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Estimation of marginal excess moments for Weibull-type distributions 魏布尔型分布的边际超额矩估计
IF 1.3 3区 数学
Extremes Pub Date : 2024-09-02 DOI: 10.1007/s10687-024-00494-0
Yuri Goegebeur, Armelle Guillou, Jing Qin
{"title":"Estimation of marginal excess moments for Weibull-type distributions","authors":"Yuri Goegebeur, Armelle Guillou, Jing Qin","doi":"10.1007/s10687-024-00494-0","DOIUrl":"https://doi.org/10.1007/s10687-024-00494-0","url":null,"abstract":"<p>We consider the estimation of the marginal excess moment (<i>MEM</i>), which is defined for a random vector (<i>X</i>, <i>Y</i>) and a parameter <span>(beta &gt;0)</span> as <span>(mathbb {E}[(X-Q_{X}(1-p))_{+}^{beta }|Y&gt; Q_{Y}(1-p)])</span> provided <span>(mathbb {E}|X|^{beta }&lt; infty )</span>, and where <span>(y_{+}:=max (0,y))</span>, <span>(Q_{X})</span> and <span>(Q_{Y})</span> are the quantile functions of <i>X</i> and <i>Y</i> respectively, and <span>(pin (0,1))</span>. Our interest is in the situation where the random variable <i>X</i> is of Weibull-type while the distribution of <i>Y</i> is kept general, the extreme dependence structure of (<i>X</i>, <i>Y</i>) converges to that of a bivariate extreme value distribution, and we let <span>(p downarrow 0)</span> as the sample size <span>(n rightarrow infty )</span>. By using extreme value arguments we introduce an estimator for the marginal excess moment and we derive its limiting distribution. The finite sample properties of the proposed estimator are evaluated with a simulation study and the practical applicability is illustrated on a dataset of wave heights and wind speeds.</p>","PeriodicalId":49274,"journal":{"name":"Extremes","volume":"51 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142182390","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Probability of entering an orthant by correlated fractional Brownian motion with drift: exact asymptotics 带漂移的相关分数布朗运动进入正交的概率：精确渐近学
IF 1.3 3区 数学
Extremes Pub Date : 2024-08-29 DOI: 10.1007/s10687-024-00489-x
Krzysztof Dȩbicki, Lanpeng Ji, Svyatoslav Novikov
{"title":"Probability of entering an orthant by correlated fractional Brownian motion with drift: exact asymptotics","authors":"Krzysztof Dȩbicki, Lanpeng Ji, Svyatoslav Novikov","doi":"10.1007/s10687-024-00489-x","DOIUrl":"https://doi.org/10.1007/s10687-024-00489-x","url":null,"abstract":"<p>For <span>({varvec{B}_{H}(t)= (B_{H,1}(t) ,ldots ,B_{H,d}(t))^{{top }},tge 0})</span>, where <span>({B_{H,i}(t),tge 0}, 1le ile d)</span> are mutually independent fractional Brownian motions, we obtain the exact asymptotics of </p><span>$$mathbb P (exists tge 0: A varvec{B}_{H}(t) - varvec{mu }t &gt;varvec{nu }u), urightarrow infty ,$$</span><p>where <i>A</i> is a non-singular <span>(dtimes d)</span> matrix and <span>(varvec{mu }=(mu _1,ldots , mu _d)^{{top }}in mathbb {R}^d)</span>, <span>(varvec{nu }=(nu _1, ldots , nu _d)^{{top }} in mathbb {R}^d)</span> are such that there exists some <span>(1le ile d)</span> such that <span>(mu _i&gt;0, nu _i&gt;0.)</span></p>","PeriodicalId":49274,"journal":{"name":"Extremes","volume":"1 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142182392","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Improving estimation for asymptotically independent bivariate extremes via global estimators for the angular dependence function 通过角依赖函数的全局估计器改进渐近独立双变量极值的估计方法
IF 1.3 3区 数学
Extremes Pub Date : 2024-08-13 DOI: 10.1007/s10687-024-00490-4
C. J. R. Murphy-Barltrop, J. L. Wadsworth, E. F. Eastoe
{"title":"Improving estimation for asymptotically independent bivariate extremes via global estimators for the angular dependence function","authors":"C. J. R. Murphy-Barltrop, J. L. Wadsworth, E. F. Eastoe","doi":"10.1007/s10687-024-00490-4","DOIUrl":"https://doi.org/10.1007/s10687-024-00490-4","url":null,"abstract":"<p>Modelling the extremal dependence of bivariate variables is important in a wide variety of practical applications, including environmental planning, catastrophe modelling and hydrology. The majority of these approaches are based on the framework of bivariate regular variation, and a wide range of literature is available for estimating the dependence structure in this setting. However, such procedures are only applicable to variables exhibiting asymptotic dependence, even though asymptotic independence is often observed in practice. In this paper, we consider the so-called ‘angular dependence function’; this quantity summarises the extremal dependence structure for asymptotically independent variables. Until recently, only pointwise estimators of the angular dependence function have been available. We introduce a range of global estimators and compare them to another recently introduced technique for global estimation through a systematic simulation study, and a case study on river flow data from the north of England, UK.</p>","PeriodicalId":49274,"journal":{"name":"Extremes","volume":"49 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142182394","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The longest edge in discrete and continuous long-range percolation 离散和连续长程渗流中的最长边缘
IF 1.3 3区 数学
Extremes Pub Date : 2024-06-18 DOI: 10.1007/s10687-024-00488-y
Arnaud Rousselle, Ercan Sönmez
{"title":"The longest edge in discrete and continuous long-range percolation","authors":"Arnaud Rousselle, Ercan Sönmez","doi":"10.1007/s10687-024-00488-y","DOIUrl":"https://doi.org/10.1007/s10687-024-00488-y","url":null,"abstract":"<p>We consider the random connection model in which an edge between two Poisson points at distance <i>r</i> is present with probability <i>g</i>(<i>r</i>). We conduct an extreme value analysis on this model, namely by investigating the longest edge with at least one endpoint within some finite observation window, as the volume of this window tends to infinity. We show that the length of the latter, after normalizing by some appropriate centering and scaling sequences, asymptotically behaves like one of each of the three extreme value distributions, depending on choices of the probability <i>g</i>(<i>r</i>). We prove our results by giving a formal construction of the model by means of a marked Poisson point process and a Poisson coupling argument adapted to this construction. In addition, we study a discrete variant of the model. We obtain parameter regimes with varying behavior in our findings and an unexpected singularity.</p>","PeriodicalId":49274,"journal":{"name":"Extremes","volume":"24 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141501119","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Correlation of powers of Hüsler–Reiss vectors and Brown–Resnick fields, and application to insured wind losses Hüsler-Reiss 矢量幂与 Brown-Resnick 场的相关性，以及在投保风力损失中的应用
IF 1.3 3区 数学
Extremes Pub Date : 2024-06-14 DOI: 10.1007/s10687-023-00474-w
Erwan Koch
{"title":"Correlation of powers of Hüsler–Reiss vectors and Brown–Resnick fields, and application to insured wind losses","authors":"Erwan Koch","doi":"10.1007/s10687-023-00474-w","DOIUrl":"https://doi.org/10.1007/s10687-023-00474-w","url":null,"abstract":"<p>Hüsler–Reiss vectors and Brown–Resnick fields are popular models in multivariate and spatial extreme-value theory, respectively, and are widely used in applications. We provide analytical formulas for the correlation between powers of the components of the bivariate Hüsler–Reiss vector, extend these to the case of the Brown–Resnick field, and thoroughly study the properties of the resulting dependence measure. The use of correlation is justified by spatial risk theory, while power transforms are insightful when taking correlation as dependence measure, and are moreover very suited damage functions for weather events such as wind extremes or floods. This makes our theoretical results worthwhile for, e.g., actuarial applications. We finally perform a case study involving insured losses from extreme wind speeds in Germany, and obtain valuable conclusions for the insurance industry.</p>","PeriodicalId":49274,"journal":{"name":"Extremes","volume":"6 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141501120","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Spatial extremes and stochastic geometry for Gaussian-based peaks-over-threshold processes 基于高斯的峰过阈值过程的空间极值和随机几何
IF 1.3 3区 数学
Extremes Pub Date : 2024-05-01 DOI: 10.1007/s10687-024-00487-z
Elena Di Bernardino, Anne Estrade, Thomas Opitz
{"title":"Spatial extremes and stochastic geometry for Gaussian-based peaks-over-threshold processes","authors":"Elena Di Bernardino, Anne Estrade, Thomas Opitz","doi":"10.1007/s10687-024-00487-z","DOIUrl":"https://doi.org/10.1007/s10687-024-00487-z","url":null,"abstract":"<p>Geometric properties of exceedance regions above a given quantile level provide meaningful theoretical and statistical characterizations for stochastic processes defined on Euclidean domains. Many theoretical results have been obtained for excursions of Gaussian processes and include expected values of the so-called Lipschitz–Killing curvatures (LKCs), such as the area, perimeter and Euler characteristic in two-dimensional Euclidean space. In this paper, we derive novel results for the expected LKCs of excursion sets of more general processes whose construction is based on location or scale mixtures of a Gaussian process, which means that the mean or the standard deviation, respectively, of a stationary Gaussian process is a random variable. We first present exact formulas for peaks-over-threshold-stable limit processes (so-called Pareto processes) arising from the use of Gaussian or log-Gaussian spectral functions in the spectral construction of max-stable processes. These peaks-over-threshold limits are known to arise for Gaussian location or scale mixtures if the mixing distributions satisfies certain regular-variation properties. As a second important result, we show that expected LKCs of excursion sets of such general mixture processes converge to the corresponding expressions of their Pareto process limits. We further provide exact subasymptotic formulas of expected LKCs for various specific choices of the distribution of the mixing variable. Finally, we discuss consistent empirical estimation of LKCs of exceedance regions and implement numerical experiments to illustrate the rate of convergence towards asymptotic expressions. An application to daily temperature data simulated by climate models for the period 1951–2005 over a regular pixel grid covering continental France showcases the practical utility of the new results.</p>","PeriodicalId":49274,"journal":{"name":"Extremes","volume":"101 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140833322","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Stochastic ordering in multivariate extremes 多元极值中的随机排序
IF 1.3 3区 数学
Extremes Pub Date : 2024-05-01 DOI: 10.1007/s10687-024-00486-0
Michela Corradini, Kirstin Strokorb
{"title":"Stochastic ordering in multivariate extremes","authors":"Michela Corradini, Kirstin Strokorb","doi":"10.1007/s10687-024-00486-0","DOIUrl":"https://doi.org/10.1007/s10687-024-00486-0","url":null,"abstract":"<p>The article considers the multivariate stochastic orders of upper orthants, lower orthants and positive quadrant dependence (PQD) among simple max-stable distributions and their exponent measures. It is shown for each order that it holds for the max-stable distribution if and only if it holds for the corresponding exponent measure. The finding is non-trivial for upper orthants (and hence PQD order). From dimension <span>(dge 3)</span> these three orders are not equivalent and a variety of phenomena can occur. However, every simple max-stable distribution PQD-dominates the corresponding independent model and is PQD-dominated by the fully dependent model. Among parametric models the asymmetric Dirichlet family and the Hüsler-Reiß family turn out to be PQD-ordered according to the natural order within their parameter spaces. For the Hüsler-Reiß family this holds true even for the supermodular order.</p>","PeriodicalId":49274,"journal":{"name":"Extremes","volume":"41 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140842299","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Generalized pareto regression trees for extreme event analysis 用于极端事件分析的广义帕累托回归树
IF 1.3 3区 数学
Extremes Pub Date : 2024-03-23 DOI: 10.1007/s10687-024-00485-1
Sébastien Farkas, Antoine Heranval, Olivier Lopez, Maud Thomas
{"title":"Generalized pareto regression trees for extreme event analysis","authors":"Sébastien Farkas, Antoine Heranval, Olivier Lopez, Maud Thomas","doi":"10.1007/s10687-024-00485-1","DOIUrl":"https://doi.org/10.1007/s10687-024-00485-1","url":null,"abstract":"<p>This paper derives finite sample results to assess the consistency of Generalized Pareto regression trees introduced by Farkas et al. (Insur. Math. Econ. 98:92–105, 2021) as tools to perform extreme value regression for heavy-tailed distributions. This procedure allows the constitution of classes of observations with similar tail behaviors depending on the value of the covariates, based on a recursive partition of the sample and simple model selection rules. The results we provide are obtained from concentration inequalities, and are valid for a finite sample size. A misspecification bias that arises from the use of a “Peaks over Threshold” approach is also taken into account. Moreover, the derived properties legitimate the pruning strategies, that is the model selection rules, used to select a proper tree that achieves a compromise between simplicity and goodness-of-fit. The methodology is illustrated through a simulation study, and a real data application in insurance for natural disasters.</p>","PeriodicalId":49274,"journal":{"name":"Extremes","volume":"16 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-03-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140197074","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On approximating dependence function and its derivatives 关于近似依存函数及其导数
IF 1.3 3区 数学
Extremes Pub Date : 2024-03-15 DOI: 10.1007/s10687-024-00484-2
Nader Tajvidi
{"title":"On approximating dependence function and its derivatives","authors":"Nader Tajvidi","doi":"10.1007/s10687-024-00484-2","DOIUrl":"https://doi.org/10.1007/s10687-024-00484-2","url":null,"abstract":"<p>Bivariate extreme value distributions can be used to model dependence of observations from random variables in extreme levels. There is no finite dimensional parametric family for these distributions, but they can be characterized by a certain one-dimensional function which is known as Pickands dependence function. In many applications the general approach is to estimate the dependence function with a non-parametric method and then conduct further analysis based on the estimate. Although this approach is flexible in the sense that it does not impose any special structure on the dependence function, its main drawback is that the estimate is not available in a closed form. This paper provides some theoretical results which can be used to find a closed form approximation for an exact or an estimate of a twice differentiable dependence function and its derivatives. We demonstrate the methodology by calculating approximations for symmetric and asymmetric logistic dependence functions and their second derivatives. We show that the theory can be even applied to approximating a non-parametric estimate of dependence function using a convex optimization algorithm. Other discussed applications include a procedure for testing whether an estimate of dependence function can be assumed to be symmetric and estimation of the concordance measures of a bivariate extreme value distribution. Finally, an Australian annual maximum temperature dataset is used to illustrate how the theory can be used to build semi-infinite and compact predictions regions.</p>","PeriodicalId":49274,"journal":{"name":"Extremes","volume":"148 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140154269","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Extremes for stationary regularly varying random fields over arbitrary index sets 任意索引集上静止规则变化随机场的极值
IF 1.3 3区 数学
Extremes Pub Date : 2024-01-25 DOI: 10.1007/s10687-023-00481-x
Riccardo Passeggeri, Olivier Wintenberger
{"title":"Extremes for stationary regularly varying random fields over arbitrary index sets","authors":"Riccardo Passeggeri, Olivier Wintenberger","doi":"10.1007/s10687-023-00481-x","DOIUrl":"https://doi.org/10.1007/s10687-023-00481-x","url":null,"abstract":"<p>We consider the clustering of extremes for stationary regularly varying random fields over arbitrary growing index sets. We study sufficient assumptions on the index set such that the limit of the point processes of the exceedances above a high threshold exists. Under the so-called anti-clustering condition, the extremal dependence is only local. Thus the index set can have a general form compared to previous literature (Basrak and Planinić in Bernoulli 27(2):1371–1408, 2021; Stehr and Rønn-Nielsen in Extremes 24(4):753–795, 2021). However, we cannot describe the clustering of extreme values in terms of the usual spectral tail measure (Wu and Samorodnitsky in Stochastic Process Appl 130(7):4470–4492, 2020) except for hyperrectangles or index sets in the lattice case. Using the recent extension of the spectral measure for star-shaped equipped space (Segers et al. in Extremes 20:539–566, 2017), the <span>(Upsilon)</span>-spectral tail measure provides a natural extension that describes the clustering effect in full generality.</p>","PeriodicalId":49274,"journal":{"name":"Extremes","volume":"9 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139552332","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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