In extreme value analysis, natural upper bounds can appear that truncate the probability tail. At other instances ultimately at the largest data, deviations from a Pareto tail behaviour become apparent. This matter is especially important when extrapolation outside the sample is required. Given that in practice one does not always know whether the distribution is truncated or not, we consider estimators for extreme quantiles both under truncated and non-truncated Pareto-type distributions. We make use of the estimator of the tail index for the truncated Pareto distribution first proposed in Aban et al. (J. Amer. Statist. Assoc. 101(473), 270–277, 2006). We also propose a truncated Pareto QQ-plot and a formal test for truncation in order to help deciding between a truncated and a non-truncated case. In this way we enlarge the possibilities of extreme value modelling using Pareto tails, offering an alternative scenario by adding a truncation point T that is large with respect to the available data. In the mathematical modelling we hence let T→∞ at different speeds compared to the limiting fraction (k/n→0) of data used in the extreme value estimation. This work is motivated using practical examples from different fields, simulation results, and some asymptotic results.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
Aban, I.B., Meerschaert, M.M.: Generalized least squares estimators for the thickness of heavy tails. Journal of Statistical Planning and Inference 119, 341–352 (2004)
Aban, I. B., Meerschaert, M.M., Panorska, A.K.: Parameter estimation for the truncated pareto distribution. J. Amer. Statist. Assoc. 101(473), 270–277 (2006)
Beirlant, J., Joossens, E., Segers, J.: Second-order refined peaks-over-threshold modelling for heavy-tailed distribution. Journal of Statistical Planning and Inference 139, 2800–2815 (2009)
Beirlant, J., Vynckier, P., Teugels, J.: Tail index estimation, Pareto quantile plots and regression diagnostics. J. Amer. Statist. Assoc. 91, 1659–1667 (1996)
Beirlant, J., Goegebeur, Y., Teugels, J., Segers, J.: Statistics of Extremes: Theory and Applications. Wiley, UK (2004)
Brilhante, M.F., Gomes, M.I., Dinis Pestana, D.: A simple generalisation of the Hill estimator. Comput. Stat. Data Anal. 57(1), 518–535 (2013)
Chakrabarty, A., Samorodnitsky, G.: Understanding heavy tails in a bounded world, or, is a truncated heavy tail heavy or not?. Stoch. Model. 28, 109–143 (2012)
Clark, D.R.: A note on the upper-truncated Pareto distribution In Proc. of the Enterprise Risk Management Symposium, April 22-24, Chicago IL (2013)
Dekkers, A., Einmahl, J., de Haan, L.: A moment estimator for the index of an extreme-value distribution. Ann. Statist. 17, 1795–1832 (1989)
Gomes, M.I., Figueiredo, F., Neves, M.M.: Adaptive estimation of heavy right tails: the bootstrap methodology in action. Extremes 15(4), 463–489 (2012)
de Haan, L., Ferreira, A.: Extreme Value Theory: an Introduction Springer Science and Business Media. LLC, New York (2006)
Hill, B.M.: A simple general approach about the tail of a distribution. Ann. Stat. 3, 1163–1174 (1975)
Mason, D.M., Turova, T.S.: Weak Convergence of the Hill Estimator Process. In: Galambos, J., Lechner, J., Simiu, E. (eds.) Extreme Value Theory and Applications: Proceedings of the Conference on Extreme Value Theory and Applications, Gaithersburg, 1993, vol. 1. Kluwer, Dordrecht (1994)
Nuyts, J.: Inference about the tail of a distribution: improvement on the Hill estimator. Int. J. Math. Math. Sci. 924013 (2010)
Reed, W.J., McKelvey, K.S.: Power-law behaviour and parametric models for the size-distributions of forest fires. Ecol. Model. 150, 239–254 (2002)
Seneta, E.: Regularly Varying Functions. Lecture Notes in Math 508. Springer-Verlag, Berlin (1976)
Weissman, I.: Estimation of parameters and large quantiles based on the k largest observations. J. Am. Stat. Assoc. 73, 812–815 (1978)
Willems, P.: A time-series tool to support the multi-criteria performance evaluation of rainfall-runoff models. Environ. Model. Softw. 24, 311–321 (2009)
About this article
Cite this article
Beirlant, J., Alves, I.F. & Gomes, I. Tail fitting for truncated and non-truncated Pareto-type distributions. Extremes 19, 429–462 (2016). https://doi.org/10.1007/s10687-016-0247-3
- Pareto-type distributions
- Extreme quantiles
AMS 2000 Subject Classifications