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Mathematics
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New Advances and Applications of Extreme Value Theory
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New Advances and Applications of Extreme Value Theory

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Probability and Statistics".

Deadline for manuscript submissions: closed (30 June 2023) | Viewed by 12138

Special Issue Editor

Prof. Dr. Natalia Markovich

E-Mail Website
Guest Editor
V.A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences, Profsoyusnaya str., 65, 117997 Moscow, Russia
Interests: mathematical statistics; applied probability; nonparametric estimation; extreme value theory; statistical analysis of heavy-tailed distributions and rare events; time series; performance analysis of random networks

Special Issue Information

Dear Colleagues,

In recent years probabilistic and statistical aspects of extreme value analysis have attracted the interest of many researchers due to their numerous applications in climate and atmospheric science, industrial risks, geosciences, hydrology, finance, economics and insurance, biosciences, physics, and telecommunications and stochastic networks.

As extremal events may lead to tremendous risks, the actuality of the intensive development of both extreme value theory and its practical outcomes is evident.

Regarding the theory of extreme values, the tail and extremal indices represent central characteristics. There are still many problems with regard to their nonparametric estimation by stationary random sequences as well as random graphs, which can be heterogeneous. These problems concern the optimal choice of tuning parameters like the threshold and the number of the largest order statistics. The extremal index as a measure of a local dependence determines the clustering structure of the underlying stochastic process. It constitutes another important characteristic of extreme value theory whose estimation has to be studied more intensively. Clusters of exceedances are present in stochastic processes due to local dependence and heavy tails. Many nontypical events can be expressed in terms of such clusters. Indeed, the study of cluster characteristics presents many open problems.

This Special Issue of Mathematics is devoted to new theoretical advances and applications in extreme value theory. It expects to receive contributions by applied mathematicians of various profiles in the form of scientific articles showing their achievements and confirming the relevance of their current and future research.

The aim of this Special Issue is to collect articles that develop methods for modeling, forecasting, and nonparametrically estimating extremal characteristics based on incomplete data and in the presence of heavy tails of the underlying distributions.

Prof. Dr. Natalia Markovich
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Published Papers (9 papers)

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Research

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14 pages, 303 KiB  
Article
The Limit Properties of Maxima of Stationary Gaussian Sequences Subject to Random Replacing
by Yuwei Li and Zhongquan Tan
Mathematics 2023, 11(14), 3155; https://doi.org/10.3390/math11143155 - 18 Jul 2023
Viewed by 571
Abstract
In applications, missing data may occur randomly and some relevant datum are often used to replace the missing ones. This article mainly explores the influence of the degree of dependence of stationary Gaussian sequences on the joint asymptotic distribution of the maximum of [...] Read more.
In applications, missing data may occur randomly and some relevant datum are often used to replace the missing ones. This article mainly explores the influence of the degree of dependence of stationary Gaussian sequences on the joint asymptotic distribution of the maximum of the Gaussian sequence and its maximum when the sequence is subject to random replacing. Full article
(This article belongs to the Special Issue New Advances and Applications of Extreme Value Theory)
16 pages, 335 KiB  
Article
Polynomial Recurrence for SDEs with a Gradient-Type Drift, Revisited
by Alexander Veretennikov
Mathematics 2023, 11(14), 3096; https://doi.org/10.3390/math11143096 - 13 Jul 2023
Viewed by 699
Abstract
In this paper, polynomial recurrence bounds for a class of stochastic differential equations with a rotational symmetric gradient type drift and an additive Wiener process are established, as well as certain a priori moment inequalities for solutions. The key feature of this paper [...] Read more.
In this paper, polynomial recurrence bounds for a class of stochastic differential equations with a rotational symmetric gradient type drift and an additive Wiener process are established, as well as certain a priori moment inequalities for solutions. The key feature of this paper is that the approach does not use Lyapunov functions because it is not clear how to construct them. The method based on Dynkin’s (nonrandom) chain of equations is applied instead. Another key feature is that the asymptotic conditions on the potential near infinity are assumed as inequalities—which allows for more flexibility compared to a single limit at infinity, making it less restrictive. Full article
(This article belongs to the Special Issue New Advances and Applications of Extreme Value Theory)
26 pages, 3052 KiB  
Article
A New Reciprocal Weibull Extension for Modeling Extreme Values with Risk Analysis under Insurance Data
by Haitham M. Yousof, Yusra Tashkandy, Walid Emam, M. Masoom Ali and Mohamed Ibrahim
Mathematics 2023, 11(4), 966; https://doi.org/10.3390/math11040966 - 13 Feb 2023
Cited by 8 | Viewed by 1234
Abstract
Probability-based distributions might be able to explain risk exposure well. Usually, one number, or at the very least, a limited number of numbers called the key risk indicators (KRIs), are used to describe the level of risk exposure. These risk exposure values, which [...] Read more.
Probability-based distributions might be able to explain risk exposure well. Usually, one number, or at the very least, a limited number of numbers called the key risk indicators (KRIs), are used to describe the level of risk exposure. These risk exposure values, which are undeniably the outcome of a specific model, are frequently referred to as essential critical risk indicators. Five key risk indicators, including value-at-risk, tail variance, tail-value-at-risk, and tail mean-variance, were also used for describing the risk exposure under the reinsurance revenues data. These measurements were created for the proposed model; hence, this paper presents a novel distribution for this purpose. Relevant statistical properties are derived, including the generating function, ordinary moments, and incomplete moments. Special attention is devoted to the applicability of the new model under extreme data sets. Three applications to real data show the usefulness and adaptability of the proposed model. The new model proved its superiority against many well-known related models. Five key risk indicators are employed for analyzing the risk level under the reinsurance revenues dataset. An application is provided along with its relevant numerical analysis and panels. Some useful results are identified and highlighted. Full article
(This article belongs to the Special Issue New Advances and Applications of Extreme Value Theory)
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26 pages, 20635 KiB  
Article
Tail Index Estimation of PageRanks in Evolving Random Graphs
by Natalia Markovich, Maksim Ryzhov and Marijus Vaičiulis
Mathematics 2022, 10(16), 3026; https://doi.org/10.3390/math10163026 - 22 Aug 2022
Cited by 3 | Viewed by 1162
Abstract
Random graphs are subject to the heterogeneities of the distributions of node indices and their dependence structures. Superstar nodes to which a large proportion of nodes attach in the evolving graphs are considered. In the present paper, a statistical analysis of the extremal [...] Read more.
Random graphs are subject to the heterogeneities of the distributions of node indices and their dependence structures. Superstar nodes to which a large proportion of nodes attach in the evolving graphs are considered. In the present paper, a statistical analysis of the extremal part of random graphs is considered. We used the extreme value theory regarding sums and maxima of non-stationary random length sequences to evaluate the tail index of the PageRanks and max-linear models of superstar nodes in the evolving graphs where existing nodes or edges can be deleted or not. The evolution is provided by a linear preferential attachment. Our approach is based on the analysis of maxima and sums of the node PageRanks over communities (block maxima and block sums), which can be independent or weakly dependent random variables. By an empirical study, it was found that tail indices of the block maxima and block sums are close to the minimum tail index of representative series extracted from the communities. The tail indices are estimated by data of simulated graphs. Full article
(This article belongs to the Special Issue New Advances and Applications of Extreme Value Theory)
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16 pages, 333 KiB  
Article
Infinite-Server Resource Queueing Systems with Different Types of Markov-Modulated Poisson Process and Renewal Arrivals
by Ekaterina Pankratova, Svetlana Moiseeva and Mais Farkhadov
Mathematics 2022, 10(16), 2962; https://doi.org/10.3390/math10162962 - 17 Aug 2022
Cited by 3 | Viewed by 1305
Abstract
In this paper, we propose models that significantly expand the scope of practical applications, namely, queueing systems with various nodes for processing heterogeneous data that require arbitrary resource capacities for their service. When a customer arrives in the system, the customer typeis randomly [...] Read more.
In this paper, we propose models that significantly expand the scope of practical applications, namely, queueing systems with various nodes for processing heterogeneous data that require arbitrary resource capacities for their service. When a customer arrives in the system, the customer typeis randomly selected according to a set of probabilities. Then the customer goes to the server of the corresponding device type, where its service is performed during a random time period with a distribution function depending on the type of customer. Moreover, each customer requires a random amount of resources, of which the distribution function also depends on the customer type, but is independent of its service time. The aim of this research was to develop a heterogeneous queueing resource system with an unlimited number of servers and an arrival process in the form of a Markov-modulated Poisson process or stationary renewal process, and with requests for a random number of heterogeneous resources. We have performed analysis under conditions of growing intensity of the arrival process. Here we formulate the theorems and prove that under high-load conditions, the joint asymptotic probability distribution of the n-dimensional process of the total amounts of the occupied resources in the system is a multidimensional Gaussian distribution with parameters that are dependent on the type of arrival process. As a result of numerical and simulation experiments, conclusions are drawn on the limits of the applicability of the obtained asymptotic results. The dependence of the convergence of experimental results on the type of distribution of the system parameters (including the distributions of the service time and of the customer capacity) are also studied. The results of the approximations may be applied to estimating the optimal total number of resources for a system with a limited amount of resources. Full article
(This article belongs to the Special Issue New Advances and Applications of Extreme Value Theory)
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28 pages, 653 KiB  
Article
Universal Local Linear Kernel Estimators in Nonparametric Regression
by Yuliana Linke, Igor Borisov, Pavel Ruzankin, Vladimir Kutsenko, Elena Yarovaya and Svetlana Shalnova
Mathematics 2022, 10(15), 2693; https://doi.org/10.3390/math10152693 - 29 Jul 2022
Cited by 11 | Viewed by 1848
Abstract
New local linear estimators are proposed for a wide class of nonparametric regression models. The estimators are uniformly consistent regardless of satisfying traditional conditions of dependence of design elements. The estimators are the solutions of a specially weighted least-squares method. The design can [...] Read more.
New local linear estimators are proposed for a wide class of nonparametric regression models. The estimators are uniformly consistent regardless of satisfying traditional conditions of dependence of design elements. The estimators are the solutions of a specially weighted least-squares method. The design can be fixed or random and does not need to meet classical regularity or independence conditions. As an application, several estimators are constructed for the mean of dense functional data. The theoretical results of the study are illustrated by simulations. An example of processing real medical data from the epidemiological cross-sectional study ESSE-RF is included. We compare the new estimators with the estimators best known for such studies. Full article
(This article belongs to the Special Issue New Advances and Applications of Extreme Value Theory)
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15 pages, 2637 KiB  
Article
A Novel Tree Ensemble Model to Approximate the Generalized Extreme Value Distribution Parameters of the PM2.5 Maxima in the Mexico City Metropolitan Area
by Alejandro Ivan Aguirre-Salado, Sonia Venancio-Guzmán, Carlos Arturo Aguirre-Salado and Alicia Santiago-Santos
Mathematics 2022, 10(12), 2056; https://doi.org/10.3390/math10122056 - 14 Jun 2022
Viewed by 1395
Abstract
We introduce a novel spatial model based on the distribution of generalized extreme values (GEVs) and tree ensemble models to analyze the maximum concentrations levels of particulate matter with a diameter of less than 2.5 microns (PM2.5) in the Mexico City metropolitan area [...] Read more.
We introduce a novel spatial model based on the distribution of generalized extreme values (GEVs) and tree ensemble models to analyze the maximum concentrations levels of particulate matter with a diameter of less than 2.5 microns (PM2.5) in the Mexico City metropolitan area during the period 2003–2021. Spatial trends were modeled through a decision tree in the context of a non-stationary GEV model. We used a tree ensemble model as a predictor of GEV parameters to approximate nonlinear trends. The decision tree was built by using a greedy stagewise approach, the objective function of which was the log-likelihood. We verified the validity of our model by means of the likelihood and Akaike’s information criterion (AIC). The maps of the generalized extreme value parameters on the spatial plane show the existence of differentiated local trends in the extreme values of PM2.5 in the study area. The results indicated strong evidence of an increase in the west–east direction of the study area. A spatial map of risk with maximum concentration levels of PM2.5 in a period of 25 years was built. Full article
(This article belongs to the Special Issue New Advances and Applications of Extreme Value Theory)
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45 pages, 1322 KiB  
Article
Branching Random Walks with Two Types of Particles on Multidimensional Lattices
by Iuliia Makarova, Daria Balashova, Stanislav Molchanov and Elena Yarovaya
Mathematics 2022, 10(6), 867; https://doi.org/10.3390/math10060867 - 09 Mar 2022
Cited by 5 | Viewed by 1391
Abstract
We consider a continuous-time branching random walk on a multidimensional lattice with two types of particles and an infinite number of initial particles. The main results are devoted to the study of the generating function and the limiting behavior of the moments of [...] Read more.
We consider a continuous-time branching random walk on a multidimensional lattice with two types of particles and an infinite number of initial particles. The main results are devoted to the study of the generating function and the limiting behavior of the moments of subpopulations generated by a single particle of each type. We assume that particle types differ from each other not only by the laws of branching, as in multi-type branching processes, but also by the laws of walking. For a critical branching process at each lattice point and recurrent random walk of particles, the effect of limit spatial clustering of particles over the lattice is studied. A model illustrating epidemic propagation is also considered. In this model, we consider two types of particles: infected and immunity generated. Initially, there is an infected particle that can infect others. Here, for the local number of particles of each type at a lattice point, we study the moments and their limiting behavior. Additionally, the effect of intermittency of the infected particles is studied for a supercritical branching process at each lattice point. Simulations are presented to demonstrate the effect of limit clustering for the epidemiological model. Full article
(This article belongs to the Special Issue New Advances and Applications of Extreme Value Theory)
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35 pages, 812 KiB  
Review
Extreme Value Statistics for Evolving Random Networks
by Natalia Markovich and Marijus Vaičiulis
Mathematics 2023, 11(9), 2171; https://doi.org/10.3390/math11092171 - 05 May 2023
Viewed by 1259
Abstract
Our objective is to survey recent results concerning the evolution of random networks and related extreme value statistics, which are a subject of interest due to numerous applications. Our survey concerns the statistical methodology but not the structure of random networks. We focus [...] Read more.
Our objective is to survey recent results concerning the evolution of random networks and related extreme value statistics, which are a subject of interest due to numerous applications. Our survey concerns the statistical methodology but not the structure of random networks. We focus on the problems arising in evolving networks mainly due to the heavy-tailed nature of node indices. Tail and extremal indices of the node influence characteristics like in-degrees, out-degrees, PageRanks, and Max-linear models arising in the evolving random networks are discussed. Related topics like preferential and clustering attachments, community detection, stationarity and dependence of graphs, information spreading, finding the most influential leading nodes and communities, and related methods are surveyed. This survey tries to propose possible solutions to unsolved problems, like testing the stationarity and dependence of random graphs using known results obtained for random sequences. We provide a discussion of unsolved or insufficiently developed problems like the distribution of triangle and circle counts in evolving networks, or the clustering attachment and the local dependence of the modularity, the impact of node or edge deletion at each step of evolution on extreme value statistics, among many others. Considering existing techniques of community detection, we pay attention to such related topics as coloring graphs and anomaly detection by machine learning algorithms based on extreme value theory. In order to understand how one can compute tail and extremal indices on random graphs, we provide a structured and comprehensive review of their estimators obtained for random sequences. Methods to calculate the PageRank and PageRank vector are shortly presented. This survey aims to provide a better understanding of the directions in which the study of random networks has been done and how extreme value analysis developed for random sequences can be applied to random networks. Full article
(This article belongs to the Special Issue New Advances and Applications of Extreme Value Theory)
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