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Ferreira M. Clustering of extreme values: estimation and application. ADVANCES IN STATISTICAL ANALYSIS : ASTA : A JOURNAL OF THE GERMAN STATISTICAL SOCIETY 2023:1-25. [PMID: 37360852 PMCID: PMC10064624 DOI: 10.1007/s10182-023-00474-y] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/02/2022] [Accepted: 03/02/2023] [Indexed: 04/08/2023]
Abstract
The extreme value theory (EVT) encompasses a set of methods that allow inferring about the risk inherent to various phenomena in the scope of economic, financial, actuarial, environmental, hydrological, climatic sciences, as well as various areas of engineering. In many situations the clustering effect of high values may have an impact on the risk of occurrence of extreme phenomena. For example, extreme temperatures that last over time and result in drought situations, the permanence of intense rains leading to floods, stock markets in successive falls and consequent catastrophic losses. The extremal index is a measure of EVT associated with the degree of clustering of extreme values. In many situations, and under certain conditions, it corresponds to the arithmetic inverse of the average size of high-value clusters. The estimation of the extremal index generally entails two sources of uncertainty: the level at which high observations are considered and the identification of clusters. There are several contributions in the literature on the estimation of the extremal index, including methodologies to overcome the aforementioned sources of uncertainty. In this work we will revisit several existing estimators, apply automatic choice methods, both for the threshold and for the clustering parameter, and compare the performance of the methods. We will end with an application to meteorological data.
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Affiliation(s)
- Marta Ferreira
- Centro de Matemática, Universidade do Minho, Braga, Portugal
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Ferreira H, Ferreira M. A new blocks estimator for the extremal index. COMMUN STAT-THEOR M 2022. [DOI: 10.1080/03610926.2022.2050405] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/03/2022]
Affiliation(s)
- Helena Ferreira
- Centro de Matemática e Aplicações (CMA-UBI), Universidade da Beira Interior, Covilhã, Portugal
| | - Marta Ferreira
- Centro de Matemática, Universidade do Minho, Braga, Portugal
- Centro de Matemática Computacional e Estocástica, Universidade de Lisboa, Lisbon, Portugal
- Centro de Estatística e Aplicações, Universidade de Lisboa, Lisbon, Portugal
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Abstract
AbstractA Markov tree is a random vector indexed by the nodes of a tree whose distribution is determined by the distributions of pairs of neighbouring variables and a list of conditional independence relations. Upon an assumption on the tails of the Markov kernels associated to these pairs, the conditional distribution of the self-normalized random vector when the variable at the root of the tree tends to infinity converges weakly to a random vector of coupled random walks called a tail tree. If, in addition, the conditioning variable has a regularly varying tail, the Markov tree satisfies a form of one-component regular variation. Changing the location of the root, that is, changing the conditioning variable, yields a different tail tree. When the tails of the marginal distributions of the conditioning variables are balanced, these tail trees are connected by a formula that generalizes the time change formula for regularly varying stationary time series. The formula is most easily understood when the various one-component regular variation statements are tied up into a single multi-component statement. The theory of multi-component regular variation is worked out for general random vectors, not necessarily Markov trees, with an eye towards other models, graphical or otherwise.
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Gomes DP, Neves MM. Extremal index blocks estimator: the threshold and the block size choice. J Appl Stat 2020; 47:2846-2861. [DOI: 10.1080/02664763.2020.1720626] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/25/2022]
Affiliation(s)
- D. Prata Gomes
- Faculdade de Ciências e Tecnologia and CMA, Universidade Nova de Lisboa, Lisboa, Portugal
| | - M. Manuela Neves
- Instituto Superior de Agronomia, and CEAUL, Universidade de Lisboa, Lisboa, Portugal
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Kulik R, Soulier P, Wintenberger O. The tail empirical process of regularly varying functions of geometrically ergodic Markov chains. Stoch Process Their Appl 2019. [DOI: 10.1016/j.spa.2018.11.014] [Citation(s) in RCA: 13] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
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Abstract
The extremes of a univariate Markov chain with regularly varying stationary marginal distribution and asymptotically linear behavior are known to exhibit a multiplicative random walk structure called the tail chain. In this paper we extend this fact to Markov chains with multivariate regularly varying marginal distributions inRd. We analyze both the forward and the backward tail process and show that they mutually determine each other through a kind of adjoint relation. In a broader setting, we will show that even for non-Markovian underlying processes a Markovian forward tail chain always implies that the backward tail chain is also Markovian. We analyze the resulting class of limiting processes in detail. Applications of the theory yield the asymptotic distribution of both the past and the future of univariate and multivariate stochastic difference equations conditioned on an extreme event.
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Yun S. The distributions of cluster functionals of extreme events in a dth-order Markov chain. J Appl Probab 2016. [DOI: 10.1239/jap/1014842266] [Citation(s) in RCA: 21] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
The paper concerns the asymptotic distributions of cluster functionals of extreme events in a dth-order stationary Markov chain {Xn, n = 1,2,…} for which the joint distribution of (X1,…,Xd+1) is absolutely continuous. Under some distributional assumptions for {Xn}, we establish weak convergence for a class of cluster functionals and obtain representations for the asymptotic distributions which are well suited for simulation. A number of examples important in applications are presented to demonstrate the usefulness of the results.
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The extremal index and clustering of high values for derived stationary sequences. J Appl Probab 2016. [DOI: 10.1017/s0021900200103456] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
Abstract
Given a sequence of independent identically distributed random variables, we derive a moving-maximum sequence (with random translations). The extremal index of the derived sequence is computed and the limiting behaviour of clusters of high values is studied. We are then given two or more independent stationary sequences whose extremal indices are known. We derive a new stationary sequence by taking either a pointwise maximum or by a mixture of the original sequences. In each case, we compute the extremal index of the derived sequence.
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Abstract
We consider extreme value theory for a class of stationary Markov chains with values in ℝd. The asymptotic distribution of M
n
, the vector of componentwise maxima, is determined under mild dependence restrictions and suitable assumptions on the marginal distribution and the transition probabilities of the chain. This is achieved through computation of a multivariate extremal index of the sequence, extending results of Smith [26] and Perfekt [21] to a multivariate setting. As a by-product, we obtain results on extremes of higher-order, real-valued Markov chains. The results are applied to a frequently studied random difference equation.
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Abstract
For arbitrary stationary sequences of random variables satisfying a mild mixing condition, distributional approximations are established for functionals of clusters of exceedances over a high threshold. The approximations are in terms of the distribution of the process conditionally on the event that the first variable exceeds the threshold. This conditional distribution is shown to converge to a nontrivial limit if the finite-dimensional distributions of the process are in the domain of attraction of a multivariate extreme-value distribution. In this case, therefore, limit distributions are obtained for functionals of clusters of extremes, thereby generalizing results for higher-order stationary Markov chains by Yun (2000).
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Abstract
An asymptotic model for the extreme behavior of certain Markov chains is the ‘tail chain’. Generally taking the form of a multiplicative random walk, it is useful in deriving extremal characteristics, such as point process limits. We place this model in a more general context, formulated in terms of extreme value theory for transition kernels, and extend it by formalizing the distinction between extreme and nonextreme states. We make the link between the update function and transition kernel forms considered in previous work, and we show that the tail chain model leads to a multivariate regular variation property of the finite-dimensional distributions under assumptions on the marginal tails alone.
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Winter HC, Tawn JA. Modelling heatwaves in central France: a case-study in extremal dependence. J R Stat Soc Ser C Appl Stat 2015. [DOI: 10.1111/rssc.12121] [Citation(s) in RCA: 19] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
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Markov Tail Chains. J Appl Probab 2014. [DOI: 10.1017/s002190020001202x] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
Abstract
The extremes of a univariate Markov chain with regularly varying stationary marginal distribution and asymptotically linear behavior are known to exhibit a multiplicative random walk structure called the tail chain. In this paper we extend this fact to Markov chains with multivariate regularly varying marginal distributions in R
d
. We analyze both the forward and the backward tail process and show that they mutually determine each other through a kind of adjoint relation. In a broader setting, we will show that even for non-Markovian underlying processes a Markovian forward tail chain always implies that the backward tail chain is also Markovian. We analyze the resulting class of limiting processes in detail. Applications of the theory yield the asymptotic distribution of both the past and the future of univariate and multivariate stochastic difference equations conditioned on an extreme event.
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Affiliation(s)
- Paola Bortot
- Dipartimento di Scienze Statistiche; Università di Bologna
| | - Carlo Gaetan
- Dipartimento di Scienze Ambientali, Informatica e Statistica; Università Ca' Foscari Venezia
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Estimation of Extreme Values by the Average Conditional Exceedance Rate Method. JOURNAL OF PROBABILITY AND STATISTICS 2013. [DOI: 10.1155/2013/797014] [Citation(s) in RCA: 20] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022] Open
Abstract
This paper details a method for extreme value prediction on the basis of a sampled time series. The method is specifically designed to account for statistical dependence between the sampled data points in a precise manner. In fact, if properly used, the new method will provide statistical estimates of the exact extreme value distribution provided by the data in most cases of practical interest. It avoids the problem of having to decluster the data to ensure independence, which is a requisite component in the application of, for example, the standard peaks-over-threshold method. The proposed method also targets the use of subasymptotic data to improve prediction accuracy. The method will be demonstrated by application to both synthetic and real data. From a practical point of view, it seems to perform better than the POT and block extremes methods, and, with an appropriate modification, it is directly applicable to nonstationary time series.
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Basrak B, Krizmanić D, Segers J. A functional limit theorem for dependent sequences with infinite variance stable limits. ANN PROBAB 2012. [DOI: 10.1214/11-aop669] [Citation(s) in RCA: 42] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Balan R, Louhichi S. Explicit Conditions for the Convergence of Point Processes Associated to Stationary Arrays. ELECTRONIC COMMUNICATIONS IN PROBABILITY 2010. [DOI: 10.1214/ecp.v15-1563] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Ledford AW, Tawn JA. Diagnostics for dependence within time series extremes. J R Stat Soc Series B Stat Methodol 2003. [DOI: 10.1111/1467-9868.00400] [Citation(s) in RCA: 87] [Impact Index Per Article: 4.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
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The distributions of cluster functionals of extreme events in a dth-order Markov chain. J Appl Probab 2000. [DOI: 10.1017/s0021900200015230] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/06/2022]
Abstract
The paper concerns the asymptotic distributions of cluster functionals of extreme events in a dth-order stationary Markov chain {X
n
, n = 1,2,…} for which the joint distribution of (X
1,…,X
d+1) is absolutely continuous. Under some distributional assumptions for {X
n
}, we establish weak convergence for a class of cluster functionals and obtain representations for the asymptotic distributions which are well suited for simulation. A number of examples important in applications are presented to demonstrate the usefulness of the results.
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The extremal index in 10 seconds. J Appl Probab 1997. [DOI: 10.1017/s0021900200101494] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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Hsing T, Hüsler J, Reiss RD. The extremes of a triangular array of normal random variables. ANN APPL PROBAB 1996. [DOI: 10.1214/aoap/1034968149] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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