Estimation of extreme risk regions under multivariate regular variation

Juan Juan Cai, John H.J. Einmahl, Laurens De Haan

Research output: Contribution to JournalArticleAcademicpeer-review


When considering d possibly dependent random variables, one is often interested in extreme risk regions, with very small probability p. We consider risk regions of the form {z ε ℝd : f (z) ≤ β}, where f is the joint density and β a small number. Estimation of such an extreme risk region is difficult since it contains hardly any or no data. Using extreme value theory, we construct a natural estimator of an extreme risk region and prove a refined form of consistency, given a random sample of multivariate regularly varying random vectors. In a detailed simulation and comparison study, the good performance of the procedure is demonstrated. We also apply our estimator to financial data.

Original languageEnglish
Pages (from-to)1803-1826
Number of pages24
JournalAnnals of Statistics
Issue number3
Publication statusPublished - 1 Jun 2011
Externally publishedYes


  • Extremes
  • Level set
  • Multivariate quantile
  • Rare event
  • Spectral density
  • Tail dependence


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