order extended regular variation. Publ Inst Math 80:109–120 CrossRefGoogle Scholar
  17. Fraga Alves MI, Gomes MI, de Haan L (2003) A new class of semi-parametric estimators of the second order parameter. Port Math 60:193–213 MathSciNetzbMATHGoogle Scholar
  18. Geluk JL, de Haan L (1987) Regular variation, extensions and Tauberian theorems. CWI tract, vol 40. Center for Mathematics and Computer Science, Amsterdam zbMATHGoogle Scholar
  19. Gnedenko BV (1943) Sur la distribution limite du terme maximum d’une série aléatoire. Ann Math 44:423–453 MathSciNetzbMATHCrossRefGoogle Scholar
  20. Goegebeur Y, Beirlant J, de Wet T (2010) Kernel estimators for the second order parameter in extreme value statistics. J Stat Plan Inference 140:2632–2652 zbMATHCrossRefGoogle Scholar
  21. Goegebeur Y, de Wet T (2011) On the estimation of higher order distributional parameters in extreme value statistics. Technical report available at http://www.stat.sdu.dk/matstat/yuri/yuri.goegebeur.htm
  22. Gomes MI, de Haan L, Peng L (2002) Semi-parametric estimation of the second order parameter in statistics of extremes. Extremes 5:387–414 MathSciNetCrossRefGoogle Scholar
  23. Gomes MI, de Haan L, Rodrigues LH (2008) Tail index estimation for heavy-tailed models: accommodation of bias in weighted log-excesses. J R Stat Soc B 70:31–52 zbMATHGoogle Scholar
  24. Gomes MI, Martins MJ (2002) “Asymptotically unbiased” estimators of the tail index based on external estimation of the second order parameter. Extremes 5:5–31 MathSciNetzbMATHCrossRefGoogle Scholar
  25. Guillou A, Hall P (2001) A diagnostic for selecting the threshold in extreme value analysis. J R Stat Soc B 63:293–305 MathSciNetzbMATHCrossRefGoogle Scholar
  26. Hall P, Welsh AH (1985) Adaptive estimates of parameters of regular variation. Ann Stat 13:331–341 MathSciNetzbMATHCrossRefGoogle Scholar
  27. Rényi A (1953) On the theory of order statistics. Acta Math Acad Sci Hung 4:191–231 zbMATHCrossRefGoogle Scholar
  28. Rodriguez RN (1977) A guide to the Burr type XII distributions. Biometrika 64:129–134 MathSciNetzbMATHCrossRefGoogle Scholar
  29. Wang XQ, Cheng SH (2005) General regular variation of n-th order and the 2nd order Edgeworth expansion of the extreme value distribution (I). Acta Math Sin 21:1121–1130 MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Sociedad de Estadística e Investigación Operativa 2011

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceUniversity of Southern DenmarkOdense MDenmark
  2. 2.Department of Statistics and Actuarial ScienceUniversity of StellenboschMatielandSouth Africa

Personalised recommendations