### Abstract

Heavy tailed phenomena are naturally analyzed by extreme value statistics. A

crucial step in such an analysis is the estimation of the extreme value index, which describes the tail heaviness of the underlying probability distribution. We consider the situation where we have next to the n observations of interest another n+m observations of one or more related variables, like, e.g., financial losses due to earthquakes and the related amounts of energy released, for a

longer period than that of the losses. Based on such a data set, we present an adapted version of the Hill estimator that shows greatly improved behavior and we establish the asymptotic normality of this estimator. For this adaptation the tail dependence between the variable of interest and the related variable(s) plays an important role. A simulation study confirms the substantially improved performance of our adapted estimator relative to the Hill estimator. We also present an application to the aforementioned earthquake losses.

crucial step in such an analysis is the estimation of the extreme value index, which describes the tail heaviness of the underlying probability distribution. We consider the situation where we have next to the n observations of interest another n+m observations of one or more related variables, like, e.g., financial losses due to earthquakes and the related amounts of energy released, for a

longer period than that of the losses. Based on such a data set, we present an adapted version of the Hill estimator that shows greatly improved behavior and we establish the asymptotic normality of this estimator. For this adaptation the tail dependence between the variable of interest and the related variable(s) plays an important role. A simulation study confirms the substantially improved performance of our adapted estimator relative to the Hill estimator. We also present an application to the aforementioned earthquake losses.

Original language | English |
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Place of Publication | Tilburg |

Publisher | CentER, Center for Economic Research |

Number of pages | 16 |

Volume | 2018-025 |

Publication status | Published - 16 Jul 2018 |

### Publication series

Name | CentER Discussion Paper |
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Volume | 2018-025 |

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### Keywords

- asymptotic normality
- heavy tail
- Hill estimator
- tail dependence
- variance reduction

### Cite this

Ahmed, H., & Einmahl, J. (2018).

*Improved Estimation of the Extreme Value Index Using Related Variables*. (CentER Discussion Paper; Vol. 2018-025). Tilburg: CentER, Center for Economic Research.