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Abstract: Abstract 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 \(\Upsilon\) -spectral tail measure provides a natural extension that describes the clustering effect in full generality. PubDate: 2024-01-25

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Abstract: Abstract In this paper, we study the asymptotical behaviour of high exceedence probabilities for centered continuous \(\mathbb {R}^n\) -valued Gaussian random field \(\varvec{X}\) with covariance matrix satisfying \(\Sigma - R ( t + s, t ) \sim \sum _{l = 1}^n B_l ( t ) \, s_l ^{\alpha _l}\) as \(s \downarrow 0\) . Such processes occur naturally as time transformations of homogenous random fields, and we present two asymptotic results of this nature as applications of our findings. The technical novelty of our proof consists in showing that the Slepian-Gordon inequality technique, essential in the univariate case, can also be successfully applied in the multivariate setup. This is noteworthy because this technique was previously believed to be inaccessible in this particular context. PubDate: 2024-01-08

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Abstract: Abstract The convergence of a sequence of point processes with dependent points, defined by a symmetric function of iid high-dimensional random vectors, to a Poisson random measure is proved. This also implies the convergence of the joint distribution of a fixed number of upper order statistics. As applications of the result a generalization of maximum convergence to point process convergence is given for simple linear rank statistics, rank-type U-statistics and the entries of sample covariance matrices. PubDate: 2023-12-20

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Abstract: Abstract Statistical methods are proposed to select homogeneous regions when analyzing spatial block maxima data, such as in extreme event attribution studies. Here, homogeneitity refers to the fact that marginal model parameters are the same at different locations from the region. The methods are based on classical hypothesis testing using Wald-type test statistics, with critical values obtained from suitable parametric bootstrap procedures and corrected for multiplicity. A large-scale Monte Carlo simulation study finds that the methods are able to accurately identify homogeneous locations, and that pooling the selected locations improves the accuracy of subsequent statistical analyses. The approach is illustrated with a case study on precipitation extremes in Western Europe. The methods are implemented in an R package that allows for easy application in future extreme event attribution studies. PubDate: 2023-12-16

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Abstract: Abstract The notion of tail adversarial stability has been proven useful in obtaining limit theorems for tail dependent time series. Its implication and advantage over the classical strong mixing framework has been examined for max-linear processes, but not yet studied for additive linear processes. In this article, we fill this gap by verifying the tail adversarial stability condition for regularly varying additive linear processes. We in addition consider extensions of the result to a stochastic volatility generalization and to a max-linear counterpart. We also address the invariance of tail adversarial stability under monotone transforms. Some implications for limit theorems in statistical context are also discussed. PubDate: 2023-12-13

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Abstract: Abstract There are many ways of measuring and modeling tail-dependence in random vectors: from the general framework of multivariate regular variation and the flexible class of max-stable vectors down to simple and concise summary measures like the matrix of bivariate tail-dependence coefficients. This paper starts by providing a review of existing results from a unifying perspective, which highlights connections between extreme value theory and the theory of cuts and metrics. Our approach leads to some new findings in both areas with some applications to current topics in risk management. We begin by using the framework of multivariate regular variation to show that extremal coefficients, or equivalently, the higher-order tail-dependence coefficients of a random vector can simply be understood in terms of random exceedance sets, which allows us to extend the notion of Bernoulli compatibility. In the special but important case of bivariate tail-dependence, we establish a correspondence between tail-dependence matrices and \(L^1\) - and \(\ell _1\) -embeddable finite metric spaces via the spectral distance, which is a metric on the space of jointly 1-Fréchet random variables. Namely, the coefficients of the cut-decomposition of the spectral distance and of the Tawn-Molchanov max-stable model realizing the corresponding bivariate extremal dependence coincide. We show that line metrics are rigid and if the spectral distance corresponds to a line metric, the higher order tail-dependence is determined by the bivariate tail-dependence matrix. Finally, the correspondence between \(\ell _1\) -embeddable metric spaces and tail-dependence matrices allows us to revisit the realizability problem, i.e. checking whether a given matrix is a valid tail-dependence matrix. We confirm a conjecture of Shyamalkumar and Tao (2020) that this problem is NP-complete. PubDate: 2023-12-01 DOI: 10.1007/s10687-023-00471-z

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Abstract: Abstract Consider two stationary time series with heavy-tailed marginal distributions. We aim to detect whether they have a causal relation, that is, if a change in one causes a change in the other. Usual methods for causal discovery are not well suited if the causal mechanisms only appear during extreme events. We propose a framework to detect a causal structure from the extremes of time series, providing a new tool to extract causal information from extreme events. We introduce the causal tail coefficient for time series, which can identify asymmetrical causal relations between extreme events under certain assumptions. This method can handle nonlinear relations and latent variables. Moreover, we mention how our method can help estimate a typical time difference between extreme events. Our methodology is especially well suited for large sample sizes, and we show the performance on the simulations. Finally, we apply our method to real-world space-weather and hydro-meteorological datasets. PubDate: 2023-10-31 DOI: 10.1007/s10687-023-00479-5

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Abstract: Abstract Users of social networks display diversified behavior and online habits. For instance, a user’s tendency to reply to a post can depend on the user and the person posting. For convenience, we group users into aggregated behavioral patterns, focusing here on the tendency to reply to or reciprocate messages. The reciprocity feature in social networks reflects the information exchange among users. We study the properties of a preferential attachment model with heterogeneous reciprocity levels, give the growth rate of model edge counts, and prove the convergence of empirical degree frequencies to a limiting distribution. This limiting distribution is not only multivariate regularly varying, but also has the property of hidden regular variation. PubDate: 2023-09-25 DOI: 10.1007/s10687-023-00478-6

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Abstract: Abstract Confounding variables are a recurrent challenge for causal discovery and inference. In many situations, complex causal mechanisms only manifest themselves in extreme events, or take simpler forms in the extremes. Stimulated by data on extreme river flows and precipitation, we introduce a new causal discovery methodology for heavy-tailed variables that allows the effect of a known potential confounder to be almost entirely removed when the variables have comparable tails, and also decreases it sufficiently to enable correct causal inference when the confounder has a heavier tail. We also introduce a new parametric estimator for the existing causal tail coefficient and a permutation test. Simulations show that the methods work well and the ideas are applied to the motivating dataset. PubDate: 2023-09-01 DOI: 10.1007/s10687-022-00456-4

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Abstract: Abstract We build a sharp approximation of the whole distribution of the sum of iid heavy-tailed random vectors, combining mean and extreme behaviors. It extends the so-called ’normex’ approach from a univariate to a multivariate framework. We propose two possible multi-normex distributions, named d-Normex and MRV-Normex. Both rely on the Gaussian distribution for describing the mean behavior, via the CLT, while the difference between the two versions comes from using the exact distribution or the EV theorem for the maximum. The main theorems provide the rate of convergence for each version of the multi-normex distributions towards the distribution of the sum, assuming second order regular variation property for the norm of the parent random vector when considering the MRV-normex case. Numerical illustrations and comparisons are proposed with various dependence structures on the parent random vector, using QQ-plots based on geometrical quantiles. PubDate: 2023-09-01 DOI: 10.1007/s10687-022-00461-7

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Abstract: Abstract Weissman extrapolation methodology for estimating extreme quantiles from heavy-tailed distributions is based on two estimators: an order statistic to estimate an intermediate quantile and an estimator of the tail-index. The common practice is to select the same intermediate sequence for both estimators. In this work, we show how an adapted choice of two different intermediate sequences leads to a reduction of the asymptotic bias associated with the resulting refined Weissman estimator. The asymptotic normality of the latter estimator is established and a data-driven method is introduced for the practical selection of the intermediate sequences. Our approach is compared to the Weissman estimator and to six bias reduced estimators of extreme quantiles on a large scale simulation study. It appears that the refined Weissman estimator outperforms its competitors in a wide variety of situations, especially in the challenging high bias cases. Finally, an illustration on an actuarial real data set is provided. PubDate: 2023-09-01 DOI: 10.1007/s10687-022-00452-8

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Abstract: Abstract The sequential tail empirical process is analyzed in a stochastic model allowing for serially dependent observations and heteroscedasticity of extremes in the sense of Einmahl et al. (J. R. Stat. Soc. Ser. B. Stat. Methodol. 78(1), 31–51, 2016). Weighted weak convergence of the sequential tail empirical process is established. As an application, a central limit theorem for an estimator of the extreme value index is proven. PubDate: 2023-08-24 DOI: 10.1007/s10687-023-00476-8

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Abstract: Abstract This review paper surveys recent development in software implementations for extreme value analyses since the publication of Stephenson and Gilleland (Extremes 8:87–109, 2006) and Gilleland et al. (Extremes 16(1):103–119, 2013). We provide a comparative review by topic and highlight differences in existing numerical routines, along with listing areas where software development is lacking. The online supplement contains two vignettes comparing implementations of frequentist and Bayesian estimation of univariate extreme value models. PubDate: 2023-08-03 DOI: 10.1007/s10687-023-00475-9

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Abstract: Abstract Extreme quantile regression provides estimates of conditional quantiles outside the range of the data. Classical quantile regression performs poorly in such cases since data in the tail region are too scarce. Extreme value theory is used for extrapolation beyond the range of observed values and estimation of conditional extreme quantiles. Based on the peaks-over-threshold approach, the conditional distribution above a high threshold is approximated by a generalized Pareto distribution with covariate dependent parameters. We propose a gradient boosting procedure to estimate a conditional generalized Pareto distribution by minimizing its deviance. Cross-validation is used for the choice of tuning parameters such as the number of trees and the tree depths. We discuss diagnostic plots such as variable importance and partial dependence plots, which help to interpret the fitted models. In simulation studies we show that our gradient boosting procedure outperforms classical methods from quantile regression and extreme value theory, especially for high-dimensional predictor spaces and complex parameter response surfaces. An application to statistical post-processing of weather forecasts with precipitation data in the Netherlands is proposed. PubDate: 2023-07-21 DOI: 10.1007/s10687-023-00473-x

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Abstract: Abstract The conditional extremes framework allows for event-based stochastic modeling of dependent extremes, and has recently been extended to spatial and spatio-temporal settings. After standardizing the marginal distributions and applying an appropriate linear normalization, certain non-stationary Gaussian processes can be used as asymptotically-motivated models for the process conditioned on threshold exceedances at a fixed reference location and time. In this work, we adapt existing conditional extremes models to allow for the handling of large spatial datasets. This involves specifying the model for spatial observations at d locations in terms of a latent \(m\ll d\) dimensional Gaussian model, whose structure is specified by a Gaussian Markov random field. We perform Bayesian inference for such models for datasets containing thousands of observation locations using the integrated nested Laplace approximation, or INLA. We explain how constraints on the spatial and spatio-temporal Gaussian processes, arising from the conditioning mechanism, can be implemented through the latent variable approach without losing the computationally convenient Markov property. We discuss tools for the comparison of models via their posterior distributions, and illustrate the flexibility of the approach with gridded Red Sea surface temperature data at over 6,000 observed locations. Posterior sampling is exploited to study the probability distribution of cluster functionals of spatial and spatio-temporal extreme episodes. PubDate: 2023-07-12 DOI: 10.1007/s10687-023-00468-8

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Abstract: Abstract Using an intrinsic approach, we study some properties of random fields which appear as tail fields of regularly varying stationary random fields. The index set is allowed to be a general locally compact Hausdorff Abelian group \({\mathbb G}\) . The values are taken in a measurable cone, equipped with a pseudo norm. We first discuss some Palm formulas for the exceedance random measure \(\xi\) associated with a stationary (measurable) random field \(Y=(Y_s)_{s\in {\mathbb G}}\) . It is important to allow the underlying stationary measure to be \(\sigma\) -finite. Then we proceed to a random field (defined on a probability space) which is spectrally decomposable, in a sense which is motivated by extreme value theory. We characterize mass-stationarity of the exceedance random measure in terms of a suitable version of the classical Mecke equation. We also show that the associated stationary measure is homogeneous, that is a tail measure. We then proceed with establishing and studying the spectral representation of stationary tail measures and with characterizing a moving shift representation. Finally we discuss anchoring maps and the candidate extremal index. PubDate: 2023-06-27 DOI: 10.1007/s10687-023-00472-y

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Abstract: Abstract Consider a stationary Poisson process in a d-dimensional hyperbolic space. For \(R>0\) define the point process \(\xi _R^{(k)}\) of exceedance heights over a suitable threshold of the hyperbolic volumes of kth nearest neighbour balls centred around the points of the Poisson process within a hyperbolic ball of radius R centred at a fixed point. The point process \(\xi _R^{(k)}\) is compared to an inhomogeneous Poisson process on the real line with intensity function \(e^{-u}\) and point process convergence in the Kantorovich-Rubinstein distance is shown. From this, a quantitative limit theorem for the hyperbolic maximum kth nearest neighbour ball with a limiting Gumbel distribution is derived. PubDate: 2023-04-20 DOI: 10.1007/s10687-023-00470-0

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Abstract: Abstract Ross Leadbetter had had a broad and deep influence on the development of probabilistic and statistical theory of extreme values and on the application of extreme-value methods. He has been an inspiration and a friend for many of us. This editorial collects thirteen personal recollections of Ross and his work. An account of his career and some of his work can be found in the IMS Obituary “Ross Leadbetter 1931–2022”. PubDate: 2023-04-10 DOI: 10.1007/s10687-023-00464-y

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Abstract: Abstract We study the joint occurrence of large values of a Markov random field or undirected graphical model associated to a block graph. On such graphs, containing trees as special cases, we aim to generalize recent results for extremes of Markov trees. Every pair of nodes in a block graph is connected by a unique shortest path. These paths are shown to determine the limiting distribution of the properly rescaled random field given that a fixed variable exceeds a high threshold. The latter limit relation implies that the random field is multivariate regularly varying and it determines the max-stable distribution to which component-wise maxima of independent random samples from the field are attracted. When the sub-vectors induced by the blocks have certain limits parametrized by Hüsler–Reiss distributions, the global Markov property of the original field induces a particular structure on the parameter matrix of the limiting max-stable Hüsler–Reiss distribution. The multivariate Pareto version of the latter turns out to be an extremal graphical model according to the original block graph. Thanks to these algebraic relations, the parameters are still identifiable even if some variables are latent. PubDate: 2023-04-04 DOI: 10.1007/s10687-023-00467-9

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Abstract: Abstract Extreme value theory motivates estimating extreme upper quantiles of a distribution by selecting some threshold, discarding those observations below the threshold and fitting a generalized Pareto distribution to exceedances above the threshold via maximum likelihood. This sharp cutoff between observations that are used in the parameter estimation and those that are not is at odds with statistical practice for analogous problems such as nonparametric density estimation, in which observations are typically smoothly downweighted as they become more distant from the value at which the density is being estimated. By exploiting the fact that the order statistics of independent and identically distributed observations form a Markov chain, this work shows how one can obtain a natural weighted composite log-likelihood function for fitting generalized Pareto distributions to exceedances over a threshold. A method for producing confidence intervals based on inverting a test statistic calibrated via parametric bootstrapping is proposed. Some theory demonstrates the asymptotic advantages of using weights in the special case when the shape parameter of the limiting generalized Pareto distribution is known to be 0. Methods for extending this approach to observations that are not identically distributed are described and applied to an analysis of daily precipitation data in New York City. Perhaps the most important practical finding is that including weights in the composite log-likelihood function can reduce the sensitivity of estimates to small changes in the threshold. PubDate: 2023-03-29 DOI: 10.1007/s10687-023-00466-w