Extremes of spherical fractional Brownian motion Abstract Let \(\{B_{\beta } (x), x \in \mathbb {\mathbb S}^{N}\}\) be a fractional Brownian motion on the N-dimensional unit sphere \(\mathbb {S}^{N}\) with Hurst index β. We study the excursion probability \(\mathbb {P}\left \{{{\sup _{x\in T} B_{\beta }(x) > u }}\right \}\) and obtain the asymptotics as u →∞, where T can be the entire sphere \(\mathbb {S}^{N}\) or a geodesic disc on \(\mathbb {S}^{N}\) .

Extremes of stationary random fields on a lattice Abstract Extremal behavior of stationary Gaussian sequences/random fields is widely investigated since it models common cluster phenomena and brings a bridge between discrete and continuous extremes. We establish extensively limit theorems of stationary random fields under certain mixing and dependence conditions, which are further illustrated by typical examples of order statistics of Gaussian random fields and skew-Gaussian random fields. The positivity of the cluster index involved and its link with the expected cluster size are discussed.

Exceedance-based nonlinear regression of tail dependence Abstract The probability and structure of co-occurrences of extreme values in multivariate data may critically depend on auxiliary information provided by covariates. In this contribution, we develop a flexible generalized additive modeling framework based on high threshold exceedances for estimating covariate-dependent joint tail characteristics for regimes of asymptotic dependence and asymptotic independence. The framework is based on suitably defined marginal pretransformations and projections of the random vector along the directions of the unit simplex, which lead to convenient univariate representations of multivariate exceedances based on the exponential distribution. Good performance of our estimators of a nonparametrically designed influence of covariates on extremal coefficients and tail dependence coefficients are shown through a simulation study. We illustrate the usefulness of our modeling framework on a large dataset of nitrogen dioxide measurements recorded in France between 1999 and 2012, where we use the generalized additive framework for modeling marginal distributions and tail dependence in large concentrations observed at pairs of stations. Our results imply asymptotic independence of data observed at different stations, and we find that the estimated coefficients of tail dependence decrease as a function of spatial distance and show distinct patterns for different years and for different types of stations (traffic vs. background).

A nonparametric method for producing isolines of bivariate exceedance probabilities Abstract We present a method for drawing isolines indicating regions of equal joint exceedance probability for bivariate data. The method relies on bivariate regular variation, a dependence framework widely used for extremes. The method we utilize for characterizing dependence in the tail is largely nonparametric. The extremes framework enables drawing isolines corresponding to very low exceedance probabilities and may even lie beyond the range of the data; such cases would be problematic for standard nonparametric methods. Furthermore, we extend this method to the case of asymptotic independence and propose a procedure which smooths the transition from hidden regular variation in the interior to the first-order behavior on the axes. We propose a diagnostic plot for assessing the isoline estimate and choice of smoothing, and a bootstrap procedure to visually assess uncertainty.

The time of ultimate recovery in Gaussian risk model Abstract We analyze the distance \(\mathcal {R}_{T}(u)\) between the first and the last passage time of {X(t) − ct : t ∈ [0, T]} at level u in time horizon T ∈ (0, ∞], where X is a centered Gaussian process with stationary increments and \(c\in {\mathbb {R}}\) , given that the first passage time occurred before T. Under some tractable assumptions on X, we find Δ(u) and G(x) such that $$\lim\limits_{u\to\infty}\mathbb{P} \left( \mathcal{R}_{T}(u)>{\Delta}(u)x \right)=G(x), $$ for x ≥ 0. We distinguish two scenarios: T < ∞ and T = ∞, that lead to qualitatively different asymptotics. The obtained results provide exact asymptotics of the ultimate recovery time after the ruin in Gaussian risk model.

On maximum of Gaussian random field having unique maximum point of its variance Abstract Gaussian random fields on Euclidean spaces whose variances reach their maximum values at unique points are considered. Exact asymptotic behaviors of probabilities of large absolute maximum of theirs trajectories have been evaluated using Double Sum Method under the widest possible conditions.

Bias-corrected estimation for conditional Pareto-type distributions with random right censoring Abstract We consider bias-reduced estimation of the extreme value index in conditional Pareto-type models with random covariates when the response variable is subject to random right censoring. The bias-correction is obtained by fitting the extended Pareto distribution locally to the relative excesses over a high threshold using the maximum likelihood method. Convergence in probability and asymptotic normality of the estimators are established under suitable assumptions. The finite sample behaviour is illustrated with a simulation experiment and the method is applied to two real datasets.

The largest order statistics for the inradius in an isotropic STIT tessellation Abstract A planar stationary and isotropic STIT tessellation at time t > 0 is observed in the window \(W_{\rho }={t^{-1}}\sqrt {\pi \ \rho }\cdot [-\frac {1}{2},\frac {1}{2}]^{2}\) , for ρ > 0. With each cell of the tessellation, we associate the inradius, which is the radius of the largest disk contained in the cell. Using the Chen-Stein method, we compute the limit distributions of the largest order statistics for the inradii of all cells whose nuclei are contained in Wρ as ρ goes to infinity.

Improving precipitation forecasts using extreme quantile regression Abstract Aiming to estimate extreme precipitation forecast quantiles, we propose a nonparametric regression model that features a constant extreme value index. Using local linear quantile regression and an extrapolation technique from extreme value theory, we develop an estimator for conditional quantiles corresponding to extreme high probability levels. We establish uniform consistency and asymptotic normality of the estimators. In a simulation study, we examine the performance of our estimator on finite samples in comparison with a method assuming linear quantiles. On a precipitation data set in the Netherlands, these estimators have greater predictive skill compared to the upper member of ensemble forecasts provided by a numerical weather prediction model.

Extremal dependence of random scale constructions Abstract A bivariate random vector can exhibit either asymptotic independence or dependence between the largest values of its components. When used as a statistical model for risk assessment in fields such as finance, insurance or meteorology, it is crucial to understand which of the two asymptotic regimes occurs. Motivated by their ubiquity and flexibility, we consider the extremal dependence properties of vectors with a random scale construction (X1,X2) = R(W1,W2), with non-degenerate R > 0 independent of (W1,W2). Focusing on the presence and strength of asymptotic tail dependence, as expressed through commonly-used summary parameters, broad factors that affect the results are: the heaviness of the tails of R and (W1,W2), the shape of the support of (W1,W2), and dependence between (W1,W2). When R is distinctly lighter tailed than (W1,W2), the extremal dependence of (X1,X2) is typically the same as that of (W1,W2), whereas similar or heavier tails for R compared to (W1,W2) typically result in increased extremal dependence. Similar tail heavinesses represent the most interesting and technical cases, and we find both asymptotic independence and dependence of (X1,X2) possible in such cases when (W1,W2) exhibit asymptotic independence. The bivariate case often directly extends to higher-dimensional vectors and spatial processes, where the dependence is mainly analyzed in terms of summaries of bivariate sub-vectors. The results unify and extend many existing examples, and we use them to propose new models that encompass both dependence classes.