scholarly journals Generalized regular variation of second order

1996 ◽  
Vol 61 (3) ◽  
pp. 381-395 ◽  
Author(s):  
Laurens de Haan ◽  
Ulrich Stadtmüller
Keyword(s):  
Asymptotic Behaviour ◽  
Extreme Value Theory ◽  
Regular Variation ◽  
Inverse Function ◽  
Limit Relation ◽  
Laplace Transforms ◽  
Auxiliary Function ◽  
Value Theory ◽  
Extreme Value ◽  
Second Order

AbstractAssume that for a measurable funcion f on (0, ∞) there exist a positive auxiliary function a(t) and some γ ∈ R such that . Then f is said to be of generalized regular variation. In order to control the asymptotic behaviour of certain estimators for distributions in extreme value theory we are led to study regular variation of second order, that is, we assume that exists non-trivially with a second auxiliary function a1(t). We study the possible limit functions in this limit relation (defining generalized regular variation of second order) and their domains of attraction. Furthermore we give the corresponding relation for the inverse function of a monotone f with the stated property. Finally, we present an Abel-Tauber theorem relating these functions and their Laplace transforms.

scholarly journals EVT-based estimation of risk capital and convergence of high quantiles

2008 ◽  
Vol 40 (03) ◽  
pp. 696-715 ◽  
Author(s):  
Matthias Degen ◽  
Paul Embrechts
Keyword(s):  
Extreme Value Theory ◽  
Value Theory ◽  
Extreme Value ◽  
Second Order ◽  
Parametric Models ◽  
Tail Behavior ◽  
Parametric Approach ◽  
Risk Capital ◽  
High Quantiles ◽  
Estimation Of Risk

We discuss some issues regarding the accuracy of a quantile-based estimation of risk capital. In this context, extreme value theory (EVT) emerges naturally. The paper sheds some further light on the ongoing discussion concerning the use of a semi-parametric approach like EVT and the use of specific parametric models such as the g-and-h. In particular, we discusses problems and pitfalls evolving from such parametric models when using EVT and highlight the importance of the underlying second-order tail behavior.

On extreme-value theory in the presence of a trend

1987 ◽  
Vol 24 (1) ◽  
pp. 62-76 ◽  
Author(s):  
L. De Haan ◽  
E. Verkade
Keyword(s):  
Asymptotic Behaviour ◽  
Extreme Value Theory ◽  
Random Variables ◽  
Value Theory ◽  
Extreme Value

Extreme-value theory is considered in the context of independent but not identically distributed random variables. The departure from the case of identical distributions comes from a trend added to i.i.d. observations. Unlike previous authors we consider trends that do not completely destroy the asymptotic behaviour from the i.i.d. case.

On extreme-value theory in the presence of a trend

1987 ◽  
Vol 24 (01) ◽  
pp. 62-76 ◽  
Author(s):  
L. De Haan ◽  
E. Verkade
Keyword(s):  
Asymptotic Behaviour ◽  
Extreme Value Theory ◽  
Random Variables ◽  
Value Theory ◽  
Extreme Value

Extreme-value theory is considered in the context of independent but not identically distributed random variables. The departure from the case of identical distributions comes from a trend added to i.i.d. observations. Unlike previous authors we consider trends that do not completely destroy the asymptotic behaviour from the i.i.d. case.

A Handbook of Second Order Expansions of Quantile Functions and Asymptotic Record Values Laws

2021 ◽  
Author(s):  
Gane Samb Lo ◽  
Moumouni Diallo ◽  
Modou Ngom
Keyword(s):  
Extreme Value Theory ◽  
Domain Of Attraction ◽  
Rates Of Convergence ◽  
Record Values ◽  
Value Theory ◽  
Extreme Value ◽  
Second Order ◽  
Record Value ◽  
Quantile Functions ◽  
The Right

In this monograph, our final objective is to provide second order expansions of quantile functions of as many probability laws as possible. Second order expansions of quantile functions are important tools for finding extreme value domain of attraction of probability laws and for discovering rates of convergence in extreme value theory. We hope that readers will make profit of the results in their works by using the right expansions of quantile functions from the monograph. In that spirit, we apply the quantiles expansions exposed here to deliver the corresponding asymptotic laws of records values. <br><br> In this first edition, fifty four distributions are concerned. For each of those probability laws, full computations for finding the expansion and the asymptotic record value theory are entirely justified. We will regularly update the handbook by adding probability laws in later editions.

scholarly journals EVT-based estimation of risk capital and convergence of high quantiles

2008 ◽  
Vol 40 (3) ◽  
pp. 696-715 ◽  
Author(s):  
Matthias Degen ◽  
Paul Embrechts
Keyword(s):  
Extreme Value Theory ◽  
Value Theory ◽  
Extreme Value ◽  
Second Order ◽  
Parametric Models ◽  
Tail Behavior ◽  
Parametric Approach ◽  
Risk Capital ◽  
High Quantiles ◽  
Estimation Of Risk

We discuss some issues regarding the accuracy of a quantile-based estimation of risk capital. In this context, extreme value theory (EVT) emerges naturally. The paper sheds some further light on the ongoing discussion concerning the use of a semi-parametric approach like EVT and the use of specific parametric models such as the g-and-h. In particular, we discusses problems and pitfalls evolving from such parametric models when using EVT and highlight the importance of the underlying second-order tail behavior.

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