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.

Convergence rates for the ultimate and pentultimate approximations in extreme-value theory

1982 ◽  
Vol 14 (04) ◽  
pp. 833-854 ◽  
Author(s):  
Jonathan P. Cohen
Keyword(s):  
Extreme Value Theory ◽  
Convergence Rates ◽  
Domain Of Attraction ◽  
Extreme Value Distribution ◽  
Rates Of Convergence ◽  
Value Theory ◽  
Extreme Value ◽  
Value Distribution ◽  
Type I ◽  
Type Ii

Let F be a distribution in the domain of attraction of the type I extreme-value distribution Λ(x). In this paper we derive uniform rates of convergence of Fn to Λfor a large class of distributions F. We also generalise the penultimate approximation of Fisher and Tippett (1928) and show that for many F a type II or type III extreme-value distribution gives a better approximation than the limiting type I distribution.

Convergence rates for the ultimate and pentultimate approximations in extreme-value theory

1982 ◽  
Vol 14 (4) ◽  
pp. 833-854 ◽  
Author(s):  
Jonathan P. Cohen
Keyword(s):  
Extreme Value Theory ◽  
Convergence Rates ◽  
Domain Of Attraction ◽  
Extreme Value Distribution ◽  
Rates Of Convergence ◽  
Value Theory ◽  
Extreme Value ◽  
Value Distribution ◽  
Type I ◽  
Type Ii

Let F be a distribution in the domain of attraction of the type I extreme-value distribution Λ(x). In this paper we derive uniform rates of convergence of Fn to Λfor a large class of distributions F. We also generalise the penultimate approximation of Fisher and Tippett (1928) and show that for many F a type II or type III extreme-value distribution gives a better approximation than the limiting type I distribution.

scholarly journals Return Period Evaluation of the Largest Possible Earthquake Magnitudes in Mainland China Based on Extreme Value Theory

Sensors ◽  
2021 ◽  
Vol 21 (10) ◽  
pp. 3519
Author(s):  
Yanbing Bai ◽  
Ning Ma ◽  
Shengwang Meng
Keyword(s):  
Earthquake Engineering ◽  
Return Period ◽  
Extreme Value Theory ◽  
Threshold Model ◽  
Mainland China ◽  
Value Theory ◽  
Extreme Value ◽  
Geographical Information ◽  
Geographical Regions ◽  
The Right

The largest possible earthquake magnitude based on geographical characteristics for a selected return period is required in earthquake engineering, disaster management, and insurance. Ground-based observations combined with statistical analyses may offer new insights into earthquake prediction. In this study, to investigate the seismic characteristics of different geographical regions in detail, clustering was used to provide earthquake zoning for Mainland China based on the geographical features of earthquake events. In combination with geospatial methods, statistical extreme value models and the right-truncated Gutenberg–Richter model were used to analyze the earthquake magnitudes of Mainland China under both clustering and non-clustering. The results demonstrate that the right-truncated peaks-over-threshold model is the relatively optimal statistical model compared with classical extreme value theory models, the estimated return level of which is very close to that of the geographical-based right-truncated Gutenberg–Richter model. Such statistical models can provide a quantitative analysis of the probability of future earthquake risks in China, and geographical information can be integrated to locate the earthquake risk accurately.

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.

Uniform rates of convergence in extreme-value theory

1982 ◽  
Vol 14 (03) ◽  
pp. 600-622 ◽  
Author(s):  
Richard L. Smith
Keyword(s):  
Extreme Value Theory ◽  
Error Term ◽  
Extreme Value Distribution ◽  
Rates Of Convergence ◽  
Value Theory ◽  
Extreme Value ◽  
Value Distribution ◽  
Convergence In Distribution ◽  
Principal Error ◽  
Close Parallel

Rates of convergence are derived for the convergence in distribution of renormalised sample maxima to the appropriate extreme-value distribution. Related questions which are discussed include the estimation of the principal error term and the optimality of the renormalising constants. Throughout the paper a close parallel is drawn with the theory of slow variation with remainder. This theory is used in proving most of the results. Some applications are discussed, including some models of importance in reliability.

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.

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