Improved estimation of the extreme value index using related variables

Research output: Contribution to journalArticleScientificpeer-review

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. For this adaptation the tail dependence between the variable of interest and the related variable(s) plays an important role. We establish the asymptotic normality of this new estimator. It shows greatly improved behavior relative to the Hill estimator, in particular the asymptotic variance is substantially reduced, whereas we can keep the asymptotic bias the same. A simulation study confirms the substantially improved performance of our adapted estimator. We also present an application to the aforementioned earthquake losses.
Original languageEnglish
JournalExtremes
DOIs
Publication statusE-pub ahead of print - Aug 2019

Fingerprint

Extreme Value Index
Hill Estimator
Earthquake
Earthquakes
Extreme Value Statistics
Tail Dependence
Estimator
Asymptotic Bias
Asymptotic Variance
Asymptotic Normality
Probability distributions
Tail
Probability Distribution
Statistics
Simulation Study
Energy
Extreme values
Observation
Hill estimator

Keywords

  • asymptotic normality
  • heavy tail
  • hill estimator
  • tail dependence
  • variance reduction

Cite this

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title = "Improved estimation of the extreme value index using related variables",
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. For this adaptation the tail dependence between the variable of interest and the related variable(s) plays an important role. We establish the asymptotic normality of this new estimator. It shows greatly improved behavior relative to the Hill estimator, in particular the asymptotic variance is substantially reduced, whereas we can keep the asymptotic bias the same. A simulation study confirms the substantially improved performance of our adapted estimator. We also present an application to the aforementioned earthquake losses.",
keywords = "asymptotic normality, heavy tail, hill estimator, tail dependence, variance reduction",
author = "Hanan Ahmed and John Einmahl",
year = "2019",
month = "8",
doi = "10.1007/s10687-019-00358-y",
language = "English",
journal = "Extremes",
issn = "1386-1999",
publisher = "Springer Netherlands",

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Improved estimation of the extreme value index using related variables. / Ahmed, Hanan; Einmahl, John.

In: Extremes, 08.2019.

Research output: Contribution to journalArticleScientificpeer-review

TY - JOUR

T1 - Improved estimation of the extreme value index using related variables

AU - Ahmed, Hanan

AU - Einmahl, John

PY - 2019/8

Y1 - 2019/8

N2 - 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. For this adaptation the tail dependence between the variable of interest and the related variable(s) plays an important role. We establish the asymptotic normality of this new estimator. It shows greatly improved behavior relative to the Hill estimator, in particular the asymptotic variance is substantially reduced, whereas we can keep the asymptotic bias the same. A simulation study confirms the substantially improved performance of our adapted estimator. We also present an application to the aforementioned earthquake losses.

AB - 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. For this adaptation the tail dependence between the variable of interest and the related variable(s) plays an important role. We establish the asymptotic normality of this new estimator. It shows greatly improved behavior relative to the Hill estimator, in particular the asymptotic variance is substantially reduced, whereas we can keep the asymptotic bias the same. A simulation study confirms the substantially improved performance of our adapted 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

U2 - 10.1007/s10687-019-00358-y

DO - 10.1007/s10687-019-00358-y

M3 - Article

JO - Extremes

JF - Extremes

SN - 1386-1999

ER -