Extremes
Journal Prestige (SJR): 1.562 Citation Impact (citeScore): 1 Number of Followers: 2 Hybrid journal (It can contain Open Access articles) ISSN (Print) 1572915X  ISSN (Online) 13861999 Published by SpringerVerlag [2469 journals] 
 On the asymptotic distribution of the scan statistic for empirical
distributions
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Abstract: Abstract This paper investigates the asymptotic behavior of several variants of the scan statistic for empirical distributions, which can be applied to detect the presence of an anomalous interval of any given length. In particular, we are interested in a Studentized scan statistic that is often preferable in practice. The main ingredients of our proof include Kolmogorov’s theorem, Poisson approximation, and the technical devices developed by Kabluchko and Wang (Stoch. Process. Their Appl. 124 (2014) 2824–2867).
PubDate: 20220901

 Heavytailed phasetype distributions: a unified approach

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Abstract: Abstract A phasetype distribution is the distribution of the time until absorption in a finite statespace timehomogeneous Markov jump process, with one absorbing state and the rest being transient. These distributions are mathematically tractable and conceptually attractive to model physical phenomena due to their interpretation in terms of a hidden Markov structure. Three recent extensions of regular phasetype distributions give rise to models which allow for heavy tails: discrete or continuousscaling; fractionaltime semiMarkov extensions; and inhomogeneous timechange of the underlying Markov process. In this paper, we present a unifying theory for heavytailed phasetype distributions for which all three approaches are particular cases. Our main objective is to provide useful models for heavytailed phasetype distributions, but any other tail behavior is also captured by our specification. We provide relevant new examples and also show how existing approaches are naturally embedded. Subsequently, two multivariate extensions are presented, inspired by the univariate construction which can be considered as a matrix version of a frailty model. We provide fully explicit EMalgorithms for all models and illustrate them using synthetic and reallife data.
PubDate: 20220901

 Limit theorems for branching processes with immigration in a random
environment
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Abstract: Abstract We investigate branching processes with immigration in a random environment. Using Goldie’s implicit renewal theory we prove that under a generalized Cramér condition the stationary distribution of such processes has a power law tail. We further show how several methods familiar in the extreme value theory provide a natural and elegant path to their mathematical analysis. In particular, we rely on the point processes theory and the concept of tail process to determine the limiting distribution for the corresponding extremes and partial sums. Since Kesten, Kozlov and Spitzer seminal 1975 paper, it is known that one class of these processes has a close relation with random walks in a random environment. Even in that well studied context, the method we follow yields new results. For instance, we are able to i) move away from the conditions used by Kesten et al., ii) provide precise form of the limiting distribution in their main theorem, and iii) characterize the long term behavior of the worst traps a random walk in random environment encounters when drifting away from the origin.
PubDate: 20220712
DOI: 10.1007/s10687022004439

 The asymptotic distribution of the condition number for random circulant
matrices
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Abstract: Abstract In this manuscript, we study the limiting distribution for the joint law of the largest and the smallest singular values for random circulant matrices with generating sequence given by independent and identically distributed random elements satisfying the socalled Lyapunov condition. Under an appropriated normalization, the joint law of the extremal singular values converges in distribution, as the matrix dimension tends to infinity, to an independent product of Rayleigh and Gumbel laws. The latter implies that a normalized \(\textit{condition number}\) converges in distribution to a Fréchet law as the dimension of the matrix increases.
PubDate: 20220704
DOI: 10.1007/s1068702200442w

 Improved interexceedancetimesbased estimator of the extremal index using
truncated distribution
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Abstract: Abstract The extremal index is an important parameter in the characterization of extreme values of a stationary sequence. This paper presents a novel approach to estimation of the extremal index based on truncation of interexceedance times. The truncated estimator based on the maximum likelihood method is derived together with its firstorder bias. The estimator is further improved using penultimate approximation to the limiting mixture distribution. In order to assess the performance of the proposed estimator, a simulation study is carried out for various stationary processes satisfying the local dependence condition \(D^{(k)}(u_n)\) . An application to daily maximum temperatures at Uccle, Belgium, is also presented.
PubDate: 20220624
DOI: 10.1007/s10687022004448

 Testing mean changes by maximal ratio statistics

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Abstract: Abstract We propose a new test statistic \(\mathrm {MR}_{\gamma ,n}\) for detecting a changed segment in the mean, at unknown dates, in a regularly varying sample. Our model supports several alternatives of shifts in the mean, including one change point, constant, epidemic and linear form of a change. Our aim is to detect a short length changed segment \(\ell ^{*}\) , assuming \(\ell^*/n\) to be small as the sample size n is large. \(\mathrm {MR}_{\gamma ,n}\) is built by taking maximal ratios of weighted moving sums statistics of four subsamples. An important feature of \(\mathrm {MR}_{\gamma ,n}\) is to be scale free. We obtain the limiting distribution of ratio statistics under the null hypothesis as well as their consistency under the alternative. These results are extended from i.i.d. samples under \(H_0\) to some dependent samples. To supplement theoretical results, empirical illustrations are provided by generating samples from symmetrized Pareto and LogGamma distributions.
PubDate: 20220601
DOI: 10.1007/s10687021004235

 Extremal lifetimes of persistent cycles

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Abstract: Abstract Persistent homology captures the appearances and disappearances of topological features such as loops and cavities when growing disks centered at a Poisson point process. We study extreme values for the lifetimes of features dying in bounded components and with birth resp. death time bounded away from the threshold for continuum percolation and the coexistence region. First, we describe the scaling of the minimal lifetimes for general feature dimensions, and of the maximal lifetimes for cavities in the Čech filtration. Then, we proceed to a more refined analysis and establish Poisson approximation for large lifetimes of cavities and for small lifetimes of loops. Finally, we also study the scaling of minimal lifetimes in the VietorisRips setting and point to a surprising difference to the Čech filtration.
PubDate: 20220601
DOI: 10.1007/s10687021004306

 Extremes of censored and uncensored lifetimes in survival data

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Abstract: Abstract We consider a random censoring model for survival analysis, allowing the possibility that only a proportion of individuals in the population are susceptible to death or failure, and the remainder are immune or cured. Susceptibles suffer the event under study eventually, but the time at which this occurs may not be observed due to censoring. Immune individuals have infinite lifetimes which are always censored in the sample. Assuming that the distribution of the susceptibles’ lifetimes as well as the censoring distribution have infinite right endpoints and are in the domain of attraction of the Gumbel distribution, we obtain asymptotic distributions, as sample size tends to infinity, of statistics relevant to testing for the possible existence of immunes in the population.
PubDate: 20220601
DOI: 10.1007/s10687021004262

 Functional strong law of large numbers for Betti numbers in the tail

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Abstract: Abstract The objective of this paper is to investigate the layered structure of topological complexity in the tail of a probability distribution. We establish the functional strong law of large numbers for Betti numbers, a basic quantifier of algebraic topology, of a geometric complex outside an open ball of radius \(R_n\) , such that \(R_n\rightarrow \infty\) as the sample size n increases. The nature of the obtained law of large numbers is determined by the decay rate of a probability density and how rapidly \(R_n\) diverges. In particular, if \(R_n\) diverges sufficiently slowly, the limiting function in the law of large numbers is crucially affected by the emergence of arbitrarily large connected components supporting topological cycles in the limit.
PubDate: 20220531
DOI: 10.1007/s1068702200441x

 Tail probabilities of random linear functions of regularly varying random
vectors
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Abstract: Abstract We provide a new extension of Breiman’s Theorem on computing tail probabilities of a product of random variables to a multivariate setting. In particular, we give a characterization of regular variation on cones in \([0,\infty )^d\) under random linear transformations. This allows us to compute probabilities of a variety of tail events, which classical multivariate regularly varying models would report to be asymptotically negligible. We illustrate our findings with applications to risk assessment in financial systems and reinsurance markets under a bipartite network structure.
PubDate: 20220526
DOI: 10.1007/s10687021004324

 Continuous simulation of storm processes

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Abstract: Abstract Storm processes constitute prototype models for spatial extremes. They are classically simulated on a finite number of points within a given domain. We propose a new algorithm that allows to perform such a task in continuous domains like hyperrectangles or balls, in arbitrary dimension. This consists in generating basic ingredients that can subsequently be used to assign a value at any point of the simulation field. Such an approach is particularly appropriate to investigate the geometrical properties of storm processes. Particular attention is paid to efficiency: by introducing and exploiting the notion of domain of influence of each storm, the running time is considerably reduced. Besides, most parts of the algorithm are designed to be parallelizable.
PubDate: 20220509
DOI: 10.1007/s10687022004386

 Adapting the Hill estimator to distributed inference: dealing with the
bias
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Abstract: Abstract The distributed Hill estimator is a divideandconquer algorithm for estimating the extreme value index when data are stored in multiple machines. In applications, estimates based on the distributed Hill estimator can be sensitive to the choice of the number of the exceedance ratios used in each machine. Even when choosing the number at a low level, a high asymptotic bias may arise. We overcome this potential drawback by designing a bias correction procedure for the distributed Hill estimator, which adheres to the setup of distributed inference. The asymptotically unbiased distributed estimator we obtained, on the one hand, is applicable to distributed stored data, on the other hand, inherits all known advantages of bias correction methods in extreme value statistics.
PubDate: 20220507
DOI: 10.1007/s1068702200440y

 Asymptotic dependence of in and outdegrees in a preferential attachment
model with reciprocity
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Abstract: Abstract Reciprocity characterizes the information exchange between users in a network, and some empirical studies have revealed that social networks have a high proportion of reciprocal edges. Classical directed preferential attachment (PA) models, though generating scalefree networks, may give networks with low reciprocity. This points out one potential problem of fitting a classical PA model to a given network dataset with high reciprocity, and indicates alternative models need to be considered. We give one possible modification of the classical PA model by including another parameter which controls the probability of adding a reciprocated edge at each step. Asymptotic analyses suggest that large in and outdegrees become fully dependent in this modified model, as a result of the additional reciprocated edges.
PubDate: 20220430
DOI: 10.1007/s10687022004395

 Environmental contours as Voronoi cells

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Abstract: Abstract Environmental contours are widely used as basis for design of structures exposed to environmental loads. The basic idea of the method is to decouple the environmental description from the structural response. This is done by establishing an envelope of joint extreme values representing critical environmental conditions, such that any structure tolerating loads on this envelope will have a failure probability smaller than a prescribed value. Specifically, given an ndimensional random variable \(\mathbf {X}\) and a target probability of failure \(p_{e}\) , an environmental contour is the boundary of a set \(\mathcal {B} \subset \mathbb {R}^{n}\) with the following property: For any failure set \(\mathcal {F} \subset \mathbb {R}^{n}\) , if \(\mathcal {F}\) does not intersect the interior of \(\mathcal {B}\) , then the probability of failure, \(P(\mathbf {X} \in \mathcal {F})\) , is bounded above by \(p_{e}\) . We work under the assumption that failure sets are convex, which is relevant for many realworld applications. In this paper, we show that such environmental contours may be regarded as boundaries of Voronoi cells. This geometric interpretation leads to new theoretical insights and suggests a simple novel construction algorithm that guarantees the desired probabilistic properties. The method is illustrated with examples in two and three dimensions, but the results extend to environmental contours in arbitrary dimensions. Inspired by the VoronoiDelaunay duality in the numerical discrete scenario, we are also able to derive an analytical representation where the environmental contour is considered as a differentiable manifold, and a criterion for its existence is established.
PubDate: 20220429
DOI: 10.1007/s10687022004377

 The tail process and tail measure of continuous time regularly varying
stochastic processes
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Abstract: Abstract The goal of this paper is to investigate the tools of extreme value theory originally introduced for discrete time stationary stochastic processes (time series), namely the tail process and the tail measure, in the framework of continuous time stochastic processes with paths in the space \(\mathcal {D}\) of càdlàg functions indexed by \(\mathbb {R}\) , endowed with Skorohod’s J1 topology. We prove that the essential properties of these objects are preserved, with some minor (though interesting) differences arising. We first obtain structural results which provide representation for homogeneous shiftinvariant measures on \(\mathcal {D}\) and then study regular variation of random elements in \(\mathcal {D}\) . We give practical conditions and study several examples, recovering and extending known results.
PubDate: 20220301
DOI: 10.1007/s10687021004173

 Modeling spatial extremes using normal meanvariance mixtures

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Abstract: Abstract Classical models for multivariate or spatial extremes are mainly based upon the asymptotically justified maxstable or generalized Pareto processes. These models are suitable when asymptotic dependence is present, i.e., the joint tail decays at the same rate as the marginal tail. However, recent environmental data applications suggest that asymptotic independence is equally important and, unfortunately, existing spatial models in this setting that are both flexible and can be fitted efficiently are scarce. Here, we propose a new spatial copula model based on the generalized hyperbolic distribution, which is a specific normal meanvariance mixture and is very popular in financial modeling. The tail properties of this distribution have been studied in the literature, but with contradictory results. It turns out that the proofs from the literature contain mistakes. We here give a corrected theoretical description of its tail dependence structure and then exploit the model to analyze a simulated dataset from the inverted Brown–Resnick process, hindcast significant wave height data in the North Sea, and wind gust data in the state of Oklahoma, USA. We demonstrate that our proposed model is flexible enough to capture the dependence structure not only in the tail but also in the bulk.
PubDate: 20220131
DOI: 10.1007/s10687021004342

 Handling missing extremes in tail estimation

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Abstract: Abstract In some data sets, it may be the case that a portion of the extreme observations is missing. This might arise in cases where the extreme observations are just not available or are imprecisely measured. For example, considering human lifetimes, a topic of recent interest, birth certificates of centenarians may not even exist and many such individuals may not even be included in the data sets that are currently available. In essence, one does not have a clear record of the largest lifetimes of human populations. If there are missing extreme observations, then the assessment of risk can be severely underestimated resulting in rare events occurring more often than originally thought. In concrete terms, this may mean a 500 year flood is in fact a 100 (or even a 20) year flood. In this paper, we present methods for estimating the number of missing extremes together with the tail index associated with tail heaviness of the data. Ignoring one or the other can severely impact the estimation of risk. Our estimates are based on the HEWE (Hill estimate without extremes) of the tail index that adjusts for missing extremes. Based on a functional convergence of this process to a limit process, we consider an asymptotic likelihoodbased procedure for estimating both the number of missing extremes and the tail index. We derive the asymptotic distribution of the resulting estimates. By artificially removing segments of extremes in the data, this methodology can be used for assessing the reliability of the underlying assumptions that are imposed on the data.
PubDate: 20211223
DOI: 10.1007/s1068702100429z

 Extremal linkage networks

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Abstract: Abstract We demonstrate how sophisticated graph properties, such as small distances and scalefree degree distributions, arise naturally from a reinforcement mechanism on layered graphs. Every node is assigned an apriori i.i.d. fitness with maxstable distribution. The fitness determines the node attractiveness w.r.t. incoming edges as well as the spatial range for outgoing edges. For maxstable fitness distributions, we thus obtain a complex spatial network, which we coin extremal linkage network.
PubDate: 20211203
DOI: 10.1007/s10687021004333

 Choquet random supmeasures with aggregations

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Abstract: Abstract A variation of Choquet random supmeasures is introduced. These random supmeasures are shown to arise as the scaling limits of empirical random supmeasures of a general aggregated model. Because of the aggregations, the finitedimensional distributions of introduced random supmeasures do not necessarily have classical extremevalue distributions. Examples include the recently introduced stableregenerative random supmeasures as a special case.
PubDate: 20210920
DOI: 10.1007/s10687021004253

 Extremes of subexponential Lévydriven random fields in the Gumbel
domain of attraction
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Abstract: Abstract We consider a spatial Lévydriven moving average with an underlying Lévy measure having a subexponential right tail, which is also in the maximum domain of attraction of the Gumbel distribution. Assuming that the left tail is not heavier than the right tail, and that the integration kernel satisfies certain regularity conditions, we show that the supremum of the field over any bounded set has a right tail equivalent to that of the Lévy measure. Furthermore, for a very general class of expanding index sets, we show that the running supremum of the field, under a suitable scaling, converges to the Gumbel distribution.
PubDate: 20210911
DOI: 10.1007/s10687021004280
