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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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We investigate the behavior of extreme values in Gaussian triangular arrays under strong dependence conditions. By extending previous results, we establish conditions for convergence to a mixture of Gaussian and Gumbel distributions without requiring stationarity. Our findings offer insights into the application of these models, particularly for analyzing air ozone concentrations. [ABSTRACT FROM AUTHOR]

We consider the random connection model in which an edge between two Poisson points at distance r is present with probability g(r). 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 g(r). 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 Υ -spectral tail measure provides a natural extension that describes the clustering effect in full generality. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 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 ξ associated with a stationary (measurable) random field Y = (Y s) s ∈ G . It is important to allow the underlying stationary measure to be σ -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. [ABSTRACT FROM AUTHOR]

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 ℓ 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 ℓ 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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". [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Motivated by the Extreme Value Analysis 2021 (EVA 2021) data challenge, we propose a method based on statistics and machine learning for the spatial prediction of extreme wildfire frequencies and sizes. This method is tailored to handle large datasets, including missing observations. Our approach relies on a four-stage, bivariate, sparse spatial model for high-dimensional zero-inflated data that we develop using stochastic partial differential equations (SPDE), allowing sparse precision matrices for the latent processes. In Stage 1, the observations are separated in zero/nonzero categories and modeled using a two-layered hierarchical Bayesian sparse spatial model to estimate the probabilities of these two categories. In Stage 2, we first obtain empirical estimates of the spatially-varying mean and variance profiles across the spatial locations for the positive observations and smooth those estimates using fixed rank kriging. This approximate Bayesian inference method is employed to avoid the high computational burden of large spatial data modeling using spatially-varying coefficients. In Stage 3, we further model the standardized log-transformed positive observations from the second stage using a sparse bivariate spatial Gaussian process. The Gaussian distribution assumption for wildfire counts developed in the third stage is computationally effective but erroneous. Thus, in Stage 4, the predicted exceedance probabilities are post-processed using Random Forests. We draw posterior inference for Stages 1 and 3 using Markov chain Monte Carlo (MCMC) sampling. We then create a cross-validation scheme for the artificially generated gaps and compare the EVA 2021 prediction scores of the proposed model to those obtained using some competitors. [ABSTRACT FROM AUTHOR]

In this paper we propose a procedure to estimate the distribution of wildfire frequency and severity using the wildfire data measured by month during 1993–2015. To this end, a spatial quantile autoregressive model (SQAR) is applied to the data with an aid of extreme value theory. Using the proposed method we are able to predict the distributional behavior of the data and identify the hidden structures beyond their mean structures. In addition, abundant interpretations are available with a regression-based model. We provide the estimated results from the wildfire data, including significant explanatory variables and some meaningful interpretations. [ABSTRACT FROM AUTHOR]

This paper details a methodology proposed for the EVA 2021 conference data challenge. The aim of this challenge was to predict the number and size of wildfires over the contiguous US between 1993 and 2015, with more importance placed on extreme events. In the data set provided, over 14% of both wildfire count and burnt area observations are missing; the objective of the data challenge was to estimate a range of marginal probabilities from the distribution functions of these missing observations. To enable this prediction, we make the assumption that the marginal distribution of a missing observation can be informed using non-missing data from neighbouring locations. In our method, we select spatial neighbourhoods for each missing observation and fit marginal models to non-missing observations in these regions. For the wildfire counts, we assume the compiled data sets follow a zero-inflated negative binomial distribution, while for burnt area values, we model the bulk and tail of each compiled data set using non-parametric and parametric techniques, respectively. Cross validation is used to select tuning parameters, and the resulting predictions are shown to significantly outperform the benchmark method proposed in the challenge outline. We conclude with a discussion of our modelling framework, and evaluate ways in which it could be extended. [ABSTRACT FROM AUTHOR]

We present our submission to the Extreme Value Analysis 2021 Data Challenge in which teams were asked to accurately predict distributions of wildfire frequency and size within spatio-temporal regions of missing data. For this competition, we developed a variant of the powerful variational autoencoder models, which we call Conditional Missing data Importance-Weighted Autoencoder (CMIWAE). Our deep latent variable generative model requires little to no feature engineering and does not necessarily rely on the specifics of scoring in the Data Challenge. It is fully trained on incomplete data, with the single objective to maximize log-likelihood of the observed wildfire information. We mitigate the effects of the relatively low number of training samples by stochastic sampling from a variational latent variable distribution, as well as by ensembling a set of CMIWAE models trained and validated on different splits of the provided data. [ABSTRACT FROM AUTHOR]

This paper details the approach of the team Kohrrelation in the 2021 Extreme Value Analysis data challenge, dealing with the prediction of wildfire counts and sizes over the contiguous US. Our approach uses ideas from extreme-value theory in a machine learning context with theoretically justified loss functions for gradient boosting. We devise a spatial cross-validation scheme and show that in our setting it provides a better proxy for test set performance than naive cross-validation. The predictions are benchmarked against boosting approaches with different loss functions, and perform competitively in terms of the score criterion, finally placing second in the competition ranking. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 first-order 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. [ABSTRACT FROM AUTHOR]

This paper devises a regression-type 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 real-data illustration reveals interesting aspects on the conditional risk of extreme losses in two leading international stock markets. [ABSTRACT FROM AUTHOR]

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 → ∞ 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 D of càdlàg functions indexed by 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 D and then study regular variation of random elements in D . We give practical conditions and study several examples, recovering and extending known results. [ABSTRACT FROM AUTHOR]

First, we consider a stationary random field indexed by an increasing sequence of subsets of ℤ d . Under certain mixing and anti–clustering conditions combined with a very broad assumption on how the sequence of spatial index sets increases, we obtain an extremal result that relates a normalized version of the distribution of the maximum of the field over the index sets to the tail distribution of the individual variables. Furthermore, we identify the limiting distribution as an extreme value distribution. Secondly, we consider a continuous, infinitely divisible random field indexed by ℝ d given as an integral of a kernel function with respect to a Lévy basis with convolution equivalent Lévy measure. When observing the supremum of this field over an increasing sequence of (continuous) index sets, we obtain an extreme value theorem for the distribution of this supremum. The proof relies on discretization and a conditional version of the technique applied in the first part of the paper, as we condition on the high activity and light–tailed part of the field. [ABSTRACT FROM AUTHOR]

In this paper we derive new results on multivariate extremes and D-norms. In particular we establish new characterizations of the multivariate max-domain of attraction property. The limit distribution of certain multivariate exceedances above high thresholds is derived, and the distribution of that generator of a D-norm on ℝ d , whose components sum up to d, is obtained. Finally we introduce exchangeable D-norms and show that the set of exchangeable D-norms is a simplex. [ABSTRACT FROM AUTHOR]

Threshold selection plays a key role in various aspects of statistical inference of rare events. In this work, two new threshold selection methods are introduced. The first approach measures the fit of the exponential approximation above a threshold and achieves good performance in small samples. The second method smoothly estimates the asymptotic mean squared error of the Hill estimator and performs consistently well over a wide range of processes. Both methods are analyzed theoretically, compared to existing procedures in an extensive simulation study and applied to a dataset of financial losses, where the underlying extreme value index is assumed to vary over time. [ABSTRACT FROM AUTHOR]

Max-stable processes are a natural extension of multivariate extreme value theory important for modeling the spatial dependence of environmental extremes. Inference for max-stable processes observed at several spatial locations is challenging due to the intractability of the full likelihood function. Composite likelihood methods avoid these difficulties by combining a number of low-dimensional likelihood objects, typically defined on pairs or triplets of spatial locations. This work develops a new truncation procedure based on ℓ1-penalty to reduce the complexity and computational cost associated with the composite likelihood function. The new method is shown to achieve a favorable trade-off between computational burden and statistical efficiency by dropping a number of noisy sub-likelihoods. The properties of the new method are illustrated through numerical simulations and an application to real extreme temperature data. [ABSTRACT FROM AUTHOR]

We study the behaviour of large values of extremal processes at small times, obtaining an analogue of the Fisher-Tippet-Gnedenko Theorem. Thus, necessary and sufficient conditions for local convergence of such maxima, linearly normalised, to the Fréchet or Gumbel distributions, are established. Weibull distributions are not possible limits in this situation. Moreover, assuming second order regular variation, we prove local asymptotic normality for intermediate order statistics, and derive explicit formulae for the normalising constants for tempered stable processes. We adapt Hill's estimator of the tail index to the small time setting and establish its asymptotic normality under second order regular variation conditions, illustrating this with simulations. Applications to the fine structure of asset returns processes, possibly with infinite variation, are indicated. [ABSTRACT FROM AUTHOR]

We show how fat tails in agricultural commodity returns arise endogenously from productivity shocks in a standard macroeconomic model. Using nearly ninety years of data, we show that the eight agricultural commodities in our sample exhibit fat-tailed return distributions. Statistical tests confirm the heavy-tailedness of price spikes for agricultural commodities. We apply extreme value theory to estimate the size and likelihood of price spikes in agricultural commodities. Back-testing verifies the validity of our risk assessment methodology. [ABSTRACT FROM AUTHOR]

We develop a method for probabilistic prediction of extreme value hot-spots in a spatio-temporal framework, tailored to big datasets containing important gaps. In this setting, direct calculation of summaries from data, such as the minimum over a space-time domain, is not possible. To obtain predictive distributions for such cluster summaries, we propose a two-step approach. We first model marginal distributions with a focus on accurate modeling of the right tail and then, after transforming the data to a standard Gaussian scale, we estimate a Gaussian space-time dependence model defined locally in the time domain for the space-time subregions where we want to predict. In the first step, we detrend the mean and standard deviation of the data and fit a spatially resolved generalized Pareto distribution to apply a correction of the upper tail. To ensure spatial smoothness of the estimated trends, we either pool data using nearest-neighbor techniques, or apply generalized additive regression modeling. To cope with high space-time resolution of data, the local Gaussian models use a Markov representation of the Matérn correlation function based on the stochastic partial differential equations (SPDE) approach. In the second step, they are fitted in a Bayesian framework through the integrated nested Laplace approximation implemented in R-INLA. Finally, posterior samples are generated to provide statistical inferences through Monte-Carlo estimation. Motivated by the 2019 Extreme Value Analysis data challenge, we illustrate our approach to predict the distribution of local space-time minima in anomalies of Red Sea surface temperatures, using a gridded dataset (11315 days, 16703 pixels) with artificially generated gaps. In particular, we show the improved performance of our two-step approach over a purely Gaussian model without tail transformations. [ABSTRACT FROM AUTHOR]

This paper describes the estimation of the extreme spatio-temporal sea surface temperature data based on the quantile factor model implemented by the SNU multiscale team. The proposed method was developed for the EVA2019 Data Challenge. Various attempts have been conducted to use factor models in spatio-temporal data analysis to find hidden factors in high-dimensional data. Factor models represent high-dimensional data as a linear combination of several factors, and hence, can describe spatially and temporally correlated data in a simple form. Meanwhile, unlike ordinary factor models, there are asymmetric norm-based factor models, such as quantile factor models or expectile dynamic semiparametric factor models, that can help understand the quantitative behavior of data beyond their mean structure. For this purpose, we apply a quantile factor model to the data to obtain significant factors explaining the quantile response of the temperatures and find quantile estimates. We develop a new method for inference of quantiles of extremal levels by extrapolating quantile estimates from the factor model with extreme value theory. The proposed method provides better performance than the benchmark, gives some interpretable insights, and shows the potential to expand the factor model with various data. [ABSTRACT FROM AUTHOR]

We describe our submission to the Extreme Value Analysis 2019 Data Challenge in which teams were asked to predict extremes of sea surface temperature anomaly within spatio-temporal regions of missing data. We present a computational framework which reconstructs missing data using convolutional deep neural networks. Conditioned on incomplete data, we employ autoencoder-like models as multivariate conditional distributions from which possible reconstructions of the complete dataset are sampled using imputed noise. In order to mitigate bias introduced by any one particular model, a prediction ensemble is constructed to create the final distribution of extremal values. Our method does not rely on expert knowledge in order to accurately reproduce dynamic features of a complex oceanographic system with minimal assumptions. The obtained results promise reusability and generalization to other domains. [ABSTRACT FROM AUTHOR]

It is long known that the distribution of a sum Sn of independent non-negative integer-valued random variables can often be approximated by a Poisson law: Sn≈πλ, where. The problem of evaluating the accuracy of such approximation has attracted a lot of attention in the past six decades. From a practical point of view, the problem has important applications in insurance, reliability theory, extreme value theory, etc.; from a theoretical point of view, it provides insights into Kolmogorov's problem. Among popular metrics considered in the literature is the Gini–Kantorovich distance dG. The task of establishing an estimate of dG(Sn;πλ) with correct (the best possible) constant at the leading term remained open for a long while. The paper presents a solution to that problem. A first-order asymptotic expansion is established as well. We show that the accuracy of approximation can be considerably better if the random variables obey an extra condition involving the first two moments. A sharp estimate of the accuracy of shifted (translated) Poisson approximation is established as well. [ABSTRACT FROM AUTHOR]

Recently, the notion of implicit extreme value distributions has been established, which is based on a given loss function f ≥ 0. From an application point of view, one is rather interested in extreme loss events that occur relative to f than in the corresponding extreme values itself. In this context, so-called f -implicit α-Fréchet max-stable distributions arise and have been used to construct independently scattered sup-measures that possess such margins. In this paper we solve an open problem in Goldbach (2016) by developing a stochastic integral of a deterministic function g ≥ 0 with respect to implicit max-stable sup-measures. The resulting theory covers the construction of max-stable extremal integrals (see Stoev and Taqqu Extremes 8, 237–266 (2005)) and, at the same time, reveals striking parallels. [ABSTRACT FROM AUTHOR]

This paper details the approach of team Lancaster to the 2019 EVA data challenge, dealing with spatio-temporal modelling of Red Sea surface temperature anomalies. We model the marginal distributions and dependence features separately; for the former, we use a combination of Gaussian and generalised Pareto distributions, while the dependence is captured using a localised Gaussian process approach. We also propose a space-time moving estimate of the cumulative distribution function that takes into account spatial variation and temporal trend in the anomalies, to be used in those regions with limited available data. The team's predictions are compared to results obtained via an empirical benchmark. Our approach performs well in terms of the threshold-weighted continuous ranked probability score criterion, chosen by the challenge organiser. [ABSTRACT FROM AUTHOR]

Classification tasks usually assume that all possible classes are present during the training phase. This is restrictive if the algorithm is used over a long time and possibly encounters samples from unknown new classes. It is therefore fundamental to develop algorithms able to distinguish between normal and abnormal test data. In the last few years, extreme value theory has become an important tool in multivariate statistics and machine learning. The recently introduced extreme value machine, a classifier motivated by extreme value theory, addresses this problem and achieves competitive performance in specific cases. We show that this algorithm has some theoretical and practical drawbacks and can fail even if the recognition task is fairly simple. To overcome these limitations, we propose two new algorithms for anomaly detection relying on approximations from extreme value theory that are more robust in such cases. We exploit the intuition that test points that are extremely far from the training classes are more likely to be abnormal objects. We derive asymptotic results motivated by univariate extreme value theory that make this intuition precise. We show the effectiveness of our classifiers in simulations and on real data sets. [ABSTRACT FROM AUTHOR]

We investigate maxima in incomplete samples from strictly stationary random sequences defined as linear models of i.i.d. random variables with heavy-tailed innovations that satisfy the tail balance condition. Using the point process approach we obtain limit theorems for the sequence of random vectors whose components are properly normalized maxima in complete and incomplete samples. [ABSTRACT FROM AUTHOR]

We consider removing lower order statistics from the classical Hill estimator in extreme value statistics, and compensating for it by rescaling the remaining terms. Trajectories of these trimmed statistics as a function of the extent of trimming turn out to be quite flat near the optimal threshold value. For the regularly varying case, the classical threshold selection problem in tail estimation is then revisited, both visually via trimmed Hill plots and, for the Hall class, also mathematically via minimizing the expected empirical variance. This leads to a simple threshold selection procedure for the classical Hill estimator which circumvents the estimation of some of the tail characteristics, a problem which is usually the bottleneck in threshold selection. As a by-product, we derive an alternative estimator of the tail index, which assigns more weight to large observations, and works particularly well for relatively lighter tails. A simple ratio statistic routine is suggested to evaluate the goodness of the implied selection of the threshold. We illustrate the favourable performance and the potential of the proposed method with simulation studies and real insurance data. [ABSTRACT FROM AUTHOR]

We acknowledge the priority on the introduction of the formula of t-lgHill estimator for the positive extreme value index. We provide a novel motivation for this estimator based on ecologically driven dynamical systems. Another motivation is given directly by applying the general t-Hill procedure to log-gamma distribution. We illustrate the good quality of t-lgHill estimator in comparison to classical Hill estimator on the novel data of the concentration of arsenic in drinking water in the rural area of the Arica and Parinacota Region, Chile. [ABSTRACT FROM AUTHOR]

In this paper we define the class of matrix Mittag-Leffler distributions and study some of its properties. We show that it can be interpreted as a particular case of an inhomogeneous phase-type distribution with random scaling factor, and alternatively also as the absorption time of a semi-Markov process with Mittag-Leffler distributed interarrival times. We then identify this class and its power transforms as a remarkably parsimonious and versatile family for the modeling of heavy-tailed risks, which overcomes some disadvantages of other approaches like the problem of threshold selection in extreme value theory. We illustrate this point both on simulated data as well as on a set of real-life MTPL insurance data that were modeled differently in the past. [ABSTRACT FROM AUTHOR]

Consider the maximum of independent and identically distributed random variables. The classical result says that the renormalized sample maximum converges to an extreme value distributions, under certain conditions on the distribution function. In the present paper, we shall study the uniform rate of the convergence with respect to the Kolmogorov distance in the framework of the Stein equations. Some typical examples are raised in the paper. [ABSTRACT FROM AUTHOR]

The extreme value dependence of regularly varying stationary time series can be described by the spectral tail process. Drees et al. (Extremes 18(3), 369–402, 2015) proposed estimators of the marginal distributions of this process based on exceedances over high deterministic thresholds and analyzed their asymptotic behavior. In practice, however, versions of the estimators are applied which use exceedances over random thresholds like intermediate order statistics. We prove that these modified estimators have the same limit distributions. This finding is corroborated in a simulation study, but the version using order statistics performs a bit better for finite samples. [ABSTRACT FROM AUTHOR]

The extremal index is an important parameter in the characterization of extreme values of a stationary sequence, since it measures short-range dependence at extreme values, and governs clustering of extremes. This paper presents a novel approach to estimation of the extremal index based on artificial censoring of inter-exceedance times. The censored estimator based on the maximum likelihood method is derived together with its variance, which is estimated from the expected Fisher information measure. In order to evaluate performance of the proposed estimator, a simulation study is carried out for various stationary processes satisfying the local dependence condition D(k)(un). An application to daily maximum temperatures at Uccle, Belgium, is also presented. [ABSTRACT FROM AUTHOR]