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Introduction

## Publications

Publications (149)

New Vapnik–Chervonenkis type concentration inequalities are derived for the empirical distribution of an independent random sample. Focus is on the maximal deviation over classes of Borel sets within a low probability region. The constants are explicit, enabling numerical comparisons.

Multivariate extreme value distributions are a common choice for modelling multivariate extremes. In high dimensions, however, the construction of flexible and parsimonious models is challenging. We propose to combine bivariate extreme value distributions into a Markov random field with respect to a tree. Although in general not an extreme value di...

Motivated by examples from extreme value theory we introduce the general notion of a cluster process as a limiting point process of returns of a certain event in a time series. We explore general invariance properties of cluster processes which are implied by stationarity of the underlying time series under minimal assumptions. Of particular intere...

Driven by several successful applications such as in stochastic gradient descent or in Bayesian computation, control variates have become a major tool for Monte Carlo integration. However, standard methods do not allow the distribution of the particles to evolve during the algorithm, as is the case in sequential simulation methods. Within the stand...

To quantify the dependence between two random vectors of possibly different dimensions, we propose to rely on the properties of the 2-Wasserstein distance. We first propose two coefficients that are based on the Wasserstein distance between the actual distribution and a reference distribution with independent components. The coefficients are normal...

Extreme U-statistics arise when the kernel of a U-statistic has a high degree but depends only on its arguments through a small number of top order statistics. As the kernel degree of the U-statistic grows to infinity with the sample size, estimators built out of such statistics form an intermediate family in between those constructed in the block...

We study the joint occurrence of large values of a Markov random field or undirected graphical model associated to a block graph. On such graphs, containing trees as special cases, we aim to generalize recent results for extremes of Markov trees. Every pair of nodes in a block graph is connected by a unique shortest path. These paths are shown to d...

New Vapnik and Chervonkis type concentration inequalities are derived for the empirical distribution of an independent random sample. Focus is on the maximal deviation over classes of Borel sets within a low probability region. The constants are explicit, enabling numerical comparisons.

A Markov tree is a probabilistic graphical model for a random vector indexed by the nodes of an undirected tree encoding conditional independence relations between variables. One possible limit distribution of partial maxima of samples from such a Markov tree is a max-stable Hüsler–Reiss distribution whose parameter matrix inherits its structure fr...

Monte Carlo integration with variance reduction by means of control variates can be implemented by the ordinary least squares estimator for the intercept in a multiple linear regression model with the integrand as response and the control variates as covariates. Even without special knowledge on the integrand, significant efficiency gains can be ob...

The empirical beta copula is a simple but effective smoother of the empirical copula. Because it is a genuine copula, from which it is particularly easy to sample, it is reasonable to expect that resampling procedures based on the empirical beta copula are expedient and accurate. In this paper, after reviewing the literature on some bootstrap appro...

To quantify the dependence between two random vectors of possibly different dimensions, we propose to rely on the properties of the 2-Wasserstein distance. We first propose two coefficients that are based on the Wasserstein distance between the actual distribution and a reference distribution with independent components. The coefficients are normal...

The angular measure on the unit sphere characterizes the first-order dependence structure of the components of a random vector in extreme regions and is defined in terms of standardized margins. Its statistical recovery is an important step in learning problems involving observations far away from the center. In the common situation when the compon...

The assumption that the elements of the cost matrix in the classical assignment problem are drawn independently from a standard Gaussian distribution motivates the study of a particular Gaussian field indexed by the symmetric permutation group. The correlation structure of the field is determined by the Hamming distance between two permutations. Th...

A Markov tree is a probabilistic graphical model for a random vector indexed by the nodes of an undirected tree encoding conditional independence relations between variables. One possible limit distribution of partial maxima of samples from such a
Markov tree is a max-stable Hüsler–Reiss distribution whose parameter matrix inherits its structure fr...

Consider the problem of learning a large number of response functions simultaneously based on the same input variables. The training data consist of a single independent random sample of the input variables drawn from a common distribution together with the associated responses. The input variables are mapped into a high-dimensional linear space, c...

Goodness-of-fit tests based on the empirical Wasserstein distance are proposed for simple and composite null hypotheses involving general multivariate distributions. This includes the important problem of testing for multivariate normality with unspecified mean vector and covariance matrix and, more generally, testing for elliptical symmetry with g...

For multivariate distributions in the domain of attraction of a max-stable distribution, the tail copula and the stable tail dependence function are equivalent ways to capture the dependence in the upper tail. The empirical versions of these functions are rank-based estimators whose inflated estimation errors are known to converge weakly to a Gauss...

Describing the complex dependence structure of extreme phenomena is particularly challenging. To tackle this issue we develop a novel statistical method that describes extremal dependence taking advantage of the inherent tree-based dependence structure of the max-stable nested logistic distribution, and that identifies possible clusters of extreme...

Monte Carlo integration with variance reduction by means of control variates can be implemented by the ordinary least squares estimator for the intercept in a multiple linear regression model with the integrand as response and the control variates as covariates. Even without special knowledge on the integrand, significant efficiency gains can be ob...

Identifying groups of variables that may be large simultaneously amounts to finding out which joint tail dependence coefficients of a multivariate distribution are positive. The asymptotic distribution of a vector of nonparametric, rank-based estimators of these coefficients justifies a stopping criterion in an algorithm that searches the collectio...

The empirical beta copula is a simple but effective smoother of the empirical copula. Because it is a genuine copula, from which, moreover, it is particularly easy to sample, it is reasonable to expect that resampling procedures based on the empirical beta copula are expedient and accurate. In this paper, after reviewing the literature on some boot...

A Markov tree is a random vector indexed by the nodes of a tree whose distribution is determined by the distributions of pairs of neighbouring variables and a list of conditional independence relations. Upon an assumption on the tails of the Markov kernels associated to these pairs, the conditional distribution of the self-normalized random vector...

The replacement of indicator functions by integrated beta kernels in the definition of the empirical stable tail dependence function is shown to produce a smoothed version of the latter estimator with the same asymptotic distribution but superior finite-sample performance. The link of the new estimator with the empirical beta copula enables a simpl...

We extend Robert McCann's treatment of the existence and uniqueness of an optimal transport map between two probability measures on a Euclidean space to a class of possibly infinite measures, finite outside neighbourhoods of the origin. For convergent sequences of pairs of such measures, we study the stability of the multivalued transport maps and...

Likelihood-based procedures are a common way to estimate tail dependence parameters. They are not applicable, however, in non-differentiable models such as those arising from recent max-linear structural equation models. Moreover, they can be hard to compute in higher dimensions. An adaptive weighted least-squares procedure matching nonparametric e...

The block maxima method in extreme-value analysis proceeds by fitting an extreme-value distribution to a sample of block maxima extracted from an observed stretch of a time series. The method is usually validated under two simplifying assumptions: the block maxima should be distributed exactly according to an extreme-value distribution and the samp...

To draw inference on serial extremal dependence within heavy-tailed Markov chains, Drees et al., (2015) proposed nonparametric estimators of the spectral tail process. The methodology can be extended to the more general setting of a stationary, regularly varying time series. The large-sample distribution of the estimators is derived via empirical p...

Multivariate peaks over thresholds modeling based on generalized Pareto distributions has up to now only been used in few and mostly low-dimensional situations. This paper contributes to the theoretical understanding, physically based models, inference tools, and simulation methods needed to support routine use, also in high dimensions. We derive a...

A recommender system based on ranks is proposed, where an expert's ranking of a set of objects and a user's ranking of a subset of those objects are combined to make a prediction of the user's ranking of all objects. The rankings are assumed to be induced by latent continuous variables corresponding to the grades assigned by the expert and the user...

Background
Due to a global warming-related increase in heatwaves, it is important to obtain detailed understanding of the relationship between heat and health. We assessed the relationship between heat and urgent emergency room admissions in the Netherlands. Methods
We collected daily maximum temperature and relative humidity data over the period 2...

The use of control variates is a well-known variance reduction technique in Monte Carlo integration. If the optimal linear combination of control variates is estimated by ordinary least squares and if the number of control variates is allowed to grow to infinity, the convergence rate can be accelerated, the new rate depending on the interplay betwe...

The vanilla method in univariate extreme-value theory consists of fitting the three-parameter Generalized Extreme-Value (GEV) distribution to a sample of block maxima. Despite claims to the contrary, the asymptotic normality of the maximum likelihood estimator has never been established. In this paper, a formal proof is given using a general result...

There exist two ways of defining regular variation of a time series in a star-shaped metric space: either by the distributions of finite stretches of the series or by viewing the whole series as a single random element in a sequence space. The two definitions are shown to be equivalent. The introduction of a norm-like function, called radius, yield...

The block maxima method in extreme value theory consists of fitting an extreme value distribution to a sample of block maxima extracted from a time series. Traditionally, the maxima are taken over disjoint blocks of observations. Alternatively, the blocks can be chosen to slide through the observation period, yielding a larger number of overlapping...

Describing the complex dependence structure of multivariate extremes is particularly challenging and requires very versatile, yet interpretable, models. To tackle this issue we explore two related approaches: clustering and dimension reduction. In particular, we develop a novel statistical algorithm that takes advantage of the inherent hierarchical...

Multivariate generalized Pareto distributions arise as the limit distributions of exceedances over multivariate thresholds of random vectors in the domain of attraction of a max-stable distribution. These distributions can be parametrized and represented in a number of different ways. Moreover, generalized Pareto distributions enjoy a number of int...

The empirical copula has proved to be useful in the construction and understanding of many statistical procedures related to dependence within random vectors. The empirical beta copula is a smoothed version of the empirical copula that enjoys better finite-sample properties. At the core lie fundamental results on the weak convergence of the empiric...

Let $L_t$ be the longest gap before time $t$ in an inhomogeneous Poisson process with rate function $\lambda_t$ proportional to $t^{\alpha-1}$ for some $\alpha\in(0,1)$. It is shown that $\lambda_tL_t-b_t$ has a limiting Gumbel distribution for suitable constants $b_t$ and that the distance of this longest gap from $t$ is asymptotically of the form...

The multivariate generalized Pareto distribution arises as the limit of a suitably normalized vector conditioned upon at least one component of that vector being extreme. Statistical modelling using multivariate generalized Pareto distributions constitutes the multivariate analogue of peaks over thresholds modelling with the univariate generalized...

Given a sample from a multivariate distribution $F$, the uniform random variates generated independently and rearranged in the order specified by the componentwise ranks of the original sample look like a sample from the copula of $F$. This idea can be regarded as a variant on Baker's [J. Multivariate Anal. 99 (2008) 2312--2327] copula construction...

To draw inference on serial extremal dependence within heavy-tailed Markov chains, Drees, Segers and Warcho\l{} [Extremes (2015) 18, 369--402] proposed nonparametric estimators of the spectral tail process. The methodology can be extended to the more general setting of a stationary, regularly varying time series. The large-sample distribution of th...

In the field of spatial extremes, stochastic processes with upper semicontinuous (usc) trajectories have been proposed as random shape functions for max-stable models. In the literature dealing with usc processes, max-stability is defined via a sequences of scaling constants, rather than functions, only. It is however not clear whether and how extr...

The vanilla method in univariate extreme-value theory consists of fitting the
three-parameter Generalized Extreme-Value (GEV) distribution to a sample of
block maxima. Despite claims to the contrary, the asymptotic normality of the
maximum likelihood estimator has never been established. In this paper, a
formal proof is given using a general result...

Likelihood-based procedures are a common way to estimate tail dependence
parameters. They are not applicable, however, in non-differentiable models such
as those arising from recent max-linear structural equation models. Moreover,
they can be hard to compute in higher dimensions. An adaptive weighted
least-squares procedure matching nonparametric e...

There is an increasing interest to understand the dependence structure of a random vector not only in the center of its distribution but also in the tails. Extreme-value theory tackles the problem of modelling the joint tail of a multivariate distribution by modelling the marginal distributions and the dependence structure separately. For estimatin...

The block maxima method in extreme-value analysis proceeds by fitting an
extreme-value distribution to a sample of block maxima extracted from an
observed stretch of a time series. The method is usually validated under two
simplifying assumptions: the block maxima should be distributed according to an
extreme-value distribution and the sample of bl...

When the copula of the conditional distribution of two random variables given
a covariate does not depend on the value of the covariate, two conflicting
intuitions arise about the best possible rate of convergence attainable by
nonparametric estimators of that copula. In the end, any such estimator must be
based on the marginal conditional distribu...

Individual risk models need to capture possible correlations as failing to do
so typically results in an underestimation of extreme quantiles of the
aggregate loss. Such dependence modelling is particularly important for
managing credit risk, for instance, where joint defaults are a major cause of
concern. Often, the dependence between the individu...

There is an increasing interest to understand the dependence structure of a
random vector not only in the center of its distribution but also in the tails.
Extreme-value theory tackles the problem of modelling the joint tail of a
multivariate distribution by modelling the marginal distributions and the
dependence structure separately. For estimatin...

Classical and more recent tests for detecting distributional changes in multivariate time series often lack power against alternatives that involve changes in the cross-sectional dependence structure. To be able to detect such changes better, a test is introduced based on a recently studied variant of the sequential empirical copula process. In con...

In the past decades, weak convergence theory for stochastic processes has become a standard tool for analyzing the asymptotic properties of various statistics. Routinely, weak convergence is considered in the space of bounded functions equipped with the supremum metric. However, there are cases when weak convergence in those spaces fails to hold. E...

The mean absolute deviation about the mean is an alternative to the standard
deviation for measuring dispersion in a sample or in a population. For
stationary, ergodic time series with a finite first moment, an asymptotic
expansion for the sample mean absolute deviation is proposed. The expansion
yields the asymptotic distribution of the sample mea...

At high levels, the asymptotic distribution of a stationary, regularly
varying Markov chain is conveniently given by its tail process. The latter
takes the form of a geometric random walk, the increment distribution depending
on the sign of the process at the current state and on the flow of time, either
forward or backward. Estimation of the tail...

A variant of the empirical copula is considered which combines an estimator
of a multivariate cumulative distribution function with estimators of its
margins that are not necessarily equal to the margins of the estimator of the
joint distribution. Such a hybrid estimator may be reasonable when there is
additional information available for some marg...

Tail dependence models for distributions attracted to a max-stable law are
fitted using observations above a high threshold. To cope with spatial,
high-dimensional data, a rank-based M-estimator is proposed relying on
bivariate margins only. A data-driven weight matrix is used to minimize the
asymptotic variance. Empirical process arguments show th...

Measures of association are suggested between two random vectors. The measures are copula-based and therefore invariant with respect to the univariate marginal distributions. The measures are able to capture positive as well as negative association. In case the random vectors are just random variables, the measures reduce to Kendall’s tau or Spearm...

The core of the classical block maxima method consists of fitting an extreme
value distribution to a sample of maxima over blocks extracted from an
underlying series. In asymptotic theory, it is usually postulated that the
block maxima are an independent random sample of an extreme value distribution.
In practice however, block sizes are finite, so...

For multivariate Gaussian copula models with unknown margins and structured
correlation matrices, a rank-based, semiparametrically efficient estimator is
proposed for the Euclidean copula parameter. This estimator is defined as a
one-step update of a rank-based pilot estimator in the direction of the
efficient influence function, which is calculate...

The extremes of a univariate Markov chain with regularly varying stationary marginal distribution and asymptotically linear behavior are known to exhibit a multiplicative random walk structure called the tail chain. In this paper we extend this fact to Markov chains with multivariate regularly varying marginal distributions in
R
d
. We analyze bo...

One of the features inherent in nested Archimedean copulas, also called
hierarchical Archimedean copulas, is their rooted tree structure. A
nonparametric, rank-based method to estimate this structure is presented. The
idea is to represent the target structure as a set of trivariate structures,
each of which can be estimated individually with ease....

Discussion of "Statistical Modeling of Spatial Extremes" by A. C. Davison, S.
A. Padoan and M. Ribatet [arXiv:1208.3378].

The spectral measure plays a key role in the statistical modeling of
multivariate extremes. Estimation of the spectral measure is a complex issue,
given the need to obey a certain moment condition. We propose a Euclidean
likelihood-based estimator for the spectral measure which is simple and
explicitly defined, with its expression being free of Lag...

Multivariate extreme-value analysis is concerned with the extremes in a
multivariate random sample, that is, points of which at least some components
have exceptionally large values. Mathematical theory suggests the use of
max-stable models for univariate and multivariate extremes. A comprehensive
account is given of the various ways in which max-s...

Many interesting processes share the property of multivariate regular variation. This property is equivalent to existence of the tail process introduced by B. Basrak and J. Segers [1] to describe the asymptotic behavior for the extreme values of a regularly varying time series. We apply this theory to multivariate MA(∞) processes with random coeffi...

A pair-copula construction is a decomposition of a multivariate copula into a
structured system, called regular vine, of bivariate copulae or pair-copulae.
The standard practice is to model these pair-copulae parametrically, which
comes at the cost of a large model risk, with errors propagating throughout the
vine structure. The empirical pair-copu...

Consider a random sample in the max-domain of attraction of a multivariate
extreme value distribution such that the dependence structure of the attractor
belongs to a parametric model. A new estimator for the unknown parameter is
defined as the value that minimizes the distance between a vector of weighted
integrals of the tail dependence function...

Starting from the characterization of extreme-value copulas based on max-stability, large-sample tests of extreme-value dependence for multivariate copulas are studied. The two key ingredients of the proposed tests are the empirical copula of the data and a multiplier technique for obtaining approximate p-values for the derived statistics. The asym...

This paper suggests five measures of association between two random vectors X
= (X_1, ..., X_p) and Y = (Y_1, ..., Y_q). They are copula based and therefore
invariant with respect to the marginal distributions of the components X_i and
Y_j. The measures capture positive as well as negative association of X and Y.
In case p = q = 1 they reduce to Sp...

Extreme-value copulas arise in the asymptotic theory for componentwise maxima
of independent random samples. An extreme-value copula is determined by its
Pickands dependence function, which is a function on the unit simplex subject
to certain shape constraints that arise from an integral transform of an
underlying measure called spectral measure. M...

The tail of a bivariate distribution function in the domain of attraction of a bivariate extreme value distribution may be approximated by that of its extreme value attractor. The extreme value attractor has margins that belong to a three-parameter family and a dependence structure which is characterized by a probability measure on the unit inter...

Inference on an extreme-value copula usually proceeds via its Pickands dependence function, which is a convex function on the unit simplex satisfying certain inequality constraints. In the setting of an i.i.d. random sample from a multivariate distribution with known margins and an unknown extreme-value copula, an extension of the Capéraà–Fougères–...

Correlation mixtures of elliptical copulas arise when the correlation parameter is driven itself by a latent random process. For such copulas, both penultimate and asymptotic tail dependence are much larger than for ordinary elliptical copulas with the same unconditional correlation. Furthermore, for Gaussian and Student t-copulas, tail dependence...

Weak convergence of the empirical copula process is shown to hold under the
assumption that the first-order partial derivatives of the copula exist and are
continuous on certain subsets of the unit hypercube. The assumption is
non-restrictive in the sense that it is needed anyway to ensure that the
candidate limiting process exists and has continuo...

Conditions are given under which the empirical copula process associated with a random sample from a bivariate continuous distribution has a smaller asymptotic covariance function than the standard empirical process based on observations from the copula. Illustrations are provided and consequences for inference are outlined.

We investigate the connections between extremal indices on the one hand and stability of Markov chains on the other hand. Both theories relate to the tail behaviour of stochastic processes, and we find a close link between the extremal index and geometric ergodicity. Our results are illustrated throughout with examples from simple MCMC chains.

The quantification of diversification benefits due to risk aggregation plays a prominent role in the (regulatory) capital management of large firms within the financial industry. However, the complexity of today’s risk landscape makes a quantifiable reduction of risk concentration a challenging task. In the present paper we discuss some of the issu...

When a spatial process is recorded over time and the observation at a given time instant is viewed as a point in a function space, the result is a time series taking values in a Banach space. To study the spatio-temporal extremal dynamics of such a time series, the latter is assumed to be jointly regularly varying. This assumption is shown to be eq...

Under an appropriate regular variation condition, the affinely normalized
partial sums of a sequence of independent and identically distributed random
variables converges weakly to a non-Gaussian stable random variable. A
functional version of this is known to be true as well, the limit process being
a stable L\'{e}vy process. The main result in th...

Being the limits of copulas of componentwise maxima in independent random samples, extreme-value copulas can be considered
to provide appropriate models for the dependence structure between rare events. Extreme-value copulas not only arise naturally
in the domain of extreme-value theory, they can also be a convenient choice to model general positiv...

Under an appropriate regular variation condition, the affinely normalized partial sums of a sequence of independent and identically distributed random variables converges weakly to a non-Gaussian stable random variable. A functional version of this is known to be true as well, the limit process being a stable L\'evy process. The main result in the...

The tail of a bivariate distribution function in the domain of attraction of
a bivariate extreme-value distribution may be approximated by the one of its
extreme-value attractor. The extreme-value attractor has margins that belong to
a three-parameter family and a dependence structure which is characterised by a
spectral measure, that is a probabil...

A complete and user-friendly directory of tails of Archimedean copulas is presented which can be used in the selection and construction of appropriate models with desired properties. The results are synthesized in the form of a decision tree: Given the values of some readily computable characteristics of the Archimedean generator, the upper and low...

Modelling excesses over a high threshold using the Pareto or generalized Pareto distribution (PD/GPD) is the most popular approach in extreme value statistics. This method typically requires high thresholds in order for the (G)PD to fit well and in such a case applies only to a small upper fraction of the data. The extension of the (G)PD proposed i...