Conditional extreme value models have been introduced by Heffernan and Resnick (Ann. Appl. Probab., 17, 537–571, 2007) to describe the asymptotic behavior of a random vector as one specific component becomes extreme. Obviously, this class of models is related to classical multivariate extreme value theory which describes the behavior of a random vector as its norm (and therefore at least one of its components) becomes extreme. However, it turns out that this relationship is rather subtle and sometimes contrary to intuition. We clarify the differences between the two approaches with the help of several illuminative (counter)examples. Furthermore, we discuss marginal standardization, which is a useful tool in classical multivariate extreme value theory but, as we point out, much less straightforward and sometimes even obscuring in conditional extreme value models. Finally, we indicate how, in some situations, a more comprehensive characterization of the asymptotic behavior can be obtained if the conditions of conditional extreme value models are relaxed so that the limit is no longer unique.

The analysis of seasonal or annual block maxima is of interest in fields such as hydrology, climatology or meteorology. In connection with the celebrated method of block maxima, we study several tests that can be used to assess whether the available series of maxima is identically distributed. It is assumed that block maxima are independent but not necessarily generalized extreme value distributed. The asymptotic null distributions of the test statistics are investigated and the practical computation of approximate p-values is addressed. Extensive Monte-Carlo simulations show the adequate finite-sample behavior of the studied tests for a large number of realistic data generating scenarios. Illustrations on several environmental datasets conclude the work.

We extend the setting of the right endpoint estimator introduced in Fraga Alves and Neves (Statist. Sinica 24:1811--1835, 2014) to the broader class of light-tailed distributions with finite endpoint, belonging to some domain of attraction induced by the extreme value theorem. This stretch enables a general estimator for the finite endpoint, which does not require estimation of the (supposedly non-positive) extreme value index. A new testing procedure for selecting max-domains of attraction also arises in connection with asymptotic properties of the general endpoint estimator. The simulation study conveys that the general endpoint estimator is a valuable complement to the most usual endpoint estimators, particularly when the true extreme value index stays above $-1/2$, embracing the most common cases in practical applications. An illustration is provided via an extreme value analysis of supercentenarian women data.

This article discusses modelling of the tail of a multivariate distribution function by means of a large deviation principle (LDP), and its application to the estimation of the probability of a multivariate extreme event from a sample of n iid random vectors, with the probability bounded by powers of sample size with exponents below -1. One way to view classical tail limits is as limits of probability ratios. In contrast, the tail LDP provides asymptotic bounds or limits for log-probability ratios. After standardising the marginals to standard exponential, dependence is represented by a homogeneous rate function. Furthermore, the tail LDP can be extended to represent both dependence and marginals, the latter implying marginal log-GW tail limits. A connection is established between the tail LDP and residual tail dependence (or hidden regular variation) and a recent extension of it. Under a smoothness assumption, they are implied by the tail LDP. Based on the tail LDP, a simple estimator for very small probabilities of extreme events is formulated. It avoids estimation of the rate function by making use of its homogeneity. Strong consistency in the sense of convergence of log-probability ratios is proven. Simulations and an application illustrate the difference between the classical approach and the LDP-based approach.

Working with Markov kernels (conditional distributions) and right-hand derivatives D+A of Pickands dependence functions A we study the way two-dimensional extreme-value copulas (EVCs) CA distribute mass. Underlining the usefulness of working directly with D+A, we give first an alternative simple proof of the fact that EVCs with piecewise linear A can be expressed as weighted geometric mean of some EVCs whose dependence functions A have at most two edges and present a generalization of this result. After showing that the discrete component of the Markov kernel of CA concentrates its mass on the graphs of some increasing homeomorphisms ft, we determine which EVC assigns maximum mass to the union of the graphs of \(f_{t_{1}},\ldots ,f_{t_{N}}\), derive the absolutely continuous component of an arbitrary EVC CA and deduce that the minimum copula M is the only (purely) singular EVC. Additionally, we prove the existence of EVCs CA which, despite their simple analytic form, exhibit the following surprisingly singular behavior: the discrete, the absolutely continuous and the singular component of the Lebesgue decomposition of the Markov kernel \(K_{C_{A}}(x,\cdot )\) of CA have full support [0,1] for every x∈[0,1].

We propose a new definition of a multivariate subexponential distribution. We compare this definition with the two existing notions of multivariate subexponentiality, and compute the asymptotic behaviour of the ruin probability in the context of an insurance portfolio, when multivariate subexponentiality holds. Previously such results were available only in the case of multivariate regularly varying claims.

For the stationary storage process {Q(t), t ≥ 0}, with \( Q(t)=\sup _{s\ge t}\left (X(s)-X(t)-c(s-t)^{\beta }\right ),\) where {X(t), t ≥ 0} is a centered Gaussian process with stationary increments, c > 0 and β > 0 is chosen such that Q(t) is finite a.s., we derive exact asymptotics of \(\mathbb {P}\left (\sup _{t\in [0,T_{u}]} Q(t)>u \right )\) and \(\mathbb {P}\left (\inf _{t\in [0,T_{u}]} Q(t)>u \right )\), as \(u\rightarrow \infty \). As a by-product we find conditions under which strong Piterbarg property holds.

A distance based measure of dependence is proposed for stable distributions that completely characterizes independence for a bivariate stable distribution. Properties of this measure are analyzed, and contrasted with the covariation and co-difference. A sample analog of the measure is defined and demonstrated on simulated and real data, including time series and distributions in the domain of attraction of a stable law.

In many practical situations exploratory plots are helpful in understanding tail behavior of sample data. The Mean Excess plot is one of the exploratory tools often used in practice to understand the right tail behavior of a data set. It is known that if the underlying distribution of a data sample is in the maximum domain of attraction of a Frechet, a Gumbel or a Weibull distributions then the ME plot of the data approaches a straight line in an appropriate sense, with positive, zero or negative slope respectively. In this paper we construct confidence intervals around the ME plots which assist us in ascertaining which particular maximum domain of attraction the data set comes from. We recall weak limit results for the Frechet domain of attraction, already obtained in Das and Ghosh (Bernoulli 19, 308–342 2013) and derive weak limits for the Gumbel and Weibull domains in order to construct confidence bounds. We demonstrate our methodology by applying them to simulated and real data sets.

We use the Stein-Chen method to study the extremal behaviour of univariate and bivariate geometric laws. We obtain a rate for the convergence, to the Gumbel distribution, of the law of the maximum of i.i.d. geometric random variables, and show that convergence is faster when approximating by a discretised Gumbel. We similarly find a rate of convergence for the law of maxima of bivariate Marshall-Olkin geometric random pairs when approximating by a discrete limit law. We introduce marked point processes of exceedances (MPPEs), both with univariate and bivariate Marshall-Olkin geometric variables as marks and determine bounds on the error of the approximation, in an appropriate probability metric, of the law of the MPPE by that of a Poisson process with same mean measure. We then approximate by another Poisson process with an easier-to-use mean measure and estimate the error of this additional approximation. This work contains and extends results contained in the second author’s PhD thesis (Feidt 2013) under the supervision of Andrew D. Barbour.