Advertisement

Estimation of extreme conditional quantiles under a general tail-first-order condition

  • Laurent Gardes
  • Armelle GuillouEmail author
  • Claire Roman
Article
  • 8 Downloads

Abstract

We consider the estimation of an extreme conditional quantile. In a first part, we propose a new tail condition in order to establish the asymptotic distribution of an extreme conditional quantile estimator. Next, a general class of estimators is introduced, which encompasses, among others, kernel or nearest neighbors types of estimators. A unified theorem of the asymptotic normality for this general class of estimators is provided under the new tail condition and illustrated on the different well-known examples. A comparison between different estimators belonging to this class is provided on a small simulation study and illustrated on a real dataset on earthquake magnitudes.

Keywords

Extreme quantile Local estimation Asymptotic normality 

Notes

Acknowledgements

The authors would like to thank the reviewers, the associate editor and editor for their helpful comments and suggestions that led to substantial improvement of the paper.

Supplementary material

10463_2019_713_MOESM1_ESM.pdf (186 kb)
Supplementary material 1 (pdf 186 KB)

References

  1. Berlinet, A., Gannoun, A., Matzner-Lober, E. (2001). Asymptotic normality of convergent estimates of conditional quantiles. Statistics—A Journal of Theoretical and Applied Statistics, 35(2), 139–169.MathSciNetzbMATHGoogle Scholar
  2. Daouia, A., Gardes, L., Girard, S. (2013). On kernel smoothing for extremal quantile regression. Bernoulli, 19(5B), 2557–2589.MathSciNetCrossRefzbMATHGoogle Scholar
  3. de Haan, L., Ferreira, A. (2006). Extreme value theory: An introduction., Springer series in operations research and financial engineering New York: Springer.Google Scholar
  4. Dony, J., Einmahl, U. (2009). Uniform in bandwidth consistency of kernel regression estimators at fixed point. In High dimensional probability V: The Luminy volume (pp. 308–325). Beachwood, OH: Institute of Mathematical Statistics.Google Scholar
  5. Dziewonski, A. M., Chou, T. A., Woodhouse, J. H. (1981). Determination of earthquake source parameters from waveform data for studies of global and regional seismicity. Journal of Geophysical Research, 86, 2825–2852.CrossRefGoogle Scholar
  6. Ekström, G., Nettles, M., Dziewonski, A. M. (2012). The global CMT project 2004–2010: Centroid-moment tensors for 13,017 earthquakes. Physics of the Earth and Planetary Interiors, 200–201, 1–9.CrossRefGoogle Scholar
  7. Embrechts, P., Klüppelberg, C., Mikosch, T. (1997). Modelling extremal events., For insurance and finance Berlin: Springer.CrossRefzbMATHGoogle Scholar
  8. Einmahl, U., Mason, D. (2005). Uniform in bandwidth consistency of kernel-type function estimators. The Annals of Statistics, 33(3), 1380–1403.MathSciNetCrossRefzbMATHGoogle Scholar
  9. Fisher, R. A., Tippett, L. H. C. (1928). Limiting forms of the frequency distribution of the largest or smallest member of a sample. Mathematical Proceedings of the Cambridge Philosophical Society, 24(2), 180–190.CrossRefzbMATHGoogle Scholar
  10. Fraga Alves, I., de Haan, L., Neves, C. (2009). A test procedure for detecting super-heavy tails. Journal of Statistical Planning and Inference, 139, 213–227.MathSciNetCrossRefzbMATHGoogle Scholar
  11. Giné, E., Nickl, R. (2015). Mathematical foundations of infinite-dimensional statistical models. New York: Cambridge University Press.zbMATHGoogle Scholar
  12. Gnedenko, B. (1943). Sur la distribution limite du terme maximum d’une série aléatoire. Annals of Mathematics, 44(3), 423–453.MathSciNetCrossRefzbMATHGoogle Scholar
  13. Goegebeur, Y., Guillou, A., Osmann, M. (2014). A local moment type estimator for the extreme value index in regression with random covariates. The Canadian Journal of Statistics, 42, 487–507.MathSciNetCrossRefzbMATHGoogle Scholar
  14. Goegebeur, Y., Guillou, A., Osmann, M. (2017). A local moment type estimator for an extreme quantile in regression with random covariates. Communication in Statistics—Theory and Methods, 46, 319–343.MathSciNetCrossRefzbMATHGoogle Scholar
  15. He, X., Ng, P. (1999). Quantile splines with several covariates. Journal of Statistical Planning and Inference, 75, 343–352.MathSciNetCrossRefzbMATHGoogle Scholar
  16. Koenker, R., Bassett, G. (1978). Regression quantiles. Econometrica, 46, 33–50.MathSciNetCrossRefzbMATHGoogle Scholar
  17. Koenker, R., Ng, P., Portnoy, S. (1994). Quantile smoothing splines. Biometrika, 81, 673–680.MathSciNetCrossRefzbMATHGoogle Scholar
  18. Nadaraya, E. A. (1964). On estimating regression. Theory of Probability and its Applications, 9, 141–142.CrossRefzbMATHGoogle Scholar
  19. Racine, J. S., Li, K. (2017). Nonparametric conditional quantile estimation: A locally weighted quantile kernel approach. Journal of Econometrics, 201, 72–94.MathSciNetCrossRefzbMATHGoogle Scholar
  20. van der Vaart, A. W., Wellner, J. A. (1996). Weak convergence and empirical processes with applications to statistics., Springer series in statistics Berlin: Springer.CrossRefzbMATHGoogle Scholar
  21. Watson, G. S. (1964). Smooth regression analysis. Sankhya A, 26, 359–372.MathSciNetzbMATHGoogle Scholar
  22. Weisstein, E. W. (2003). CRC concise encyclopedia of mathematics2nd ed. Boca Raton: Chapman and Hall.Google Scholar

Copyright information

© The Institute of Statistical Mathematics, Tokyo 2019

Authors and Affiliations

  1. 1.Institut de Recherche Mathématique Avancée, UMR 7501Université de Strasbourg & CNRSStrasbourg CedexFrance

Personalised recommendations