Skip to main content
SpringerLink
Log in

Local robust and asymptotically unbiased estimation of conditional Pareto-type tails

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

Abstract

We introduce a non-parametric robust and asymptotically unbiased estimator for the tail index of a conditional Pareto-type response distribution in presence of random covariates. The estimator is obtained from local fits of the extended Pareto distribution to the relative excesses over a high threshold using an adjusted minimum density power divergence estimation technique. We derive the asymptotic properties of the proposed estimator under some mild regularity conditions, and also investigate its finite sample performance with a small simulation experiment. The practical applicability of the methodology is illustrated on a dataset of calcium content measurements of soil samples.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  • Basu A, Harris IR, Hjort NL, Jones MC (1998) Robust and efficient estimation by minimizing a density power divergence. Biometrika 85:549–559

    Article  MATH  MathSciNet  Google Scholar 

  • Beirlant J, Dierckx G, Goegebeur Y, Matthys G (1999) Tail index estimation and an exponential regression model. Extremes 2:177–200

    Article  MATH  MathSciNet  Google Scholar 

  • Beirlant J, Goegebeur Y (2004) Local polynomial maximum likelihood estimation for Pareto-type distributions. J Multivar Anal 89:97–118

    Article  MATH  MathSciNet  Google Scholar 

  • Beirlant J, Goegebeur Y, Segers J, Teugels J (2004) Statistics of extremes—theory and applications. Wiley series in probability and statistics. Wiley, New York

  • Beirlant J, Joossens E, Segers J (2009) Second-order refined peaks-over-threshold modelling for heavy-tailed distributions. J Stat Plan Inference 139:2800–2815

    Article  MATH  MathSciNet  Google Scholar 

  • Byrd RH, Lu P, Nocedal J, Zhu C (1995) A limited memory algorithm for bound constrained optimization. SIAM J Sci Comput 16:1190–1208

    Article  MATH  MathSciNet  Google Scholar 

  • Daouia A, Gardes L, Girard S (2013) On kernel smoothing for extremal quantile regression. Bernoulli 19:2557–2589

    Google Scholar 

  • Daouia A, Gardes L, Girard S, Lekina A (2011) Kernel estimators of extreme level curves. Test 20:311–333

    Article  MATH  MathSciNet  Google Scholar 

  • de Haan L, Ferreira A (2006) Extreme value theory: an introduction. Springer, New York

    Google Scholar 

  • Dierckx G, Goegebeur Y, Guillou A (2013) An asymptotically unbiased minimum density power divergence estimator for the Pareto-tail index. J Multivar Anal 121:70–86

    Google Scholar 

  • Feuerverger A, Hall P (1999) Estimating a tail exponent by modelling departure from a Pareto distribution. Ann Stat 27:760–781

    Article  MATH  MathSciNet  Google Scholar 

  • Fraga Alves MI, Gomes MI, de Haan L (2003) A new class of semi-parametric estimators of the second order parameter. Portugaliae Mathematica 60:193–213

    MATH  MathSciNet  Google Scholar 

  • Goegebeur Y, Guillou A, Schorgen A (2013) Nonparametric regression estimation of conditional tails—the random covariate case. Statistics (to appear). doi:10.1080/02331888.2013.800064

  • Goegebeur Y, Planchon V, Beirlant J, Oger R (2005) Quality assessment of pedochemical data using extreme value methodology. J Appl Sci 5:1092–1102

    Article  Google Scholar 

  • Gomes MI, Martins MJ, Neves M (2007) Improving second order reduced bias extreme value index estimation. REVSTAT 5:177–207

    MATH  MathSciNet  Google Scholar 

  • Hall P (1982) On some simple estimates of an exponent of regular variation. J R Stat Soc Ser B 44:37–42

    MATH  Google Scholar 

  • Hill BM (1975) A simple general approach to inference about the tail of a distribution. Ann Stat 3:1163–1174

    Article  MATH  Google Scholar 

  • Juárez SF, Schucany WR (2004) Robust and efficient estimation for the generalized Pareto distribution. Extremes 7:237–251

    Article  MATH  MathSciNet  Google Scholar 

  • Kim M, Lee S (2008) Estimation of a tail index based on minimum density power divergence. J Multivar Anal 99:2453–2471

    Article  MATH  Google Scholar 

  • Laroche J, Oger R (1999) Base de données sols. Première synthèse. Faculté Universitaire des Sciences Agronomique de Gembloux, Unité de Géopédology, Requasud asbl, Gembloux

  • Lehmann EL, Casella G (1998) Theory of point estimation. Springer, New York

    MATH  Google Scholar 

  • Pickands J (1975) Statistical inference using extreme order statistics. Ann Stat 3:119–131

    Article  MATH  MathSciNet  Google Scholar 

  • Piessens R, de Doncker-Kapenga E, Uberhuber C, Kahaner D (1983) Quadpack: a subroutine package for automatic integration. Springer, New York

    Book  MATH  Google Scholar 

  • van der Vaart AW (2007) Asymptotic statistics. Cambridge series in statistical and probabilistic mathematics. Cambridge University Press, Cambridge

  • Wang H, Li D, He X (2012) Estimation of high conditional quantiles for heavy-tailed distributions. J Am Stat Assoc 107:1453–1464

    Article  MATH  MathSciNet  Google Scholar 

  • Wang H, Tsai CL (2009) Tail index regression. J Am Stat Assoc 104:1233–1240

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

The authors are very grateful to the two anonymous referees for their very constructive comments on the paper. The suggestions have definitely improved the presentation of the material.

Author information

Authors and Affiliations

  1. KU Leuven, Faculty of Economics and Business, Campus Brussel, Research Centre for Mathematical Economics, Econometrics and Statistics, Warmoesberg 26, 1000 , Brussels, Belgium

    Goedele Dierckx

  2. Department of Mathematics and Computer Science, University of Southern Denmark, Campusvej 55, 5230 , Odense M, Denmark

    Yuri Goegebeur

  3. Institut Recherche Mathématique Avancée, UMR 7501, Université de Strasbourg et CNRS, 7 rue René Descartes, 67084 , Strasbourg Cedex, France

    Armelle Guillou

Authors
  1. Goedele Dierckx
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yuri Goegebeur
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Armelle Guillou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yuri Goegebeur.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (pdf 270 KB)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dierckx, G., Goegebeur, Y. & Guillou, A. Local robust and asymptotically unbiased estimation of conditional Pareto-type tails. TEST 23, 330–355 (2014). https://doi.org/10.1007/s11749-013-0350-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11749-013-0350-6

Keywords

Mathematics Subject Classification (2000)

Search

Navigation