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

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 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.

We propose a new class of extreme-value copulas which are extreme-value limits of conditional normal models. Conditional normal models are generalizations of conditional independence models, where the dependence among observed variables is modeled using one unobserved factor. Conditional on this factor, the distribution of these variables is given by the Gaussian copula. This structure allows one to build flexible and parsimonious models for data with complex dependence structures, such as data with spatial dependence or factor structure. We study the extreme-value limits of these models and show some interesting special cases of the proposed class of copulas. We develop estimation methods for the proposed models and conduct a simulation study to assess the performance of these algorithms. Finally, we apply these copula models to analyze data on monthly wind maxima and stock return minima.

We derive the exact asymptotics of $ {\mathbb {P} \left \{ \underset {t\ge 0}{\sup } \left (X_{1}(t) - \mu _{1} t\right )> u, \underset {s\ge 0}{\sup } \left (X_{2}(s) - \mu _{2} s\right )> u \right \} }, u\to \infty , $ where (X1(t), X2(s))t, s≥ 0 is a correlated two-dimensional Brownian motion with correlation ρ ∈ [− 1,1] and μ1, μ2 > 0. It appears that the play between ρ and μ1, μ2 leads to several types of asymptotics. Although the exponent in the asymptotics as a function of ρ is continuous, one can observe different types of prefactor functions depending on the range of ρ, which constitute a phase-type transition phenomena.

Among bivariate tail dependence measures, the tail dependence coefficient has emerged as the popular choice. Akin to the correlation matrix, a multivariate dependence measure is constructed using these bivariate measures, and this is referred to in the literature as the tail dependence matrix (TDM). While the problem of determining whether a given d × d matrix is a correlation matrix is of the order O(d3) in complexity, determining if a matrix is a TDM (the realization problem) is believed to be non-polynomial in complexity. Using a linear programming (LP) formulation, we show that the combinatorial structure of the constraints is related to the intractable max-cut problem in a weighted graph. This connection provides an avenue for constructing parametric classes admitting a polynomial in d algorithm for determining membership in its constraint polytope. The complexity of the general realization problem is justifiably of much theoretical interest. Since in practice one resorts to lower dimensional parametrization of the TDMs, we posit that it is rather the complexity of the realization problem restricted to parametric classes of TDMs, and algorithms for it, that are more practically relevant. In this paper, we show how the inherent symmetry and sparsity in a parametrization can be exploited to achieve a significant reduction in the LP formulation, which can lead to polynomial complexity of such realization problems - some parametrizations even resulting in the constraint polytope being independent of d. We also explore the use of a probabilistic viewpoint on TDMs to derive the constraint polytopes.

We consider stationary time series $\{X_j, j \in Z\} whose finite dimensional distributions are regularly varying with extremal independence. We assume that for each $h \geq 1$, conditionally on $X_0$ to exceed a threshold tending to infinity, the conditional distribution of $X_h$ suitably normalized converges weakly to a non degenerate distribution. We consider in this paper the estimation of the normalization and of the limiting distribution.

Preferential attachment is an appealing edge generating mechanism for modeling social networks. It provides both an intuitive description of network growth and an explanation for the observed power laws in degree distributions. However, there are often limitations in fitting parametric network models to data due to the complex nature of real-world networks. In this paper, we consider a semi-parametric estimation approach by looking at only the nodes with large in- or out-degrees of the network. This method examines the tail behavior of both the marginal and joint degree distributions and is based on extreme value theory. We compare it with the existing parametric approaches and demonstrate how it can provide more robust estimates of parameters associated with the network when the data are corrupted or when the model is misspecified.

Motivated by theoretical similarities between the classical Hill estimator of the tail index of a heavy-tailed distribution and one of its pseudo-estimator versions featuring a non-random threshold, we show a novel asymptotic representation of a class of empirical average excesses above a high random threshold, expressed in terms of order statistics, using their counterparts based on a suitable non-random threshold, which are sums of independent and identically distributed random variables. As a consequence, the analysis of the joint convergence of such empirical average excesses essentially boils down to a combination of Lyapunov’s central limit theorem and the Cramér-Wold device. We illustrate how this allows to improve upon, as well as produce conceptually simpler proofs of, very recent results about the joint convergence of marginal Hill estimators for a random vector with heavy-tailed marginal distributions. These results are then applied to the proof of a convergence result for a tail index estimator when the heavy-tailed variable of interest is randomly right-truncated. New results on the joint convergence of conditional tail moment estimators of a random vector with heavy-tailed marginal distributions are also obtained.

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.