# Extreme TiDE Seminar

## 25 November 2019 (at Delft University of Technology)

### 15:00-15:45 Yi He (University of Amsterdam)

Title: Rethinking Extreme Value Statistics

Abstract: I discuss three papers and invite rethinking of the fundamental designs in extreme value statistics. First, the tail dependence inference formulas for a diverging threshold is often redundant with that for an adaptive threshold, and the latter is more relevant in practice. Second, the univariate peaks-over-threshold method is no better than fitting a generalized Pareto model beyond an unknown finite threshold where naive bootstrapping is valid with a flexible choice of threshold. Finally, we show how nonrandom heterogeneous data generate heavy tails and propose novel modeling techniques for high dimensional datasets.

### 16:00-17:00 Anne Sabourin (Télécom Paris, Institut polytechnique de Paris)

Title: Nonasymptotic analysis of the empirical angular measure for multivariate extremes, with applications to classification and minimum volume set estimation

Abstract: In multivariate extreme value theory, the angular measure characterizes the first order dependence structure of multivariate heavy-tailed variables. In the case where the components have different tail indices, standardization using the rank transformation (empirical distribution function) is a common practice. We provide a nonasymptotic bound for the uniform deviations of an the empirical angular measure evaluated on rectangles of the unit sphere. Our bound scales as the squared root of the number of observations used for inference log(k)/√k up to a logarithmic factor. This nonasymptotic study is, to the best of our knowledge, the first of its kind in this domain. In addition we propose a modification of the classical empirical estimator based on the rank-transformed sample, based on intermediate data, i.e. upon data which norm rank among the largest of the observed sample, but not among the very largest. In other word we discard the very largest data. Our error bound for this modified estimator does not suffer from a logarithmic factor, but includes a multiplicative term depending on the truncation level. The relative merits of both versions of the empirical measure are illustrated by numerical experiments. As an application, we provide finite sample guarantees for classification in extreme regions and anomaly detection via minimum-volume sets estimation on the sphere. This is a joint work with Stéphan Clémençon, Hamid Jalalzai and Johan Segers

## 28 February 2019 (at Erasmus University Rotterdam)

### 15:00-15:45 Kate Saunders (TU Delft)

Title: A regionalisation approach for rainfall based on extremal dependence

Abstract: To mitigate the risk posed by extreme rainfall events, we require statistical models that reliably capture extremes in continuous space with dependence. However, assuming a stationary dependence structure in such models is often erroneous, particularly over large geographical domains. Furthermore, there are limitations on the ability to fit existing models, such as max-stable processes, to a large number of locations. To address these modelling challenges, we present a regionalisation method that partitions stations into regions of similar extremal dependence using clustering. To demonstrate our regionalisation approach, we consider a study region of Australia and discuss the results with respect to known climate and topographic features. To visualise and evaluate the effectiveness of the partitioning, we also fit max-stable models to the sub-regions. This work serves as a prelude to how one might consider undertaking a project where spatial dependence is non-stationary and is modelled on a continental scale.

### 16:00-17:00 Paul Embrechts (ETH Zürich)

Title: Risk-sharing, Robustness and Regulation

Abstract: In this talk I will summarize some recent work on the concept of risk-sharing, discuss in particular properties like robustness, moral hazard and equilibrium pricing and finally highlight consequences for capital regulation for the financial and insurance industry.

## 12 December 2018 (at Tilburg University)

### 15:00-15:45 Cees de Valk (KNMI)

Title: Towards estimation of the 10 million year wind speed

Abstract: Assessment of the reliability of flood protection in the Netherlands requires quantiles of wind speed for return periods of up to 10 million years. This is a major challenge, given that records of reliable wind measurements do not go back further than about 70 years. At KNMI, we are simultaneously working on different ideas for making this feasible. One idea is the utilisation of large datasets generated by numerical weather prediction models. However, this leaves a considerable gap in return period to overcome. Therefore, we also explore the use of models of the tail which are specifically designed for extrapolation over a wide range of return periods. Two large sets of ECMWF seasonal ensemble forecast wind data each representing more than 5000 years of data were used to check and compare estimates of the 10-million year wind speed based on different models of the tail.
A related issue is the variation in time of estimates of the tail of wind speed as inferred from measurements at Schiphol. A fluctuation on long time-scales is observed, which appears to be related to the variation in mean wind speed. Consequences for the estimation of high quantiles are discussed

### 16:00-17:00 Kirstin Strokorb (Cardiff Univeristy)

Title: The geometry of tail dependence

Abstract: The stable tail dependence function (stdf; Huang 1992, Drees and Huang 1998) is a well-known dependence function in multivariate extreme value analysis that appears naturally in different contexts (e.g. as part of MGEVs or MGPDs). Ressel (2013) gave the first complete set of conditions that a function has to fulfill in order to be a stdf. In this talk I will show how such conditions can be reinterpreted geometrically and in a spatial context and how this interpretation leads to new insights and connections between extreme value theory, stochastic geometry and the theory of risk measures.

The seminar will take place at room S8, Simon Building, Tilburg University. Full address: Warandelaan 2, 5037 AB Tilburg.

## 08 October 2018 (at Delft University of Technology)

### 15:00-15:45 Hanna Ahmed (Tilburg University)

Title: Improved estimation of the extreme value index using a related variables

Abstract: Heavy tailed phenomena are naturally analyzed by extreme value statistics. A crucial step in such an analysis is the estimation of the extreme value index, which describes the tail heaviness of the underlying probability distribution. We consider the situation where we have next to the n observations of interest another n + m observations of one or more related variables, like, e.g., financial losses due to earthquakes and the related amounts of energy released, for a longer period than that of the losses. Based on such a data set, we present an adapted version of the Hill estimator that shows greatly improved behavior and we establish the asymptotic normality of this estimator. For this adaptation the tail dependence between the variable of interest and the related variable(s) plays an important role. A simulation study confirms the substantially improved performance of our adapted estimator relative to the Hill estimator. We also present an application to the aforementioned earthquake losses.

This is a joint work with John H.J. Einmahl.

### 16:00-17:00 Clément Dombry(University of Franche-Comté)

Title: Analysis of the proportional tail model for extreme quantile regression via a coupling approachÂ

Abstract Extreme quantile regression is a fundamental problem in extreme value theory. Assume that we observe an $n$-sample $(x_1,y_1),\ldots,(x_n,y_n)$ of a random variable $Y\in\mathbb{R}$ together with covariates $X\in\mathbb{R}^p$. Our goal is to estimate the conditional quantile of order $1-p$ of $Y$ given $X=x$. When $p$ is small, there is not enough observations and extrapolation further in the tail distribution is needed. We face an extreme value problem.

The purpose of the talk is to present an ongoing joint work with B.Bobbia and D.Varron on the proportional tail model where we assume that the conditional tails are asymptotically proportional to the unconditional tail, that is
$P(Y>y\mid X=x)\sim \sigma(x)P(Y>y)$ as $y\to y^*$, the upper endpoint of the distribution. This framework was introduced in the slightly different context of heteroscedastic extremes in Einmahl et al. (JRSSB 2016) and the function $\sigma$ was coined the skedasis function. Assuming an extreme value condition for $Y$ together with the proportional tail model, the extreme quantile regression is reduced to the estimation of the skedasis function and the extreme value index. We present our results for such an estimation. Interestingly, we introduce different techniques for the proof as Einmahl et al.: we introduce coupling arguments relying on total variation and Wasserstein distances, whereas the original proof relies mostly on empirical process theory.

## 06 June 2018 (at Tilburg University)

### 15:00-15:45 Xuan Leng (Erasmus University Rotterdam)

Title: Bias correction for the maximum likelihood estimator of the extreme value index

Abstract: This paper conducts bias correction for the maximum likelihood estimator (MLE) of the extreme value index. Compared to the original MLE, the bias corrected estimator allows for using a larger fraction of observations in tail region for estimation, which results in a lower asymptotic variance. The bias correction is achieved by subtracting the asymptotic bias from the original MLE, which is estimated by a two-step approach. We prove the asymptotic behavior of the proposed bias-corrected estimator. Extensive simulations show the superiority of the bias-corrected estimator compared to existing estimators of the extreme value index. We apply the bias-corrected MLE to test whether human life span is unlimited.

### 16:00-17:00 Philippe Naveau (CNRS-France)

Title:Analysis of extreme climate events by combining multivariate extreme values theory and causality theory

Abstract: Multiple changes in Earthâ€™s climate system have been observed over the past decades. Determining how likely each of these changes are to have been caused by human influence, is important for decision making on mitigation and adaptation policy. This is particularly true for extreme events, e.g. the 2003 European heatwave. To quantity these issues, we combine causal counterfactual theory (Pearl 2000) with multivariate extreme value theory. In particular, we take advantage of recent advances in the modeling of the multivariate generalized Pareto distributions to propose a conceptual framework to deal with climate-related events attribution.

(Joint work with Anna Kiriliouk and Alexis Hannart and Julien Worms).

## 22 March 2018 (at Erasmus University of Rotterdam)

### Pasquale Cirillo (Delft University of Technology)

Title: Risk concentration and the inequality of tail

Abstract: Refurbishing the well-known Gini index as a measure of tail risk, I discuss a brand new set of inequality-based tools for the study of fat tails. As a side result, I will also deal with the estimation of the Gini index in the case of infinite variance, showing why its commonly used nonparametric estimator should be avoided under extremely fat tails. I will naturally explain the main theoretical aspects and properties, but I will also discuss heuristics and applications on interesting actual data, not so commonly found in the literature, related to war casualties, terrorism, op losses and bit coins.

### Axel Bücher (Ruhr-University Bochum)

Title: On a pseudo-maximum likelihood estimator for the extremal index

Abstract: The extremes of a stationary time series typically occur in clusters. A primary measure for this phenomenon is the extremal index, representing the reciprocal of the expected cluster size. Both a disjoint and a sliding blocks estimator for the extremal index, essentially due to Northrop (2015) [An efficient semiparametric maxima estimator of the extremal index. Extremes 18, 585â€“603], are analyzed in detail. In contrast to many competitors, the estimators only depend on the choice of one parameter sequence. We derive an asymptotic expansion, prove asymptotic normality and show consistency of an estimator for the asymptotic variance. Explicit calculations in certain models and a finite-sample Monte Carlo simulation study reveal that the sliding blocks estimator outperforms other blocks estimators, and that it is competitive to runs- and inter-exceedance estimators in various models. The methods are applied to a variety of financial time series.

## 11 December 2017 (at Delft University of Technology)

### Jan Beirlant (Catholic University of Leuven)

Title: Bias reduced estimation of the extreme value index

Abstract: A lot of attention has been paid to bias reduced estimation of the extreme value index in case of heavy-tailed distributions. In this talk we present some proposals for all max-domains of attraction. A first method is based on ridge regression for generalized quantiles. Secondly we discuss the use of Bernstein polynomials for estimating the bias in the Peaks over Threshold method.

### Chen Zhou (De Nederlandsche Bank and Erasmus University of Rotterdam)

Title: Trends in extreme value indices

Abstract: We consider extreme value analysis for independent but non-identically distributed observations. In particular, the observations do not share the same extreme value index. This situation is related to, but differs from, heteroscedastic extremes in Einmahl et al. (2016). Compared to the heteroscedastic extremes, our model allows for a broader class in which tails of the probability distributions of different observations are of different order. In other words, we are dealing with distributions that differ much more than the heteroscedastic extremes. Assuming continuously changing extreme value indices, we provide a non-parametric estimate for the functional extreme value index. Besides estimating the extreme value index locally, we also provide a global estimator for the trend and its joint asymptotic property. The global asymptotic property can be used for testing a pre-specified parametric trend in the extreme value indices. In particular, it can be applied to test whether the extreme value index remains at a constant level across all observations.