### Abstract

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 |

### Fingerprint

### Keywords

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

### Cite this

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

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**Improved Estimation of the Extreme Value Index Using Related Variables.** / Ahmed, Hanan; Einmahl, John.

Research output: Working paper › Discussion paper › Other research output

TY - UNPB

T1 - Improved Estimation of the Extreme Value Index Using Related Variables

AU - Ahmed, Hanan

AU - Einmahl, John

PY - 2018/7/16

Y1 - 2018/7/16

N2 - Heavy tailed phenomena are naturally analyzed by extreme value statistics. Acrucial 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 alonger 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.

AB - Heavy tailed phenomena are naturally analyzed by extreme value statistics. Acrucial 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 alonger 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.

KW - asymptotic normality

KW - heavy tail

KW - Hill estimator

KW - tail dependence

KW - variance reduction

M3 - Discussion paper

VL - 2018-025

T3 - CentER Discussion Paper

BT - Improved Estimation of the Extreme Value Index Using Related Variables

PB - CentER, Center for Economic Research

CY - Tilburg

ER -