Electronic Journal of Statistics

Adaptive confidence intervals for the tail coefficient in a wide second order class of Pareto models

Alexandra Carpentier and Arlene K. H. Kim

Full-text: Open access

Abstract

We study the problem of constructing uniform and adaptive confidence intervals for the tail coefficient in a second order Pareto model, when the second order coefficient is unknown. This problem is translated into a testing problem on the second order parameter. By constructing an appropriate model and an associated test statistic, we provide a uniform and adaptive confidence interval for the first order parameter. We also provide an almost matching lower bound, which proves that the result is minimax optimal up to a logarithmic factor.

Article information

Source
Electron. J. Statist., Volume 8, Number 2 (2014), 2066-2110.

Dates
First available in Project Euclid: 29 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1414588187

Digital Object Identifier
doi:10.1214/14-EJS944

Mathematical Reviews number (MathSciNet)
MR3273619

Zentralblatt MATH identifier
1305.62198

Keywords
Confidence intervals minimax testing Pareto model extreme value theory

Citation

Carpentier, Alexandra; Kim, Arlene K. H. Adaptive confidence intervals for the tail coefficient in a wide second order class of Pareto models. Electron. J. Statist. 8 (2014), no. 2, 2066--2110. doi:10.1214/14-EJS944. https://projecteuclid.org/euclid.ejs/1414588187


Export citation

References

  • Beirlant, J., Figueiredo, F., Gomes, M. I., and Vandewalle, B. (2008). Improved reduced-bias tail index and quantile estimators., J. Statist. Plann. Inference 138(3), 1851–1870.
  • Boucheron, S., Lugosi, G., and Massart, P. (2013)., Concentration Inequalities: A Non-asymptotic Theory of Independence. Oxford University Press.
  • Bousquet, O. (2003). Concentration inequalities for sub-additive functions using the entropy method. in: Stochastic inequalities and applications, Birkhäuser, Basel., Progr. Probab. 56(1), 213–247.
  • Bull, A. D. and Nickl, R. (2013). Adaptive confidence sets in $l_2$., Probab. Theory Related Fields 156(3), 889–919.
  • Carpentier, A. (2013). Honest and adaptive confidence sets in $l_p$., Electron. J. Stat. 7, 2875–2923.
  • Carpentier, A. and Kim, A. K. H. (2013). Adaptive and minimax optimal estimation of the tail coefficient. Available at, http://arxiv.org/abs/1309.2585.
  • Cheng, S. and Peng, L. (2001). Confidence intervals for the tail index., Bernoulli 7(5), 751–760.
  • Danielsson, J., de Haan, L., Peng, L., and de Vries, C. G. (2001). Using a bootstrap method to choose the sample fraction in tail index estimation., J. Multivariate Anal. 2, 226–248.
  • Dekkers, A. and De Haan, L. (1993). Optimal choice of sample fraction in extreme-value estimation., Journal of Multivariate Analysis 47(2), 173–195.
  • Drees, H. (1998). Optimal rates of convergence for estimates of the extreme value index., Ann. Statist. 26(1), 434–448.
  • Drees, H. (2001). Minimax risk bounds in extreme value theory., Ann. Statist. 29(1), 266–294.
  • Drees, H. and Kaufmann, E. (1998). Selecting the optimal sample fraction in univariate extreme value estimation., Stochastic Process. Appl. 75, 149–172.
  • Giné, E. and Nickl, R. (2010). Confidence bands in density estimation., Ann. Statist. 38(2), 1122–1170.
  • Gomes, I. M., De Haan, L., and Rodrigues, L. H. (2008). Tail index estimation for heavy-tailed models: accommodation of bias in weighted log-excesses., J. R. Stat. Soc. Ser. B Stat. Methodol. 91, 31–52.
  • Gomes, M. I., Figueiredo, F., and Neves, M. (2012). Adaptive estimation of heavy right tails: resampling-based methods in action., Extremes 15, 463–489.
  • Haeusler, E. and Segers, J. (2007). Assessing confidence intervals for the tail index by edgeworth expansions for the hill estimator., Bernoulli 13(1), 175–194.
  • Hall, P. (1982). On some simple estimates of an exponent of regular variation., J. R. Stat. Soc. Ser. B Stat. Methodol. 44(1), 37–42.
  • Hall, P. and Welsh, A. H. (1984). Best attainable rates of convergence for estimates of parameters of regular variation., Ann. Statist. 12(3), 1079–1084.
  • Hall, P. and Welsh, A. H. (1985). Adaptive estimates of parameters of regular variation., Ann. Statist. 75(1), 331–341.
  • Hill, B. M. (1975). A simple general approach to inference about the tail of a distribution., Ann. Statist. 3(5), 1163–1174.
  • Hoffmann, M. and Nickl, R. (2011). On adaptive inference and confidence bands., Ann. Statist. 39(5), 2383–2409.
  • Juditsky, A. and Lambert-Lacroix, S. (2003). Non-parametric confidence set estimation., Math. Methods Statist. 12(4), 410–428.
  • Low, M. G. (1997). On nonparametric confidence intervals., Ann. Statist. 25(6), 2547–2554.
  • Novak, S. Y. (2014). Lower bounds to the accuracy of inference on heavy tails., Bernoulli 20(2), 979–989.
  • Pickands, J. (1975). Statistical inference using extreme order statistics., Ann. Statist. 3(1), 119–131.
  • Robin, J. and van der Vaart, A. (2006). Adaptive non-parametric confidence sets., Ann. Statist. 34(1), 229–253.
  • Talagrand, M. (1996). New concentration inequalities in product spaces., Invent. Math. 126(3), 506–563.
  • van der Vaart, A. and Wellner, J. A. (1996)., Weak Convergence and Empirical Processes. Springer-Verlag, New York.