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 [2467 journals] 
 Running minimum in the bestchoice problem

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Abstract: Abstract The fullinformation best choice problem asks one to find a strategy maximising the probability of stopping at the minimum (or maximum) of a sequence \(X_1,\cdots ,X_n\) of i.i.d. random variables with continuous distribution. In this paper we look at more general models, where independent \(X_j\) ’s may have different distributions, discrete or continuous. A central role in our study is played by the running minimum process, which we first employ to revisit the classic problem and its limit Poisson counterpart. The approach is further applied to two explicitly solvable models: in the first the distribution of the jth variable is uniform on \(\{j,\cdots ,n\}\) , and in the second it is uniform on \(\{1,\cdots , n\}\) .
PubDate: 20221129

 Publisher Correction: Integral Functionals and the Bootstrap for the Tail
Empirical Process
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PubDate: 20221128

 Extremal characteristics of conditional models

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Abstract: Abstract Conditionally specified models are often used to describe complex multivariate data. Such models assume implicit structures on the extremes. So far, no methodology exists for calculating extremal characteristics of conditional models since the copula and marginals are not expressed in closed forms. We consider bivariate conditional models that specify the distribution of X and the distribution of Y conditional on X. We provide tools to quantify implicit assumptions on the extremes of this class of models. In particular, these tools allow us to approximate the distribution of the tail of Y and the coefficient of asymptotic independence \(\eta\) in closed forms. We apply these methods to a widely used conditional model for wave height and wave period. Moreover, we introduce a new condition on the parameter space for the conditional extremes model of Heffernan and Tawn (Journal of the Royal Statistical Society: Series B (Methodology) 66(3), 497547, 2004), and prove that the conditional extremes model does not capture \(\eta\) , when \(\eta <1\) .
PubDate: 20221110

 Asymptotic behavior of an intrinsic rankbased estimator of the Pickands
dependence function constructed from Bsplines
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Abstract: Abstract A bivariate extremevalue copula is characterized by its Pickands dependence function, i.e., a convex function defined on the unit interval satisfying boundary conditions. This paper investigates the largesample behavior of a nonparametric estimator of this function due to Cormier et al. (Extremes 17:633–659, 2014). These authors showed how to construct this estimator through constrained quadratic median Bspline smoothing of pairs of pseudoobservations derived from a random sample. Their estimator is shown here to exist whatever the order \(m \ge 3\) of the Bspline basis, and its consistency is established under minimal conditions. The largesample distribution of this estimator is also determined under the additional assumption that the underlying Pickands dependence function is a Bspline of given order with a known set of knots.
PubDate: 20221109

 Conditions for finiteness and bounds on moments of record values from iid
continuous life distributions
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Abstract: Abstract We consider the standard and kth record values arising in sequences of independent identically distributed continuous and positive random variables with finite expectations. We determine necessary and sufficient conditions on the type of record k, its number n and moment order r so that the rth moment of the n value of kth record is finite for every parent distribution. Under the conditions we present the optimal upper bounds on these moments expressed in the scale units being the respective powers of the first population moment. The theoretical results are illustrated by some numerical evaluations.
PubDate: 20221109

 Palm theory for extremes of stationary regularly varying time series and
random fields
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Abstract: Abstract The tail process \(\varvec{Y}=(Y_{\varvec{i}})_{\varvec{i}\in \mathbb {Z}^d}\) of a stationary regularly varying random field \(\varvec{X}=(X_{\varvec{i}})_{\varvec{i}\in \mathbb {Z}^d}\) represents the asymptotic local distribution of \(\varvec{X}\) as seen from its typical exceedance over a threshold u as \(u\rightarrow \infty\) . Motivated by the standard Palm theory, we show that every tail process satisfies an invariance property called exceedancestationarity and that this property, together with the spectral decomposition of the tail process, characterizes the class of all tail processes. We then restrict to the case when \(Y_{\varvec{i}}\rightarrow 0\) as \( \varvec{i} \rightarrow \infty\) and establish a couple of Palmlike dualities between the tail process and the socalled anchored tail process which, under suitable conditions, represents the asymptotic distribution of a typical cluster of extremes of \(\varvec{X}\) . The main message is that the distribution of the tail process is biased towards clusters with more exceedances. Finally, we use these results to determine the distribution of a typical cluster of extremes for moving average processes with random coefficients and heavytailed innovations.
PubDate: 20221024

 Regressiontype analysis for multivariate extreme values

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Abstract: Abstract This paper devises a regressiontype model for the situation where both the response and covariates are extreme. The proposed approach is designed for the setting where the response and covariates are modeled as multivariate extreme values, and thus contrarily to standard regression methods it takes into account the key fact that the limiting distribution of suitably standardized componentwise maxima is an extreme value copula. An important target in the proposed framework is the regression manifold, which consists of a family of regression lines obeying the latter asymptotic result. To learn about the proposed model from data, we employ a Bernstein polynomial prior on the space of angular densities which leads to an induced prior on the space of regression manifolds. Numerical studies suggest a good performance of the proposed methods, and a finance realdata illustration reveals interesting aspects on the conditional risk of extreme losses in two leading international stock markets.
PubDate: 20221021

 Integral Functionals and the Bootstrap for the Tail Empirical Process

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Abstract: Abstract The tail empirical process (TEP) generated by an i.i.d. sequence of regularly varying random variables is key to investigating the behaviour of extreme value statistics such as the Hill and harmonic moment estimators of the tail index. The main contribution of the paper is to prove that Efron’s bootstrap produces versions of the estimators that exhibit the same asymptotic behaviour, including possible bias. In addition, the bootstrap provides new estimators of the tail index based on variability. Further, the asymptotic behaviour of the bootstrap variance estimators is shown to be unaffected by bias.
PubDate: 20221014

 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
DOI: 10.1007/s10687021004351

 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
DOI: 10.1007/s10687022004368

 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

 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
