Fixed-k Asymptotic Inference About Tail Properties

Ulrich K. Müller, Yulong Wang

Research output: Contribution to journalArticlepeer-review

11 Scopus citations


We consider inference about tail properties of a distribution from an iid sample, based on extreme value theory. All of the numerous previous suggestions rely on asymptotics where eventually, an infinite number of observations from the tail behave as predicted by extreme value theory, enabling the consistent estimation of the key tail index, and the construction of confidence intervals using the delta method or other classic approaches. In small samples, however, extreme value theory might well provide good approximations for only a relatively small number of tail observations. To accommodate this concern, we develop asymptotically valid confidence intervals for high quantile and tail conditional expectations that only require extreme value theory to hold for the largest k observations, for a given and fixed k. Small-sample simulations show that these “fixed-k” intervals have excellent small-sample coverage properties, and we illustrate their use with mainland U.S. hurricane data. In addition, we provide an analytical result about the additional asymptotic robustness of the fixed-k approach compared to kn → ∞ inference.

Original languageEnglish (US)
Pages (from-to)1334-1343
Number of pages10
JournalJournal of the American Statistical Association
Issue number519
StatePublished - Jul 3 2017
Externally publishedYes


  • Extreme quantiles
  • extreme value distribution
  • tail conditional expectations

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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