## Abstract

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 ℝ^{d} : 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 language | English |
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Pages (from-to) | 1803-1826 |

Number of pages | 24 |

Journal | Annals of Statistics |

Volume | 39 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1 Jun 2011 |

Externally published | Yes |

## Keywords

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