43
Views
4
CrossRef citations to date
0
Altmetric
Original Articles

Location-invariant reduced-bias tail index estimation under a third-order framework

ORCID Icon &
Pages 206-230 | Received 15 Dec 2016, Accepted 11 Jun 2017, Published online: 21 Jul 2017

References

  • Araújo Santos, P., M. I. Fraga Alves, and M. I. Gomes. 2006. Peaks over random threshold methodology for tail index and quantile estimation. Revstat 4 (3):227–47.
  • Beirlant, J., G. Dierckx, Y. Goegebeur, and G. Matthys. 1999. Tail index estimation and an exponential regression model. Extremes 2 (2):177–200.
  • Beirlant, J., F. Caeiro, and M. I. Gomes. 2012. An overview and open research topics in the field of statistics of univariate extremes. Revstat 10 (1):1–31.
  • Brilhante, F., M. I. Gomes, and D. Pestana. 2013. A simple generalization of the Hill estimator. Computational Statistics and Data Analysis 57 (1):518–35.
  • Caeiro, F., and M. I. Gomes. 2006. A new class of estimators of a “scale” second order parameter. Extremes 9:193–211.
  • Caeiro, F., and M. I. Gomes. 2015. Threshold selection in extreme value analysis. In Extreme value modeling and risk analysis, ed. D. Dey and J. Yan, 69–87. Boca Raton, FL: Chapman-Hall/CRC.
  • Caeiro, F., M. I. Gomes, and D. D. Pestana. 2005. Direct reduction of bias of the classical Hill estimator. Revstat 3 (2):111–136.
  • Caeiro, F., M.I. Gomes, and L. Henriques-Rodrigues. 2009. Reduced-bias tail index estimators under a third-order framework. Communications in Statistics—Theory and Methods 38 (7):1019–40.
  • Caeiro, F., M. I. Gomes, J. Beirlant, and T. de Wet. 2016. Mean-of-order-p reduced-bias extreme value index estimation under a third-order framework. Extremes 19 (4):561–89.
  • Dekkers, A., J. Einmahl, and L. de Haan. 1989. A moment estimator for the index of an extreme-value distribution. Annals of Statistics 17:1833–55.
  • Drees, H. 2003. Extreme quantile estimation for dependent data, with applications to finance. Bernoulli 9 (4):617–57.
  • Feuerverger, A., and P. Hall. 1999. Estimating a tail exponent by modelling departure from a Pareto distribution. Annals of Statistics 27 (2):760–81.
  • Figueiredo, F., M. I. Gomes, L. Henriques-Rodrigues, and C. Miranda. 2012. A computational study of a quasi-PORT methodology for VaR based on second-order reduced-bias estimation. Journal of Statistical Computing and Simulation 82 (4):587–02.
  • Figueiredo, F., M. I. Gomes, and L. Henriques-Rodrigues. 2017. Value-at-risk estimation and the PORT mean-of-order-p methodology. Revstat 15 (2):187–204.
  • Fraga Alves, M. I., M. I. Gomes, and L. de Haan. 2003. A new class of semi-parametric estimators of the second order parameter. Portugaliae Mathematica 60 (2):194–213.
  • Fraga Alves, M. I., L. de Haan, and T. Lin. 2006. Third order extended regular variation. Publications de l’Institut Mathématique 80 (94):109–120.
  • Geluk, J., and L. de Haan. 1987. Regular variation, extensions and tauberian theorems. CWI Tract 40. Center for Mathematics and Computer Science, Amsterdam, The Netherlands.
  • Gnedenko, B. V. 1943. Sur la distribution limite du terme maximum d’une série aléatoire. Annals of Mathematics 44:423–53.
  • Gomes, M. I., and A. Guillou. 2015. Extreme value theory and statistics of univariate extremes: A review. International Statistical Review 83:263–92.
  • Gomes, M. I., and L. Henriques-Rodrigues. 2012. Adaptive PORT-MVRB estimation of the extreme value index. In Studies in theoretical and applied statistics: Subseries B: Recent developments in modeling and applications in statistics. ed. P. Oliveira, M. G. Temido, C. Henriques, and M. Vichi, 117–25, Springer-Verlag Berlin Heidelberg.
  • Gomes, M. I., and M. J. Martins. 2002. “Asymptotically unbiased” estimators of the tail index based on external estimation of the second order parameter. Journal of Statistics, Planning and Inference 93:161–80.
  • Gomes, M. I., and D. D. Pestana. 2007a. A simple second order reduced-bias tail index estimator. Journal of Statistical Computing and Simulation 77 (6):487–504.
  • Gomes, M. I., and D. Pestana. 2007b. A sturdy reduced-bias extreme quantile (VaR) estimator. Journal of the American Statistical Association 102 (477):280–92.
  • Gomes, M. I., M. J. Martins, and M. M. Neves. 2000. Alternatives to a semi-parametric estimator of parameters of rare events— The Jackknife methodology. Extremes 3 (3):207–29.
  • Gomes, M. I., L. de Haan, and L. Peng. 2002. Semi-parametric estimation of the second order parameter—asymptotic and finite sample behaviour. Extremes 5 (4):387–414.
  • Gomes, M. I., M. J. Martins, and M. M. Neves. 2007a. Improving second order reduced-bias tail index estimation. Revstat 5 (2):177–207.
  • Gomes, M. I., R.-D. Reiss, and M. Thomas. 2007b. Reduced-bias estimation. In Statistical analysis of extreme values with applications to insurance, finance, hydrology and other fields, ed. R.-D. Reiss, and M. Thomas, 3rd ed., 189–204, Basel, Switzerland: Birkhäuser Verlag.
  • Gomes, M. I., L. Canto e Castro, M. I. Fraga Alves, and D. Pestana. 2008a. Statistics of extremes for iid data and breakthroughs in the estimation of the extreme value index: Laurens de Haan leading contributions. Extremes 11 (1):3–34.
  • Gomes, M. I., M. I. Fraga Alves, and P. Araújo Santos. 2008b. PORT Hill and moment estimators for heavy-tailed models. Communications in Statistics—Simulation and Computation 37:1281–306.
  • Gomes, M. I., L. de Haan, and L. Henriques Rodrigues. 2008c. Tail index estimation for heavy- tailed models: Accommodation of bias in weighted log-excesses. Journal of the Royal Statistical Society B70 (1):31–52.
  • Gomes, M. I., L. Henriques-Rodrigues, and C. Miranda. 2011a. Reduced-bias location-invariant extreme value index estimation: A simulation study. Communications in Statistics—Simulation and Computation 40 (3):424–447.
  • Gomes, M. I., S. Mendonça, and D. Pestana. 2011b. Adaptive reduced-bias tail index and VaR estimation via the bootstrap methodology. Commication in Statistics—Theory and Methods 40 (16):2946–68.
  • Gomes, M. I., F. Figueiredo, and M. Neves. 2012. Adaptive estimation of heavy right tails: resampling-based methods in action. Extremes 15:463–89.
  • Gomes, M. I., L. Henriques-Rodrigues, M. I. Fraga Alves, and B. G. Manjunath. 2013. Adaptive PORT-MVRB estimation: an empirical comparison of two heuristic algorithms. Journal Statistical Computing and Simulation 83 (6):1129–44.
  • Gomes, M. I., F., Caeiro, L. Henriques-Rodrigues, and B. G. Manjunath. 2016a. Bootstrap methods in statistics of extremes. In Handbook of extreme value theory and its applications to finance and insurance. ed. F. Longin, 117–38. Handbook Series in Financial Engineering and Econometrics (Ruey Tsay Adv.Ed.). Hoboken, NJ: John Wiley and Sons.
  • Gomes, M. I., L. Henriques-Rodrigues, and B. G. Manjunath. 2016b. Mean-of-order-p location invariant extreme value index estimation. Revstat 14 (3):273–96.
  • de Haan, L., 1984. Slow variation and characterization of domains of attraction. In Statistical extremes and applications, ed. T. de Oliveira, 31–48. Dordrecht, The Netherlands: D. Reidel.
  • de Haan, L., and L. Peng. 1998. Comparison of tail index estimators. Statistica Neerlandica 52:60–70.
  • Henriques-Rodrigues, L., and M. I. Gomes. 2009. High quantile estimation and the PORT methodology. Revstat 7 (3):245–64.
  • Henriques-Rodrigues, L., and M. I. Gomes. 2012. A note on the PORT methodology in the estimation of a shape second-order parameter. In Studies in theoretical and applied statistics: Subseries B: Recent developments in modeling and applications in statistics. ed. P. Oliveira, M. G. Temido, C. Henriques, and M. Vichi, 127–137. Springer-Verlag Berlin Heidelberg.
  • Henriques-Rodrigues, L., M. I. Gomes, M. I. Fraga Alves, and C. Neves. 2014. PORT-estimation of a shape second-order parameter. Revstat 12 (3):299–328.
  • Henriques-Rodrigues, L., M. I. Gomes, and B. G. Manjuntah. 2015. Estimation of a scale second-order parameter related to the PORT methodology. Journal of Statistical Theory and Practice 9 (3):571–99.
  • Hill, B. 1975. A simple general approach to inference about the tail of a distribution. Annals of Statistics 3:1163–74.
  • Jones, M. C., and M. J. Faddy. 2003. A skew extension of the t-distribution, with applications. Journal of the Royal Statistics Society B65 (1):159–74.
  • McNeil, A., and R. Frey. 2000. Estimation of tail-related risk measures for heteroscedastic financial times series: An extreme value approach. Journal of Empirical Finance 7:271–300.
  • Reiss, R.-D., and M. Thomas. 2007. Statistical analysis of extreme values, with application to insurance, finance, hydrology and other fields, 3rd eds. Basel, Switzerland: Birkhäuser Verlag.
  • van der Vaart, A. W. 1998. Asymptotic statistics. New York, NY: Cambridge University Press.
  • Wang, X., and S. Cheng. 2005. General regular variation of n-th order and the 2nd order Edgeworth expansion of the extreme value distribution. Acta Mathematica Sinica 21 (5):1121–30.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.