Extreme value theory and statistics

Abstract

We consider the class of r-Pareto processes defined on [0, 1] whose max-stable counterparts are Brown–Resnick processes. The aim of this paper is to propose a test whether the extreme value index function of a r-Pareto process of this class remains constant over [0, 1]. We assume that we observe several independent r -Pareto processes over equispaced grids of [0, 1] but with possibly heterogeneous grid resolutions. We build a test based on the normalized approximate total variations of the paths of these processes. We provide its properties under infill asymptotics and assess its finite sample performance  through simulation experiments. We discuss how to proceed for random processes that belong to the domain of attraction of Brown–Resnick r -Pareto processes and illustrate the approach with an application to wind speed data from Germany.

Abstract

We consider expected performances based on max-stable random fields and we are interested in their derivatives with respect to the spatial dependence parameters of those fields. Max-stable fields, such as the Brown–Resnick and Smith fields, are very popular in spatial extremes. We focus on the two most popular unbiased stochastic derivative estimation approaches: the likelihood ratio method (LRM) and the infinitesimal perturbation analysis (IPA). LRM requires the multivariate density of the max-stable field to be explicit, and IPA necessitates the computation of the derivative with respect to the parameters for each simulated value. We propose convenient and tractable conditions ensuring the validity of LRM and IPA in the cases of the Brown–Resnick and Smith field, respectively. Obtaining such conditions is intricate owing to the very structure of max-stable fields. Then we focus on risk and dependence measures, which constitute one of the several frameworks where our theoretical results can be useful. We perform a simulation study which shows that both LRM and IPA perform well in various configurations, and provide a real case study that is valuable for the insurance industry.

Abstract

We consider a continuous Gaussian random field living on a compact set 𝑇 ⊂ℝ𝑑. We are interested in designing an asymptotically efficient estimator of the probability that the integral of the exponential of the Gaussian process over T exceeds a large threshold u. We propose an Asmussen–Kroese conditional Monte Carlo type estimator and discuss its asymptotic properties according to the assumptions on the first and second moments of the Gaussian random field. We also provide a simulation study to illustrate its effectiveness and compare its performance with the importance sampling type estimator of Liu and Xu (2014a).

Abstract

We consider the class of simple Brown-Resnick max-stable processes whose spectral processes are continuous exponential martingales. We develop the asymptotic theory for the realized power variations of these max-stable processes, that is, sums of powers of absolute increments. We consider an infill asymptotic setting, where the sampling frequency converges to zero while the time span remains fixed. More specifically we obtain a biased central limit theorem whose bias depends on the local times of the differences between the logarithms of the underlying spectral processes.

Abstract

Max-stable processes are central models for spatial extremes. In this paper, we focus on some space–time max-stable models introduced in Embrechts et al. (2016). The processes considered induce discrete-time Markov chains taking values in the space of continuous functions from the unit sphere of ℝ3 to (0,∞). We show that these Markov chains are geometrically ergodic. An interesting feature lies in the fact that the state space is not locally compact, making the classical methodology inapplicable. Instead, we use the fact that the state space is Polish and apply results presented in Hairer (2010).

Abstract

Max-stable random fields are very appropriate for the statistical modelling of spatial extremes. Hence, integrals of functions of max-stable random fields over a given region can play a key role in the assessment of the risk of natural disasters, meaning that it is relevant to improve our understanding of their probabilistic behaviour. For this purpose, in this paper, we propose a general central limit theorem for functions of stationary max-stable random fields on ℝ𝑑. Then, we show that appropriate functions of the Brown–Resnick random field with a power variogram and of the Smith random field satisfy the central limit theorem. Another strong motivation for our work lies in the fact that central limit theorems for random fields on ℝ𝑑 have been barely considered in the literature. As an application, we briefly show the usefulness of our results in a risk assessment context.

Abstract

We consider the Voronoi tessellation based on a homogeneous Poisson point process in an Euclidean space. For a geometric characteristic of the cells (e.g., the inradius, the circumradius, the volume), we investigate the point process of the nuclei of the cells with large values. Conditions are obtained for the convergence in distribution of this point process of exceedances to a homogeneous compound Poisson point process. We provide a characterization of the asymptotic cluster size distribution which is based on the Palm version of the point process of exceedances. This characterization allows us to compute efficiently the values of the extremal index and the cluster size probabilities by simulation for various geometric characteristics. The extension to the Poisson–Delaunay tessellation is also discussed.

Abstract

In this paper, we compare two numerical methods for approximating the probability that the sum of dependent regularly varying random variables exceeds a high threshold under Archimedean copula models. The first method is based on conditional Monte Carlo. We present four estimators and show that most of them have bounded relative errors. The second method is based on analytical expressions of the multivariate survival or cumulative distribution functions of the regularly varying random variables and provides sharp and deterministic bounds of the probability of exceedance. We discuss implementation issues and illustrate the accuracy of both procedures through numerical studies.

Abstract

Multivariate extreme value statistical analysis is concerned with observations on several variables which are thought to possess some degree of tail dependence. The main approaches to inference for multivariate extremes consist in approximating either the distribution of block component-wise maxima or the distribution of the exceedances over a high threshold. Although the expressions of the asymptotic density functions of these distributions may be characterized, they cannot be computed in general. In this paper, we study the case where the spectral random vector of the multivariate max-stable distribution has known conditional distributions. The asymptotic density functions of the multivariate extreme value distributions may then be written through univariate integrals that are easily computed or simulated. The asymptotic properties of two likelihood estimators are presented, and the utility of the method is examined via simulation.

Abstract

We introduce the systemic tail risk distribution of a financial market to characterize the asset return linkages during financial crisis. This distribution provides the probabilities that several

assets of a market lose a large part of their nominal value given that the price of at least one of them collapses. It introduces a new way of assessing the stability of a financial market during

potential systemic risk events. We propose a new type of multivariate extreme value distribution for high-dimensional vectors to model the extremal dependence between asset prices, and

we use efficient likelihood inference methods to estimate the parameters of the systemic tail risk distribution. Our real data show that the empirical static systemic tail risk distribution is U-shaped, while the empirical conditional distribution is L-shaped.

Abstract

Natural disasters may have considerable impact on society as well as on the (re-)insurance industry. Max-stable processes are ideally suited for the modelling of the spatial extent of such extreme events, but it is often assumed that there is no temporal dependence. Only a few papers have introduced spatiotemporal max-stable models, extending the Smith, Schlather and Brown‒Resnick spatial processes. These models suffer from two major drawbacks: time plays a similar role to space and the temporal dynamics are not explicit. In order to overcome these defects, we introduce spatiotemporal max-stable models where we partly decouple the influence of time and space in their spectral representations. We introduce both continuous- and discrete-time versions. We then consider particular Markovian cases with a max-autoregressive representation and discuss their properties. Finally, we briefly propose an inference methodology which is tested through a simulation study.

Abstract

In this paper, we study the asymptotic distribution of the number of exceedances of multiplicative factor models under the rare-event condition that the probability the number of exceedances is positive tends to zero. We show that the asymptotic conditional distribution (given that there is at least one exceedance), as well as the condition such that the probability of the conditional event tends to zero, depend on the distribution of the multiplicative disturbances. When the distribution of the disturbances is not a heavy-tailed distribution, it is necessary to normalize the number of exceedances to get a non-degenerate asymptotic distribution.

Abstract

Assume that claims in a portfolio of insurance contracts are described by independent and identically distributed random variables with regularly varying tails and occur according to a near mixed Poisson process. We provide a collection of results pertaining to the joint asymptotic Laplace transforms of the normalised sums of the smallest and largest claims, when the length of the considered time interval tends to infinity. The results crucially depend on the value of the tail index of the claim distribution, as well as on the number of largest claims under consideration.

Abstract

This paper is concerned with the efficient simulation of ℙ⁡(𝑆𝑛>𝑠) in situations where 𝑠 is large and 𝑆𝑛 is the sum of 𝑛 i.i.d. heavy-tailed random variables 𝑋1,…,𝑋𝑛. The most efficient and simplest estimators introduced in the rare event simulation literature are those proposed by Asmussen and Kroese (2006) and Asmussen and Kortschak (2012). Although the main techniques for facing the rare event problem are importance sampling and splitting, the estimators of Asmussen, Kortschak and Kroese combine exchangeability arguments with conditional Monte-Carlo to construct estimators whose relative errors go to 0 as 𝑠 →∞. In this paper, we decompose ℙ⁡(𝑆𝑛>𝑠) as the sum of ℙ⁡(𝑀𝑛>𝑠) and ℙ⁡(𝑆𝑛>𝑠,𝑀𝑛<𝑠) as proposed by Juneja (2007) because ℙ⁡(𝑀𝑛>𝑠) is known in closed form and is asymptotically equivalent to ℙ⁡(𝑆𝑛>𝑠). We construct new efficient estimators of ℙ⁡(𝑆𝑛>𝑠,𝑀𝑛<𝑠) by splitting up it again and then by using the same type of reliable methods as in Asmussen and Kroese (2006). We show that these new estimators have smaller relative errors than the estimators of Asmussen, Kortschak and Kroese. The conclusion of the numerical study is that our estimators compare extremely favorably with previous ones.

Abstract

We present two new classes of stationary max-stable random fields. For the first class, we use the spectral representation due to Schlather (2002) and assume that the stationary process used in the representation is proportional to a power of a max-stable random field. We derive the finite dimensional distributions, explain the relationship between distributions of both max-stable random fields and give sufficient conditions for the sample paths to be 𝛾-Hölder continuous functions for some 𝛾 ∈(0,1). For the second class, we consider a multiplicative factor model and a Poisson–Voronoï tessellation of ℝ𝑑 to construct new max-stable random fields. We provide explicit expressions for the pairwise distribution function.

Abstract

The analysis of events with low probability but disastrous impact entails understanding how they cluster in time. We present an automatic three-step procedure for identifying clusters, estimating the cluster size distribution and constructing confidence intervals for the extremal index, which measures the degree of clustering of rare events. The third step combines empirical likelihood and parametric likelihood approaches. Simulations show that our new procedure performs very well for finite samples and outperforms previous methods in constructing confidence intervals for the extremal index when there is clustering in the data, as well as in estimating probabilities for small clusters.

Abstract

This paper provides extensions of the work on subsampling by Bertail et al. in J. Econ. 120:295–326 (2004) for strongly mixing case to weakly dependent case by application of the results of Doukhan and Louhichi in Stoch. Proc. Appl. 84:313–342 (1999). We investigate properties of smooth and rough subsampling estimators for sampling distributions of converging and extreme statistics when the underlying time series is η- or λ-weakly dependent.

Abstract

The observed extremes of a stationary sequence subject to random failure or censoring depend on the failure or censoring mechanism and may differ in an important way with the extremes of the stationary sequence itself. In this work we develop theoretical results which show how to take into account the effects of the random failures or censorships on the maximum of a sample from a stationary sequence.

Abstract

In extreme value statistics for stationary sequences, blocks estimators are usually constructed by using disjoint blocks because exceedances over high thresholds of different blocks can be assumed asymptotically independent. In this paper we focus on the estimation of the extremal index which measures the degree of clustering of extremes. We consider disjoint and sliding blocks estimators and compare their asymptotic properties. In particular we show that the sliding blocks estimator is more efficient than the disjoint version and has a smaller asymptotic bias. Moreover we propose a method to reduce its bias when considering sufficiently large block sizes.

Abstract

Any limiting point process for the time normalized exceedances of high levels by a stationary sequence is necessarily compound Poisson under appropriate long range dependence conditions. Typically exceedances appear in clusters. The underlying Poisson points represent the cluster positions and the multiplicities correspond to the cluster sizes. In the present paper we introduce estimators of the limiting cluster size probabilities, which are constructed through a recursive algorithm. We derive estimators of the extremal index which plays a key role in determining the intensity of cluster positions. We study the asymptotic properties of the estimators and investigate their finite sample behavior on simulated data.

Abstract

Under appropriate long range dependence conditions, the point process of exceedances of a stationary sequence weakly converges to a homogeneous compound Poisson point process. This limiting point process can be characterized by the extremal index and the cluster-size probabilities. In this paper we address the problem of estimating these quantities and we consider the intervals estimators introduced in Ferro and Segers [2003. Inference for clusters of extreme values. J. Roy. Statist. Soc. Ser. B 545–556] and in Ferro [2004. Statistical methods for clusters of extreme values. Ph.D. Thesis, Lancaster University]. We establish asymptotic weak convergence to Gaussian random variables and we give their asymptotic variance.

Abstract

The multivariate extremal index function relates the asymptotic distribution of the vector of pointwise maxima of a multivariate stationary sequence to that of the independent sequence from the same stationary distribution. It also measures the degree of clustering of extremes in the multivariate process. In this paper, we construct nonparametric estimators of this function and prove their asymptotic normality under long-range dependence and moment conditions. The results are illustrated by means of a simulation study.

Abstract

The tail of the distribution of a sum of a random number of independent and identically distributed nonnegative random variables depends on the tails of the number of terms and of the terms themselves. This situation is of interest in the collective risk model, where the total claim size in a portfolio is the sum of a random number of claims. If the tail of the claim number is heavier than the tail of the claim sizes, then under certain conditions the tail of the total claim size does not change asymptotically if the individual claim sizes are replaced by their expectations. The conditions allow the claim number distribution to be of consistent variation or to be in the domain of attraction of a Gumbel distribution with a mean excess function that grows to infinity sufficiently fast. Moreover, the claim number is not necessarily required to be independent of the claim sizes.

Abstract

Natural catastrophes cause insurance losses in several different lines of business. An approach to modelling the dependence in loss severities is to assume that they are related to the intensity of the natural disaster. In this paper we introduce a factor model and investigate the extreme dependence. We derive a specific extreme dependence structure when considering an heavy-tailed intensity. Estimation procedures are presented and their moderate sample properties are compared in a simulation study. We also motivate our approach by an illustrative example from storm insurance. 

Abstract

Let (Y n , N n )n≥1 be independent and identically distributed bivariate random variables such that the N n are positive with finite mean ν and the Y n have a common heavy-tailed distribution F. We consider the process (Z n )n≥1 defined by Z n = Y n - Σn-1, where  It is shown that the probability that the maximum M = maxn≥1 Z n exceeds x is approximately  as x → ∞, where F' := 1 - F. Then we study the integrated tail of the maximum of a random walk with long-tailed increments and negative drift over the interval [0, σ], defined by some stopping time σ, in the case in which the randomly stopped sum is negative. Finally, an application to risk theory is considered. 

Abstract

This article presents a study of extreme price movements of Alcatel stock over 1 minute, 5 minutes and 30 minutes time intervals during a trading section. This movements are encountered after the opening, at lunch time around 12:30 a.m. and before the closing. According to extreme value theory, the form of the distribution of extreme movements is precisely known; empirically, the extreme price variations obey the Gumbel distribution. Moreover, we observe that the larger the time interval, the thinner the distribution tails.