Special Issue "Extreme Value Theory"

A special issue of Entropy (ISSN 1099-4300). This special issue belongs to the section "Statistical Physics".

Deadline for manuscript submissions: closed (20 December 2021) | Viewed by 4706

Special Issue Editors

Prof. Dr. Sandro Vaienti
E-Mail Website
Guest Editor
Aix Marseille Université, Université de Toulon, CNRS, CPT, 13009 Marseille, France
Interests: dynamical systems; limit theorems; random dynamical systems; extreme value theory and time series
Prof. Dr. Jorge Milhazes Freitas
E-Mail Website
Guest Editor
Centro de Matemática and Faculdade de Ciências da Universidade do Porto Rua do Campo Alegre, 687, 4169-007 Porto, Portugal
Interests: statistical properties of dynamical systems; extreme value theory; recurrence and hitting times; limit theorems
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Special Issue Information

Dear colleagues,

Extreme value theory (EVT) has become a powerful tool in the last few years to analyze the statistical properties of dynamical systems, even under random perturbations. Its connection with quantitative recurrence properties of systems has propelled this subject to become an established area of interest. On the other hand, dynamical systems theory has enriched EVT by allowing it to obtain new and rigorous results, for instance, to study point processes, nonstationary systems, statistics of records, etc. These latter achievements have allowed new applications in different areas, such as physics, biology, and finance.

Prof. Sandro Vaienti
Dr. Jorge Milhazes Freitas
Guest Editors

Manuscript Submission Information

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Keywords

  • extreme events
  • point processes
  • recurrence
  • clustering
  • extremal index
  • tail index
  • dimension

Published Papers (5 papers)

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Research

Article
On Max-Semistable Laws and Extremes for Dynamical Systems
Entropy 2021, 23(9), 1192; https://doi.org/10.3390/e23091192 - 09 Sep 2021
Viewed by 566
Abstract
Suppose (f,X,μ) is a measure preserving dynamical system and ϕ:XR a measurable observable. Let Xi=ϕfi1 denote the time series of observations on the system, and [...] Read more.
Suppose (f,X,μ) is a measure preserving dynamical system and ϕ:XR a measurable observable. Let Xi=ϕfi1 denote the time series of observations on the system, and consider the maxima process Mn:=max{X1,,Xn}. Under linear scaling of Mn, its asymptotic statistics are usually captured by a three-parameter generalised extreme value distribution. This assumes certain regularity conditions on the measure density and the observable. We explore an alternative parametric distribution that can be used to model the extreme behaviour when the observables (or measure density) lack certain regular variation assumptions. The relevant distribution we study arises naturally as the limit for max-semistable processes. For piecewise uniformly expanding dynamical systems, we show that a max-semistable limit holds for the (linear) scaled maxima process. Full article
(This article belongs to the Special Issue Extreme Value Theory)
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Article
Extreme Value Theory for Hurwitz Complex Continued Fractions
Entropy 2021, 23(7), 840; https://doi.org/10.3390/e23070840 - 30 Jun 2021
Viewed by 639
Abstract
The Hurwitz complex continued fraction is a generalization of the nearest integer continued fraction. In this paper, we prove various results concerning extremes of the modulus of Hurwitz complex continued fraction digits. This includes a Poisson law and an extreme value law. The [...] Read more.
The Hurwitz complex continued fraction is a generalization of the nearest integer continued fraction. In this paper, we prove various results concerning extremes of the modulus of Hurwitz complex continued fraction digits. This includes a Poisson law and an extreme value law. The results are based on cusp estimates of the invariant measure about which information is still limited. In the process, we obtained several results concerning the extremes of nearest integer continued fractions as well. Full article
(This article belongs to the Special Issue Extreme Value Theory)
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Article
Potential Well in Poincaré Recurrence
Entropy 2021, 23(3), 379; https://doi.org/10.3390/e23030379 - 23 Mar 2021
Cited by 1 | Viewed by 724
Abstract
From a physical/dynamical system perspective, the potential well represents the proportional mass of points that escape the neighbourhood of a given point. In the last 20 years, several works have shown the importance of this quantity to obtain precise approximations for several recurrence [...] Read more.
From a physical/dynamical system perspective, the potential well represents the proportional mass of points that escape the neighbourhood of a given point. In the last 20 years, several works have shown the importance of this quantity to obtain precise approximations for several recurrence time distributions in mixing stochastic processes and dynamical systems. Besides providing a review of the different scaling factors used in the literature in recurrence times, the present work contributes two new results: (1) For ϕ-mixing and ψ-mixing processes, we give a new exponential approximation for hitting and return times using the potential well as the scaling parameter. The error terms are explicit and sharp. (2) We analyse the uniform positivity of the potential well. Our results apply to processes on countable alphabets and do not assume a complete grammar. Full article
(This article belongs to the Special Issue Extreme Value Theory)
Article
A Method for Confidence Intervals of High Quantiles
Entropy 2021, 23(1), 70; https://doi.org/10.3390/e23010070 - 04 Jan 2021
Viewed by 809
Abstract
The high quantile estimation of heavy tailed distributions has many important applications. There are theoretical difficulties in studying heavy tailed distributions since they often have infinite moments. There are also bias issues with the existing methods of confidence intervals (CIs) of high quantiles. [...] Read more.
The high quantile estimation of heavy tailed distributions has many important applications. There are theoretical difficulties in studying heavy tailed distributions since they often have infinite moments. There are also bias issues with the existing methods of confidence intervals (CIs) of high quantiles. This paper proposes a new estimator for high quantiles based on the geometric mean. The new estimator has good asymptotic properties as well as it provides a computational algorithm for estimating confidence intervals of high quantiles. The new estimator avoids difficulties, improves efficiency and reduces bias. Comparisons of efficiencies and biases of the new estimator relative to existing estimators are studied. The theoretical are confirmed through Monte Carlo simulations. Finally, the applications on two real-world examples are provided. Full article
(This article belongs to the Special Issue Extreme Value Theory)
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Article
Portfolio Tail Risk: A Multivariate Extreme Value Theory Approach
Entropy 2020, 22(12), 1425; https://doi.org/10.3390/e22121425 - 17 Dec 2020
Cited by 1 | Viewed by 1040
Abstract
This paper develops a method for assessing portfolio tail risk based on extreme value theory. The technique applies separate estimations of univariate series and allows for closed-form expressions for Value at Risk and Expected Shortfall. Its forecasting ability is tested on a portfolio [...] Read more.
This paper develops a method for assessing portfolio tail risk based on extreme value theory. The technique applies separate estimations of univariate series and allows for closed-form expressions for Value at Risk and Expected Shortfall. Its forecasting ability is tested on a portfolio of U.S. stocks. The in-sample goodness-of-fit tests indicate that the proposed approach is better suited for portfolio risk modeling under extreme market movements than comparable multivariate parametric methods. Backtesting across multiple quantiles demonstrates that the model cannot be rejected at any reasonable level of significance, even when periods of stress are included. Numerical simulations corroborate the empirical results. Full article
(This article belongs to the Special Issue Extreme Value Theory)
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