A simple class of reduced bias kernel estimators of the extreme value index

Frederico Caeiro, Lígia Henriques-Rodrigues, Dora Prata Gomes

Research output: Contribution to journalConference article

Abstract

In Statistics of Extremes we often have to deal with the estimation of the extreme value index, a key parameter of extreme events. The adequate estimation of this parameter is of crucial importance in the estimation of other parameters of extreme events, such as an extreme quantile, a small exceedance probability or the return period of a high level. In the paper, we first analyze a class of kernel estimators that generalize the classical Hill estimator of the extreme value index. Then, to improve the accuracy of the estimation, we also propose a new class of reduced bias kernel estimators, parameterized with a tuning parameter that allow us to change the asymptotic mean squared error. Under suitable conditions, such class of estimators is consistent and has asymptotic normal distribution with a null dominant component of asymptotic bias. As a result, we show that further bias reduction is possible with an adequate choice of the tuning parameter. Additionally, semi‐parametric reduced‐bias extreme quantiles estimators based on kernel estimators of the extreme value index are also put forward. Under adequate conditions on the underlying model, we establish the consistency and asymptotic normality of these extreme quantile estimators. Finally, we analyze the log‐returns of the BOVESPA stock market index, collected from 2004 to 2016.
Original languageEnglish
Pages (from-to)1-12
JournalComputational and Mathematical Methods
DOIs
Publication statusE-pub ahead of print - 18 Mar 2019
Event18th International Conference on Computational and Mathematical Methods
 in Science and Engineering, CMMSE 2018 - Cádiz, Spain
Duration: 9 Jul 201813 Jul 2018

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Extreme Value Index
Kernel Estimator
Extreme Quantiles
Extreme Events
Parameter Tuning
Estimator
Statistics of Extremes
Hill Estimator
Bias Reduction
Asymptotic Bias
Exceedance
Stock Market
Asymptotic Normality
Mean Squared Error
Asymptotic distribution
Null
Gaussian distribution
Generalise
Class

Cite this

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title = "A simple class of reduced bias kernel estimators of the extreme value index",
abstract = "In Statistics of Extremes we often have to deal with the estimation of the extreme value index, a key parameter of extreme events. The adequate estimation of this parameter is of crucial importance in the estimation of other parameters of extreme events, such as an extreme quantile, a small exceedance probability or the return period of a high level. In the paper, we first analyze a class of kernel estimators that generalize the classical Hill estimator of the extreme value index. Then, to improve the accuracy of the estimation, we also propose a new class of reduced bias kernel estimators, parameterized with a tuning parameter that allow us to change the asymptotic mean squared error. Under suitable conditions, such class of estimators is consistent and has asymptotic normal distribution with a null dominant component of asymptotic bias. As a result, we show that further bias reduction is possible with an adequate choice of the tuning parameter. Additionally, semi‐parametric reduced‐bias extreme quantiles estimators based on kernel estimators of the extreme value index are also put forward. Under adequate conditions on the underlying model, we establish the consistency and asymptotic normality of these extreme quantile estimators. Finally, we analyze the log‐returns of the BOVESPA stock market index, collected from 2004 to 2016.",
author = "Frederico Caeiro and L{\'i}gia Henriques-Rodrigues and {Prata Gomes}, Dora",
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doi = "10.1002/cmm4.1025",
language = "English",
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journal = "Computational and Mathematical Methods",
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}

A simple class of reduced bias kernel estimators of the extreme value index. / Caeiro, Frederico; Henriques-Rodrigues, Lígia; Prata Gomes, Dora.

In: Computational and Mathematical Methods, 18.03.2019, p. 1-12.

Research output: Contribution to journalConference article

TY - JOUR

T1 - A simple class of reduced bias kernel estimators of the extreme value index

AU - Caeiro, Frederico

AU - Henriques-Rodrigues, Lígia

AU - Prata Gomes, Dora

PY - 2019/3/18

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N2 - In Statistics of Extremes we often have to deal with the estimation of the extreme value index, a key parameter of extreme events. The adequate estimation of this parameter is of crucial importance in the estimation of other parameters of extreme events, such as an extreme quantile, a small exceedance probability or the return period of a high level. In the paper, we first analyze a class of kernel estimators that generalize the classical Hill estimator of the extreme value index. Then, to improve the accuracy of the estimation, we also propose a new class of reduced bias kernel estimators, parameterized with a tuning parameter that allow us to change the asymptotic mean squared error. Under suitable conditions, such class of estimators is consistent and has asymptotic normal distribution with a null dominant component of asymptotic bias. As a result, we show that further bias reduction is possible with an adequate choice of the tuning parameter. Additionally, semi‐parametric reduced‐bias extreme quantiles estimators based on kernel estimators of the extreme value index are also put forward. Under adequate conditions on the underlying model, we establish the consistency and asymptotic normality of these extreme quantile estimators. Finally, we analyze the log‐returns of the BOVESPA stock market index, collected from 2004 to 2016.

AB - In Statistics of Extremes we often have to deal with the estimation of the extreme value index, a key parameter of extreme events. The adequate estimation of this parameter is of crucial importance in the estimation of other parameters of extreme events, such as an extreme quantile, a small exceedance probability or the return period of a high level. In the paper, we first analyze a class of kernel estimators that generalize the classical Hill estimator of the extreme value index. Then, to improve the accuracy of the estimation, we also propose a new class of reduced bias kernel estimators, parameterized with a tuning parameter that allow us to change the asymptotic mean squared error. Under suitable conditions, such class of estimators is consistent and has asymptotic normal distribution with a null dominant component of asymptotic bias. As a result, we show that further bias reduction is possible with an adequate choice of the tuning parameter. Additionally, semi‐parametric reduced‐bias extreme quantiles estimators based on kernel estimators of the extreme value index are also put forward. Under adequate conditions on the underlying model, we establish the consistency and asymptotic normality of these extreme quantile estimators. Finally, we analyze the log‐returns of the BOVESPA stock market index, collected from 2004 to 2016.

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JO - Computational and Mathematical Methods

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SN - 2577-7408

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