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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 sub-samples. 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 Log-Gamma distributions. PubDate: 2022-06-01

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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 Vietoris-Rips setting and point to a surprising difference to the Čech filtration. PubDate: 2022-06-01

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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: 2022-06-01

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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 hyper-rectangles 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: 2022-05-09

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Abstract: The distributed Hill estimator is a divide-and-conquer 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: 2022-05-07

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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 scale-free 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 out-degrees become fully dependent in this modified model, as a result of the additional reciprocated edges. PubDate: 2022-04-30

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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 n-dimensional 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 real-world 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 Voronoi-Delaunay 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: 2022-04-29

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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 shift-invariant 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: 2022-03-01 DOI: 10.1007/s10687-021-00417-3

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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: 2022-02-21 DOI: 10.1007/s10687-021-00435-1

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Abstract: Abstract A phase-type distribution is the distribution of the time until absorption in a finite state-space time-homogeneous 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 phase-type distributions give rise to models which allow for heavy tails: discrete- or continuous-scaling; fractional-time semi-Markov extensions; and inhomogeneous time-change of the underlying Markov process. In this paper, we present a unifying theory for heavy-tailed phase-type distributions for which all three approaches are particular cases. Our main objective is to provide useful models for heavy-tailed phase-type 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 EM-algorithms for all models and illustrate them using synthetic and real-life data. PubDate: 2022-02-16 DOI: 10.1007/s10687-022-00436-8

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Abstract: Abstract Classical models for multivariate or spatial extremes are mainly based upon the asymptotically justified max-stable 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 mean-variance 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: 2022-01-31 DOI: 10.1007/s10687-021-00434-2

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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 likelihood-based 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: 2021-12-23 DOI: 10.1007/s10687-021-00429-z

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Abstract: Abstract We demonstrate how sophisticated graph properties, such as small distances and scale-free degree distributions, arise naturally from a reinforcement mechanism on layered graphs. Every node is assigned an a-priori i.i.d. fitness with max-stable distribution. The fitness determines the node attractiveness w.r.t. incoming edges as well as the spatial range for outgoing edges. For max-stable fitness distributions, we thus obtain a complex spatial network, which we coin extremal linkage network. PubDate: 2021-12-03 DOI: 10.1007/s10687-021-00433-3

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Abstract: Abstract In the context of bivariate random variables \(\left (Y^{(1)},Y^{(2)}\right )\) , the marginal expected shortfall, defined as \(\mathbb {E}\left (Y^{(1)} Y^{(2)} \ge Q_{2}(1-p)\right )\) for p small, where Q2 denotes the quantile function of Y(2), is an important risk measure, which finds applications in areas like, e.g., finance and environmental science. Our paper pioneers the statistical modeling of this risk measure when the random variables of main interest \(\left (Y^{(1)},Y^{(2)}\right )\) are observed together with a random covariate X, leading to the concept of the conditional marginal expected shortfall. The asymptotic behavior of an estimator for this conditional marginal expected shortfall is studied for a wide class of conditional bivariate distributions, with heavy-tailed marginal conditional distributions, and where p tends to zero at an intermediate rate. The finite sample performance is evaluated on a small simulation experiment. The practical applicability of the proposed estimator is illustrated on flood claim data. PubDate: 2021-12-01 DOI: 10.1007/s10687-020-00403-1

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Abstract: Abstract It is well known that the distribution of extreme values of strictly stationary sequences differ from those of independent and identically distributed sequences in that extremal clustering may occur. Here we consider non-stationary but identically distributed sequences of random variables subject to suitable long range dependence restrictions. We find that the limiting distribution of appropriately normalized sample maxima depends on a parameter that measures the average extremal clustering of the sequence. Based on this new representation we derive the asymptotic distribution for the time between consecutive extreme observations and construct moment and likelihood based estimators for measures of extremal clustering. We specialize our results to random sequences with periodic dependence structure. PubDate: 2021-12-01 DOI: 10.1007/s10687-021-00418-2

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Abstract: Abstract We investigate the asymptotic of ruin probabilities when the company combines the life- and non-life insurance businesses and invests its reserve into a risky asset with stochastic volatility and drift driven by a two-state Markov process. Using the technique of the implicit renewal theory we obtain the rate of convergence to zero of the ruin probabilities. PubDate: 2021-12-01 DOI: 10.1007/s10687-021-00420-8

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Abstract: Abstract Modelling of multiple simultaneous failures in insurance, finance and other areas of applied probability is important especially from the point of view of pandemic-type events. A benchmark limiting model for the analysis of multiple failures is the classical d-dimensional Brownian risk model (Brm), see Delsing et al. (Methodol. Comput. Appl. Probab. 22(3), 927–948 2020). From both theoretical and practical point of view, of interest is the calculation of the probability of multiple simultaneous failures in a given time horizon. The main findings of this contribution concern the approximation of the probability that at least k out of d components of Brm fail simultaneously. We derive both sharp bounds and asymptotic approximations of the probability of interest for the finite and the infinite time horizon. Our results extend previous findings of Dȩbicki et al. (J. Appl. Probab. 57(2), 597–612 2020) and Dȩbicki et al. (Stoch. Proc. Appl. 128(12), 4171–4206 2018). PubDate: 2021-09-22 DOI: 10.1007/s10687-021-00424-4

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Abstract: Abstract A variation of Choquet random sup-measures is introduced. These random sup-measures are shown to arise as the scaling limits of empirical random sup-measures of a general aggregated model. Because of the aggregations, the finite-dimensional distributions of introduced random sup-measures do not necessarily have classical extreme-value distributions. Examples include the recently introduced stable-regenerative random sup-measures as a special case. PubDate: 2021-09-20 DOI: 10.1007/s10687-021-00425-3

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Abstract: Abstract Motivated by a seminal paper of Kesten et al. (Ann. Probab., 3(1), 1–31, 1975) we consider a branching process with a conditional geometric offspring distribution with i.i.d. random environmental parameters An, n ≥ 1 and with one immigrant in each generation. In contrast to above mentioned paper we assume that the environment is long-tailed, that is that the distribution F of \(\xi _{n}:=\log ((1-A_{n})/A_{n})\) is long-tailed. We prove that although the offspring distribution is light-tailed, the environment itself can produce extremely heavy tails of the distribution of the population size in the n th generation which becomes even heavier with increase of n. More precisely, we prove that, for all n, the distribution tail \(\mathbb {P}(Z_{n} \ge m)\) of the n th population size Zn is asymptotically equivalent to \(n\overline F(\log m)\) as m grows. In this way we generalise Bhattacharya and Palmowski (Stat. Probab. Lett., 154, 108550, 2019) who proved this result in the case n = 1 for regularly varying environment F with parameter α > 1. Further, for a subcritical branching process with subexponentially distributed ξn, we provide the asymptotics for the distribution tail \(\mathbb {P}(Z_{n}>m)\) which are valid uniformly for all n, and also for the stationary tail distribution. Then we establish the “principle of a single atypical environment” which says that the main cause for the number of particles to be large is the presence of a single very small environmental parameter Ak. PubDate: 2021-09-20 DOI: 10.1007/s10687-021-00427-1

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Abstract: Abstract We consider a spatial Lévy-driven 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: 2021-09-11 DOI: 10.1007/s10687-021-00428-0