### Abstract

Consider a random sample from a bivariate distribution function

*F*in the max-domain of attraction of an extreme-value distribution function*G*. This*G*is characterized by two extreme-value indices and a spectral measure, the latter determining the tail dependence structure of*F*. A major issue in multivariate extreme-value theory is the estimation of the spectral measure Φ_{p }with respect to the L*norm. For every*_{p}*p*∈ [1,∞], a nonparametric maximum empirical likelihood estimator is proposed for Φ_{p}. The main novelty is that these estimators are guaranteed to satisfy the moment constraints by which spectral measures are characterized. Asymptotic normality of the estimators is proved under conditions that allow for tail independence. Moreover, the conditions are easily verifiable as we demonstrate through a number of theoretical examples. A simulation study shows a substantially improved performance of the new estimators. Two case studies illustrate how to implement the methods in practice.Original language | English |
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Pages (from-to) | 2953-2989 |

Journal | The Annals of Statistics |

Volume | 37 |

Issue number | 5B |

Publication status | Published - 2009 |

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## Cite this

Einmahl, J. H. J., & Segers, J. J. J. (2009). Maximum empirical likelihood estimation of the spectral measure of an extreme-value distribution.

*The Annals of Statistics*,*37*(5B), 2953-2989. http://www.imstat.org/aos/