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## Spatial risk measures and applications to max-stable processes

### Extremes (2017-09-01) 20: 635-670 , September 01, 2017

The risk of extreme environmental events is of great importance for both the authorities and the insurance industry. This paper concerns risk measures in a spatial setting, in order to introduce the spatial features of damages stemming from environmental events into the measure of the risk. We develop a new concept of spatial risk measure, based on the spatially aggregated loss over the region of interest, and propose an adapted set of axioms for these spatial risk measures. These axioms quantify the sensitivity of the risk measure with respect to the space and are especially linked to spatial diversification. The proposed model for the cost underlying our definition of spatial risk measure involves applying a damage function to the environmental variable considered. We build and theoretically study concrete examples of spatial risk measures based on the indicator function of max-stable processes exceeding a given threshold. Some interpretations in terms of insurance are provided.

## Maxima of moving maxima of continuous functions

### Extremes (2012-09-01) 15: 267-297 , September 01, 2012

Maxima of moving maxima of continuous functions (CM3) are max-stable processes aimed at modelling extremes of continuous phenomena over time. They are defined as Smith and Weissman’s M4 processes with continuous functions rather than vectors. After standardization of the margins of the observed process into unit-Fréchet, CM3 processes can model the remaining spatio-temporal dependence structure. CM3 processes have the property of joint regular variation. The spectral processes from this class admit particularly simple expressions given here. Furthermore, depending on the speed with which the parameter functions tend toward zero, CM3 processes fulfill the finite-cluster condition and the strong mixing condition. Processes enjoying these three properties also enjoy a simple expression for their extremal index. Next a method to fit CM3 processes to data is investigated. The first step is to estimate the length of the temporal dependence. Then, by selecting a suitable number of blocks of extremes of this length, clustering algorithms are used to estimate the total number of different profiles. The parameter functions themselves are estimated thanks to the output of the partitioning algorithms. The full procedure only requires one parameter which is the range of variation allowed among the different profiles. The dissimilarity between the original CM3 and the estimated version is evaluated by means of the Hausdorff distance between the graphs of the parameter functions.

## Clustering of high values in random fields

### Extremes (2017-03-30): 1-32 , March 30, 2017

The asymptotic results that underlie applications of extreme random fields often assume that the variables are located on a regular discrete grid, identified with $\mathbb {Z}^{2}$ , and that they satisfy stationarity and isotropy conditions. Here we extend the existing theory, concerning the asymptotic behavior of the maximum and the extremal index, to non-stationary and anisotropic random fields, defined over discrete subsets of $\mathbb {R}^{2}$ . We show that, under a suitable coordinatewise mixing condition, the maximum may be regarded as the maximum of an approximately independent sequence of submaxima, although there may be high local dependence leading to clustering of high values. Under restrictions on the local path behavior of high values, criteria are given for the existence and value of the spatial extremal index which plays a key role in determining the cluster sizes and quantifying the strength of dependence between exceedances of high levels. The general theory is applied to the class of max-stable random fields, for which the extremal index is obtained as a function of well-known tail dependence measures found in the literature, leading to a simple estimation method for this parameter. The results are illustrated with non-stationary Gaussian and 1-dependent random fields. For the latter, a simulation and estimation study is performed.

## Distribution of the height of local maxima of Gaussian random fields

### Extremes (2015-06-01) 18: 213-240 , June 01, 2015

Let {*f*(*t*) : *t* ∈ *T*} be a smooth Gaussian random field over a parameter space *T*, where *T* may be a subset of Euclidean space or, more generally, a Riemannian manifold. We provide a general formula for the distribution of the height of a local maximum ℙ{*f*(*t*_{0}) > *u*|*t*_{0} is a local maximum of *f*(*t*)} when *f* is non-stationary. Moreover, we establish asymptotic approximations for the overshoot distribution of a local maximum ℙ{*f*(*t*_{0}) > *u*+*v*|*t*_{0} is a local maximum of *f*(*t*) and *f*(*t*_{0}) > *v*} as
$v\to \infty $
. Assuming further that *f* is isotropic, we apply techniques from random matrix theory related to the Gaussian orthogonal ensemble to compute such conditional probabilities explicitly when *T* is Euclidean or a sphere of arbitrary dimension. Such calculations are motivated by the statistical problem of detecting peaks in the presence of smooth Gaussian noise.

## Exact simulation of Brown-Resnick random fields at a finite number of locations

### Extremes (2015-06-01) 18: 301-314 , June 01, 2015

We propose an exact simulation method for Brown-Resnick random fields, building on new representations for these stationary max-stable fields. The main idea is to apply suitable changes of measure.

## On tail trend detection: modeling relative risk

### Extremes (2015-06-01) 18: 141-178 , June 01, 2015

The climate change dispute is about changes over time of environmental characteristics (such as rainfall). Some people say that a possible change is not so much in the mean but rather in the extreme phenomena (that is, the average rainfall may not change much but heavy storms may become more or less frequent). The paper studies changes over time in the probability that some high threshold is exceeded. The model is such that the threshold does not need to be specified, the results hold for any high threshold. For simplicity a certain linear trend is studied depending on one real parameter. Estimation and testing procedures (is there a trend?) are developed. Simulation results are presented. The method is applied to trends in heavy rainfall at 18 gauging stations across Germany and The Netherlands. A tentative conclusion is that the trend seems to depend on whether or not a station is close to the sea.

## A weak law of large numbers for maxima

### Extremes (2011-09-01) 14: 325-341 , September 01, 2011

A weak law of large numbers related to the classical Gnedenko results for maxima (see Gnedenko, Ann Math 44:423–453, 1943) is established.

## Extreme value analysis for the sample autocovariance matrices of heavy-tailed multivariate time series

### Extremes (2016-09-01) 19: 517-547 , September 01, 2016

We provide some asymptotic theory for the largest eigenvalues of a sample covariance matrix of a *p*-dimensional time series where the dimension *p* = *p*_{n} converges to infinity when the sample size *n* increases. We give a short overview of the literature on the topic both in the light- and heavy-tailed cases when the data have finite (infinite) fourth moment, respectively. Our main focus is on the heavy-tailed case. In this case, one has a theory for the point process of the normalized eigenvalues of the sample covariance matrix in the iid case but also when rows and columns of the data are linearly dependent. We provide limit results for the weak convergence of these point processes to Poisson or cluster Poisson processes. Based on this convergence we can also derive the limit laws of various function als of the ordered eigenvalues such as the joint convergence of a finite number of the largest order statistics, the joint limit law of the largest eigenvalue and the trace, limit laws for successive ratios of ordered eigenvalues, etc. We also develop some limit theory for the singular values of the sample autocovariance matrices and their sums of squares. The theory is illustrated for simulated data and for the components of the S&P 500 stock index.

## Multivariate extremes and the aggregation of dependent risks: examples and counter-examples

### Extremes (2009-06-01) 12: 107-127 , June 01, 2009

Properties of risk measures for extreme risks have become an important topic of research. In the present paper we discuss sub- and superadditivity of quantile based risk measures and show how multivariate extreme value theory yields the ideal modeling environment. Numerous examples and counter-examples highlight the applicability of the main results obtained.

## Extremal dependence measure and extremogram: the regularly varying case

### Extremes (2012-06-01) 15: 231-256 , June 01, 2012

The dependence of large values in a stochastic process is an important topic in risk, insurance and finance. The idea of risk contagion is based on the idea of large value dependence. The Gaussian copula notoriously fails to capture this phenomenon. Two notions in a process or vector context which summarize extremal dependence in a function comparable to a correlation function are the *extremal dependence measure* (EDM) and the *extremogram*. We review these ideas and compare the two tools and end with a central limit theorem for a natural estimator of the EDM which allows drawing confidence bands comparable to those provided by Bartlett’s formula in a classical context of sample correlation functions.