The Annals of Statistics

Estimating Tails of Probability Distributions

Richard L. Smith

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We study the asymptotic properties of estimators of the tail of a distribution based on the excesses over a threshold. A key idea is the use of Pickands' generalised Pareto distribution and its fitting, in most cases, by the method of maximum likelihood. The results cover all three limiting types of extreme value theory. We propose a new estimator for an index of regular variation and show that it often performs better than Hill's estimator. We give new results for estimating the endpoint of a distribution, extending earlier work by Hall and by Smith and Weissman. Finally, we give detailed results for the domain of attraction of $\exp(-e^{-x})$ and show that, in most cases, our proposed estimator is more efficient than two others, one based on the exponential distribution and the other due to Davis and Resnick. We also touch briefly on the problem of large deviations from a statistical point of view. The results make extensive use of existing work on rates of convergence.

Article information

Ann. Statist. Volume 15, Number 3 (1987), 1174-1207.

First available in Project Euclid: 12 April 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62G05: Estimation
Secondary: 62F10: Point estimation 62G30: Order statistics; empirical distribution functions 62F12: Asymptotic properties of estimators

Extreme value theory generalised Pareto distribution maximum likelihood


Smith, Richard L. Estimating Tails of Probability Distributions. Ann. Statist. 15 (1987), no. 3, 1174--1207. doi:10.1214/aos/1176350499.

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