, Volume 19, Issue 3, pp 429–462 | Cite as

Tail fitting for truncated and non-truncated Pareto-type distributions

  • Jan BeirlantEmail author
  • Isabel Fraga Alves
  • Ivette Gomes


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.


Pareto-type distributions Truncation Extreme quantiles Endpoint QQ-plots 

AMS 2000 Subject Classifications

62G32 62G05 62G10 62G20 


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Jan Beirlant
    • 1
    • 2
    Email author
  • Isabel Fraga Alves
    • 3
  • Ivette Gomes
    • 3
  1. 1.Department of MathematicsKU LeuvenLeuvenBelgium
  2. 2.Department of Mathematical Statistics and Actuarial ScienceUniversity of the Free StateBloemfonteinSouth Africa
  3. 3.Department of Statistics and Operations Research and CEAULUniversity of LisbonLisboaPortugal

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