Skip to main content

Weak properties and robustness of t-Hill estimators

Abstract

We describe a novel method of heavy tails estimation based on transformed score (t-score). Based on a new score moment method we derive the t-Hill estimator, which estimates the extreme value index of a distribution function with regularly varying tail. t-Hill estimator is distribution sensitive, thus it differs in e.g. Pareto and log-gamma case. Here, we study both forms of the estimator, i.e. t-Hill and t-lgHill. For both estimators we prove weak consistency in moving average settings as well as the asymptotic normality of t-lgHill estimator in iid setting. In cases of contamination with heavier tails than the tail of original sample, t-Hill outperforms several robust tail estimators, especially in small samples. A simulation study emphasizes the fact that the level of contamination is playing a crucial role. The larger the contamination, the better are the t-score moment estimates. The reason for this is the bounded t-score of heavy-tailed distributions (and, consequently, bounded influence functions of the estimators). We illustrate the developed methodology on a small sample data set of stake measurements from Guanaco glacier in Chile.

References

  • Abermann, J., Kinnard, C., MacDonell, S.: Albedo variations and the impact of clouds on glaciers in the Chilean semi-arid andes. J. Glaciol. 60(219), 183–191 (2014)

    Article  Google Scholar 

  • Alves, M.F.: A location invariant hill-type estimator. Extremes 4(3), 199–217 (2001)

    MathSciNet  Article  MATH  Google Scholar 

  • Beran, J., Schell, D.: On robust tail index estimation. Comput. Stat. Data Anal. 56(11), 3430–3443 (2012). doi:10.1016/j.csda.2010.05.028

    MathSciNet  Article  MATH  Google Scholar 

  • Beran, J., Schell, D., Stehlík, M.: The harmonic moment tail index estimator: asymptotic distribution and robustness. Ann. Inst. Stat. Math. 66, 193–220 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  • Billingsley, P.: Convergence of probability measures. John Wiley & Sons (2013)

  • Brilhante, M., Gomes, M., Pestana, D.: A simple generalization of the hill estimator. Comput. Stat. Data Anal. 57, 518–535 (2013)

    MathSciNet  Article  Google Scholar 

  • Brzezinski, M.: Robust estimation of the pareto tail index: a monte carlo analysis. Empir. Econ. (2015)

  • Cline, D.: Infinite Series of Random Variables with Regularly Varying Tails. Tech. Rep No 83-24. Institute of Applied Mathematics and Statistics, University of British Columbia (1983)

    Google Scholar 

  • Dekkers, A.L., de Haan L.: On the estimation of the extreme-value index and large quantile estimation. Ann Statist 17(4), 1795–1832 (1989). doi:10.1214/aos/1176347396

    MathSciNet  Article  MATH  Google Scholar 

  • Dekkers, A.L., Einmahl, J.H., de Haan, L.: A moment estimator for the index of an extreme-value distribution. Ann. Stat., 1833–1855 (1989)

  • Drees, H., de Haan, L., Resnick, S.: How to make a hill plot. Ann. Stat. 28, 254–274 (2000)

    MathSciNet  Article  MATH  Google Scholar 

  • Dupuis, D.J., Morgenthaler, S.: Robust weighted likelihood estimators with an application to bivariate extreme value problems. The Canadian Journal of Statistics/La Revue Canadienne de Statistique, 17–36 (2002)

  • Dupuis, D.J., Victoria-Feser, M.P.: A robust prediction error criterion for pareto modelling of upper tails. Can. J. Stat. 34(4), 639–658 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  • Edgeworth, F.: On the probable errors of frequency constants. J. R. Stat. Soc. 71, 381–397 (1908a)

    Article  Google Scholar 

  • Edgeworth, F.: On the probable errors of frequency constants (contd). J. R. Stat. Soc. 71, 499–512 (1908b)

    Article  Google Scholar 

  • Embrechts, P., Klueppelberg C., Mikosch T.: Modeling extremal events for insurance and finance. Springer-Verlag (1997)

  • Fabián, Z.: Induced cores and their use in robust parametric estimation. Communication in Statistics Theory Methods 30, 537–556 (2001)

    MathSciNet  Article  MATH  Google Scholar 

  • Fabián, Z.: Estimation of simple characteristics of samples from skewed and heavy-tailed distribution. Recent Advances in Stochastic Modeling and Data Analysis, 43–50 (2007)

  • Fabián, Z.: Score Moment Estimators. In: Lechavallier, Y., Saporta, G. (eds.) Proceedings of Conference COMPSTAT’2010. Springer, Physica-Verlag (2010)

  • Fabián, Z.: Score function of distribution and revival of the moment method. Communications in Statistics - Theory and Methods 0(ja), 00–00 (2015). doi:10.1080/03610926.2013.857688

    Google Scholar 

  • Fabián, Z., Stehlík, M.: On robust and distribution sensitive hill like method. Tech. rep., IFAS Reasearch Paper Series 43(4) (2009)

  • Finkelstein, M., Tucker, H.G., Alan, V.J.: Pareto tail index estimation revisited. North American actuarial journal 10(1), 1–10 (2006)

    MathSciNet  Article  Google Scholar 

  • Fisher, R.: Statistical Methods for Research Workers. Cosmo Publications (1925). http://books.google.at/books?id=4bTttAJR5kEC

  • Francou, B., Pouyaud, B.: Métodos De Observación De Glaciares En Los Andes Tropicales, Mediciones De Campo Y Procesamiento De Datos Versión 1. Great-Ice, IRD (2004)

  • Gascoin, S., Kinnard, C., Ponce, R., Macdonell, S., Lhermitte, S., Rabatel, A.: Glacier contribution to streamflow in two headwaters of the huasco river, dry andes of Chile. The Cryosphere 5, 1099–1113 (2011)

    Article  Google Scholar 

  • Geluk, J., de Haan, L., Resnick, S., Staarica, C.: Second-order regular variation, convolution and the central limit theorem. Stoch. Process. Appl. 69, 139–159 (1997)

    MathSciNet  Article  MATH  Google Scholar 

  • Gomes, M.I., Oliveira, O.: Censoring estimators of a positive tail index. Statistics & Probability Letters 65, 147–159 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  • de Haan L.: On regular variation and its application to the weak convergence of sample extremes. Mathematisch Centrum (1970)

  • de Haan, L., Ferreira, A.: Extreme Value Theory: An Introduction Springer Series in Operations Research and Financial Engineering. Springer, New York (2006)

    Book  Google Scholar 

  • de Haan, L., Stadtmüller, U.: Generalized regular variation of second order. J. Aust. Math. Soc. 61, 381–395 (1996)

    MathSciNet  Article  MATH  Google Scholar 

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

    MathSciNet  Article  MATH  Google Scholar 

  • Hosking, J.R., Wallis, J.R.: Parameter and quantile estimation for the generalized pareto distribution. Technometrics 29(3), 339–349 (1987)

    MathSciNet  Article  MATH  Google Scholar 

  • Huber, P.J., Ronchetti, E.: Robust statistics, series in probability and mathematical statistics (1981)

  • Jordanova, P., Pancheva, E.: Weak asymptotic results for t-hill estimator. Comptes rendus de l’académie bulgare des sciences 65(12), 1649–1656 (2012)

    MathSciNet  MATH  Google Scholar 

  • Kratz, M., Resnick, S.I.: The qq-estimator and heavy tails. Stoch. Model. 12(4), 699–724 (1996)

    MathSciNet  Article  MATH  Google Scholar 

  • Li, J., Peng, Z., Nadarajah, S.: Asymptotic normality of location invariant heavy tail index estimator. Extremes 13, 269–290 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  • Oyarzún, J., Oyarzún, R.: Sustainable development threats, inter-sector conflicts and environmental policy requirements in the arid, mining rich, northern Chile territory. Sustain. Dev. 19(4), 263–274 (2011)

    Article  Google Scholar 

  • Paulauskas, V., Vaiciulis, M.: On the improvement of hill and some other estimators. Lith. Math. J. 53(3), 336–355 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  • Pearson, K., Filon, L.: Mathematical contributions to the theory of evolution. iv. on the probable errors of frequency constants and on the influence of random selection on variation and correlation. Philosophical Transactions of the Royal Society of London A: Mathematical Physical and Engineering Sciences 191, 229–311 (1898). doi:10.1098/rsta.1898.0007

    Article  MATH  Google Scholar 

  • Rabatel, A., Castebrunet, H., Favier, V., Nicholson, L., Kinnard, C.: Glacier changes in the pascua-lama region, chilean andes (29 s): recent mass balance and 50 yr surface area variations. Cryosphere 5(4), 1029–1041 (2011)

    Article  Google Scholar 

  • Resnick, S., Starica, C.: Consistency of hill’s estimator for dependent data. Tech. rep., Technical report No. 1077, School of Operational Research and Industrial Engeneering Cornell University Itaca NY 14853 (1993)

  • Rousseeuw, F.H.E.R.P., Stahel, W.: Robust Statistics: the Approach Based on Influence Functions. J Wiley, New York (1986)

    MATH  Google Scholar 

  • Schultze, J., Steinebach, J.: On least squares estimates of an exponential tail coefficient. Statistics & Risk Modeling 14(4), 353–372 (1996)

    MathSciNet  Article  MATH  Google Scholar 

  • Stehlík, M., Hermann, P.: Letter to the editor. Ann. Appl. Stat. 9, 2051 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  • Stehlík, M., Potocký, R., Fabián, Z.: On the favourable estimation of fitting heavy tailed data. Comput. Stat. 25, 485–503 (2010a)

    Article  MATH  Google Scholar 

  • Stehlík, M., Potocký, R., Waldl H., Fabián, Z.: On the favourable estimation of fitting heavy tailed data. Tech. rep., IFAS res. report Nr. 32 JKU Linz (2010b)

  • Stehlík, M., Fabián, Z., Strelec, L.: Small sample robust testing for norMality against pareto tails. Communications in Statistics - Simulation and Computation 41, 1167–1194 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  • Vandewalle, B., Beirlant, J., Christmann, A., Hubert, M.: A robust estimator for the tail index of pareto-type distributions. Computational Statistics & Data Analysis 51, 6252–6268 (2007)

    MathSciNet  Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Milan Stehlík.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Jordanova, P., Fabián, Z., Hermann, P. et al. Weak properties and robustness of t-Hill estimators. Extremes 19, 591–626 (2016). https://doi.org/10.1007/s10687-016-0256-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10687-016-0256-2

Keywords

  • Point estimation
  • Asymptotic properties of estimators
  • t-Hill estimator
  • t-lgHill estimator

AMS 2000 Subject Classifications

  • 62F10
  • 62F12