Smooth tail-index estimation

Samuel Müller*, Kaspar Rufibach

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

16 Citations (Scopus)


The two parametric distribution functions appearing in the extreme-value theory - the generalized extremevalue distribution and the generalized Pareto distribution - have log-concave densities if the extreme-value index γ ε [-1,0]. Replacing the order statistics in tail-index estimators by their corresponding quantiles from the distribution function that is based on the estimated log-concave density f̂n leads to novel smooth quantile and tail-index estimators. These new estimators aim at estimating the tail index especially in small samples. Acting as a smoother of the empirical distribution function, the log-concave distribution function estimator reduces estimation variability to a much greater extent than it introduces bias. As a consequence, Monte Carlo simulations demonstrate that the smoothed version of the estimators are well superior to their non-smoothed counterparts, in terms of mean-squared error.

Original languageEnglish
Pages (from-to)1155-1167
Number of pages13
JournalJournal of Statistical Computation and Simulation
Issue number9
Publication statusPublished - Sept 2009
Externally publishedYes


  • 'extreme-value' theory
  • log-concave density estimation
  • negative Hill estimator
  • Pickands estimator
  • tail-index estimation
  • small-sample performance


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