Title: Stochastical inference in the model of extreme events Author: Jan Dienstbier Department/Institute: Department of probability and mathematical statistics Supervisor of the doctoral thesis: Doc. RNDr. Jan Picek, CSc. Abstract: The thesis deals with extremal aspects of linear models. We provide a brief explanation of extreme value theory. The attention is then turned to linear models Yn×1 = Xn×pβp×1 + En×1 with the errors Ei ∼ F, i = 1, . . . , n fulfilling the do- main of attraction condition. We examine the properties of the regression quantiles of Koenker and Basset (1978) under this setting we develop theory dealing with extremal characteristics of linear models. Our methods are based on an approximation of the regression quantile process for α ∈ [0, 1] expanding older results of Gutenbrunner et al. (1993). Our result holds in [α∗ n, 1 − α∗ n] with a better rate of α∗ n → 0 than the other approximations described previously in the literature. Consecutively we provide an ap- proximation of the tails of regression quantile. The approximations of the tails enable to develop theory of the smooth functionals, which are used to establish a new class of estimates of extreme value index. We prove T(F−1 n (1 − knt/n)) is consistent and asymp- totically normal estimate of extreme for any T member of the class....

In dieser Arbeit wurde erstmalig die Anwendung der Extremwerttheorie auf Wartezeiten eines lichtsignalgesteuerten Verkehrsknotenpunktes untersucht. Anhand der Verkehrsstärken eines realen Knotenpunktes wurden mit der Simulationssoftware PTV Vissim 100 Datensätze mit individuellen Wartezeiten erzeugt. Als Referenz wurde eine zweite Simulationsreihe durchgeführt. Diese erfolgte mit 15 % höherer Verkehrsstärke. Mittels der Blockmaximum-Methode wurden aus den erzeugten Datensätzen die Maxima ausgewählt, welche mit der Maximum-Likelihood Methode an eine Extremwertverteilung angepasst wurden. Die Bewertung der Schätzung wurde mit dem Kolmogorov-Smirnov Test vorgenommen. Anschließend wurde die Wahrscheinlichkeit, dass bestimmte Wartezeiten überschritten werden (Value at Risk) berechnet. Im Ergebnis konnten 22 % der geschätzten Extremwertverteilungen mit ausreichender Güte angepasst werden. Für die restlichen Datensätze sollte nach Alternativen zur angemessenen Beschreibung gesucht werden.

[pt] A partir da década de 90, a metodologia Value at Risk (VaR) se difundiu pelo mundo, tanto em instituições financeiras quanto em não financeiras, como uma boa prática de mensuração de riscos. Em geral, abordagens paramétricas são muito utilizadas pelo mercado, apesar de freqüentemente não levarem em conta uma característica muito encontrada nas distribuições dos retornos de ativos financeiros: a presença de caudas pesadas. Uma abordagem baseada na Teoria dos Valores Extremos (TVE) é uma boa solução quando se deseja modelar caudas de distribuições probabilísticas que possuem tal característica. Em contra partida, poucos são os trabalhos que procuram desenvolver a TVE aplicada a ativos de renda-fixa. Com base nisto, este estudo propõe uma abordagem de simples implementação de cálculo de VaR para ativos de renda-fixa baseado na Teoria dos Valores Extremos. / [en] Since the 90 decade, the use of Value at Risk (VaR) methodology has been disseminated among both financial and non-financial institutions around the world, as a good practice in terms of risks management. In spite of the fact that it does not take into account one of the most important characteristics of financial assets returns distribution - fat tails (excess of kurtosis), the parametric approach is the most used method for Value at Risk measurement. The Extreme Value Theory (EVT) is an alternative method that could be used to avoid the underestimation of Value at Risk, properly modeling the characteristics of probability distribution tails. However, there are few works that applied EVT to fixed-income market. Based on that, this study implements a simple approach to VaR calculation, in which the Extreme Value Theory is applied to fixed-income assets.

[pt] Os métodos clássicos para estudo de valores extremos de séries temporais se apóiam nas chamadas distribuições de extremos. Uma alternativa é o método P.O.T. (Peaks Over Threshold), desenvolvido por hidrologistas, o qual estuda apenas os valores da série que excedem um dado patamar. Esses procedimentos são baseados em hipóteses restritivas. Nesse trabalho desenvolvemos modelos sobre extremos que podem ser utilizados em situações mais gerais. Eles são essencialmente modelos lineares dinâmicos com inferência Bayesiana, nos quais as observações têm um distribuição de extremos. Embora essas distribuições não sejam da famí­lia exponencial, toda a análise é feita explicitamente, sem aproximações numéricas. Tratamos ainda da construção de distribuições a priori não informáticas. Finalmente, a partir desses modelos retomamos problemas clássicos de previsão de extremos. / [en] The classical approaches for extreme values studies make use the so called Extreme Values Distribution. An alternative approach, known as P.O.T. (Peaks Over Threshold) developed by hydrologists considers only excedances over a given threshold value. All the existing approaches are in a sense, based on constrained hupothesis. In this thesis we developed forecasting models for extreme values that are dynamic linear model as the underlying formulation, and the Bayesian inference. Although the process observation follows an extreme values distribution and, therefore not a member of the exponential family, we were able to formulate explicitly the model with no use of numerical approximations throughout, Concerning the parameter priors, we use in the model formulation the Jeffery`s non informative prior.

This diploma thesis studies extreme value theory and its application in finan- cial risk management, when focusing on computation of well-known risk measure - Value at Risk (VaR). The first part of the thesis reviews theoretical background. In particular, it rigorously discusses the extreme value theory when emphasi- zing fundamentals theorems and their consequences followed by the summary of methods based on this theory, specifically, Block Maxima method, Hill met- hod and Peaks over Threshold method. Moreover, specific issues that may arise in such applications and ways how to deal with these problems are described. The second part of the thesis contains extensive empirical study, which together with theoretical foundings applies each of the examined method to real market data of the closing prices of Dow Jones Industrial Average stock index, stocks of JPMorgan and stock index Russell 2000 in order to compare methods based on extreme value theory together with the classic methodology RiskMetrics. 1

(None) Technical Summary of Thesis: The topic of my thesis is currency crisis. Currency crises have been a recurrent feature of the international economy from the invention of paper money. They are not confined to particular economies or specific region. They take place in developed, emerging, and developing countries and are spread all over the globe. Countries that experience currency crises face economic losses that can be huge and disruptive. However, the exacted toll is not only financial and economic, but also human, social, and political. It is clear that the currency crisis is a real threat to financial stability and economic prosperity. The main objective of this thesis is to analyze the determinants of currency crises for twenty OECD countries and South Africa from 1970 through 1998. It systematically examines the role of economic fundamentals and contagion in the origins of currency crises and empirically attempts to identify the channels through which the crises are being transmitted. It also examines the links between the incidence of currency crises and the choice of exchange rate regimes as well as the impact of capital market liberalization policies on the occurrence of currency crises. The first chapter identifies the episodes of currency crisis in our data set. Determining true crisis periods is a vital step in the empirical studies and has direct impact on the reliability of their estimations and the relevant policy implications. We define a period as a crisis episode when the Exchange Market Pressure (EMP) index, which consists of changes in exchange rates, reserves, and interest rates, exceeds a threshold. In order to minimize the concerns regarding the accuracy of identified crisis episodes, we apply extreme value theory, which is a more objective approach compared to other methods. In this chapter, we also select the reference country, which a country’s currency pressure index should be built around, in a more systematic way rather than by arbitrary choice or descriptive reasoning. The second chapter studies the probability of a currency exiting a tranquil state into a crisis state. There is an extensive literature on currency crises that empirically evaluate the roots and causes of the crises. Despite the interesting results of the current empirical literature, only very few of them account for the influence of time on the probability of crises. We use duration models that rigorously incorporate the time factor into the likelihood functions and allow us to investigate how the amount of time that a currency has already spent in the tranquil state affects the stability of a currency. Our findings show that high values of volatility of unemployment rates, inflation rates, contagion factors (which mostly work through trade channels), unemployment rates, real effective exchange rate, trade openness, and size of economy increases the hazard of a crisis. We make use of several robustness checks, including running our models on two different crisis episodes sets that are identified based on monthly and quarterly type spells. The third chapter examines the links between the incidence of currency crises and the choice of exchange rate regimes as well as the impact of capital market liberalization policies on the occurrence of currency crises. As in our previous paper, duration analysis is our methodology to study the probability of a currency crisis occurrence under different exchange rate regimes and capital mobility policies. The third chapter finds that there is a significant link between the choice of exchange rate regime and the incidence of currency crises in our sample. Nevertheless, the results are sensitive to the choice of the de facto exchange rate system. Moreover, in our sample, capital control policies appear to be helpful in preventing low duration currency crises. The results are robust to a wide variety of sample and models checks.

Depuis le début de la décennie 2000, le marché du maïs connaît un changement profond. D'une part, le prix enregistre une volatilité extrême sans précédent. D'autre part, ce marché bénéficie d'un déferlement massif des investisseurs financiers. Il offre des opportunités d'investissements financiers rentables en raison des crises récurrentes sur le marché boursier. Il est intéressant pour des investisseurs (spéculateurs, fondamentalistes, arbitragistes) d'avoir connaissance des résultats d'analyse des variations extrêmes du prix du marché du maïs. La maîtrise des variations extrêmes du prix permet une meilleure gestion du risque. Des études ont déjà été menées dans cette direction en utilisant des techniques du type « VaR ». Cependant, les différents modèles de gestion du risque par la VaR souffrent de certaines limites. Ils supposent l'hypothèse de la normalité des distributions. Or, la distribution des rendements du maïs montre des valeurs extrêmes. Cela ne permet pas une bonne appréciation du risque. Afin de contribuer à l'analyse des variations extrêmes de prix sur le marché du maïs, nous faisons appel aux modèles GARCH et à la théorie des valeurs extrêmes. Puis, dans un cadre multi-varié, le lien entre rendements spots et futures exprime le degré de la dépendance. Il permet ainsi d'analyser l'effet de la spéculation. Pour cela, nous utilisons la théorie des valeurs extrêmes couplée à la mesure de la dépendance qu'on appelle « copule » pour cerner les mouvements extrêmes des variations du prix au delà d'un seuil. En effet, la théorie des copules propose toute une gamme de fonctions capable de mesurer la dépendance asymétrique aux queues de la distribution des rendements spots et futures du maïs. / Since the early 2000s, the corn market is undergoing a profound change. On the one hand, the price has experienced unprecedented extreme volatility. Moreover, this market has a massive outpouring of financial investors. The corn market offers profitable financial investments due to recurrent crises in the stock market opportunities. It is interesting for investors (speculators, fundamentalists, arbitrageurs) to be aware of the analysis of extreme price changes in corn results. The mastery of extreme price changes provides better risk management. Studies have already been conducted in this direction by using techniques such as "VaR". However, the different models of risk management VaR suffer from certain limitations. They assume the assumption of normality of distributions. However, the distribution of return corn shows extreme values. This does not allow a proper assessment of risk. To contribute to the analysis of extreme price changes in the corn market, we use the GARCH models and the theory of extreme values. Then, in a multi-varied context, the link between returns and future spots expresses the degree of dependence. It allows analyzing the effect of speculation. We use extreme value theory coupled to the measure of dependence called "copula" to identify extreme movements of price changes beyond a threshold. Indeed, copula theory offers a range of features that can measure the asymmetric dependence tails of the distribution of spot return and futures of corn.

Connaître la probabilité d'occurrence des aléas océano-météorologiques extrêmes est fondamental pour prévenir les risques de submersion marine en zone côtière ou concevoir des aménagements côtiers, portuaires ou des plate-formes offshore. Notamment, le concept de niveau de retour est fréquemment utilisé en ingénierie côtière pour dimensionner des ouvrages de protection. Ces niveaux, dont les périodes de retour d'intérêt se situent généralement entre 100 et 1000 ans, sont habituellement estimés par une analyse statistique locale, à partir de données observées en un site unique. Cependant, la période d'observation est généralement limitée, de sorte que les incertitudes associées aux niveaux de retour élevés sont importantes. L'analyse régionale représente une solution possible pour réduire les incertitudes inhérentes aux analyses locales. Le principe est d'exploiter l'information de sites d'observation provenant d'une région homogène, où les extrêmes sont supposés avoir un comportement probabiliste similaire. L'analyse régionale peut ainsi estimer les niveaux de retour de manière plus fiable qu'une analyse locale. Cependant, son application dans le domaine maritime étant relativement limitée et récente, différentes questions méthodologiques de meurent non-Résolues, comme la formation des régions homogènes ou le traitement de la dépendance entre sites. L'objectif scientifique de la thèse est donc d'approfondir certains points méthodologiques de l'analyse régionale, dans le cadre des aléas maritimes extrêmes. Les points suivants sont abordés en particulier :• Échantillonnage des extrêmes pour l'analyse régionale, à partir des tempêtes détectées via une procédure de declustering spatio-Temporel.• Formation de régions homogènes à partir d'une méthode basée sur l'identification des empreintes typiques des tempêtes.• Prise en compte de la dépendance entre sites d'observation, à travers la construction d'un modèle permettant par exemple d'évaluer la durée effective régionale d'observation ou la période de retour régionale d'une tempête.• Spécification et estimation de la loi régionale, avec incorporation des co-variables influentes, comme la saison d'occurrence ou la direction de provenance pour les vagues.• Comparaison entre analyses locale et régionale, notamment à travers les incertitudes sur les estimations des extrêmes et la capacité à modéliser les horsains présumés.Ces aspects sont illustrés sur des données de hauteurs significatives de vagues et de surcotes de pleine mer, dans la zone Atlantique Nord-Est, Manche et Mer du Nord.Parallèlement, l'objectif applicatif de ces travaux est de contribuer à garantir la sûreté des ouvrages EDF contre le risque de submersion marine. Ceci peut être réalisé grâce à l'exploration de nouvelles techniques d'estimation des aléas maritimes extrêmes telles que l'analyse régionale, qui permet notamment une meilleure prise en compte des horsains. / The knowledge of the probability of occurrence of oceano-Meteorological extremes is essential to prevent risks of coastal flooding or to build coastal protections or off-Shore structures. In particular, the concept of return level is frequently used in coastal engineering to design protection structures. These levels, whose return periods of interest generally lie between 100 and 1000 years, are usually estimated by a local statistical analysis, from data observed at a unique site. However, the period of observation is generally limited, which can imply high uncertainties for high return levels. Regional frequency analysis is a possible solution to reduce uncertainties inherent to local analyses. The principle is to exploit the information of sites of observation from a homogeneous region, where extremes are supposed to share a similar probabilistic behavior. Thus, regional frequency analysis can estimate return levels more accurately than a local analysis. However, its application to the marine field being relatively limited and recent, several methodological questions are still unsolved, such as the formation of homogeneous regions or the dependence between sites. The scientific objective of this thesis is thus to develop some methodological points of regional frequency analysis, in the framework of extreme marine hazards. The following questions are tackled:• Sampling of extremes for regional analysis, from the storms detected through a spatiotemporal declustering procedure.• Formation of homogeneous regions from a method based on the identification of the typical storms footprints.• Consideration of the dependence between sites of observation, through the building of a model allowing, for example, to assess the regional effective duration or the regional return period of a storm.• Specification and estimation of the regional distribution, with the incorporation of influent covariables, such as the season of occurrence or the direction for waves.• Comparison between regional and local analyses, especially through the uncertainties on the estimated extremes and the ability to model the potential outliers. These aspects are illustrated on significant wave height data and skew surge data located in the Northeast Atlantic, the Eastern Channel and the North Sea. At the same time, the industrial objective of this work is to contribute to guarantee the safety of EDF structures against the risk of coastal flooding. This can be achieved through the exploration of new techniques of estimation of extreme marine hazards such as regional frequency analysis, which allows in particular a better representation of outliers

The thesis deals with natural disasters from the statistical point of view and treats them as extremal observations. Basics of classical extreme value theory will be summarized and new approach based on maximum entropy principle will be proposed. Both methods will be used in order to analyze real discharge data observed at the river Vltava.

The thesis deals with extreme value distributions and censored samples. Theoretical part describes a maximum likelihood method, types of censored samples and introduce a extreme value distributions. In the thesis are derived likelihood equations for censored samples from exponential, Weibull, lognormal, Gumbel and generalized extreme value distribution. For these distributions are also derived asymptotic interval estimates and is made simulation studies on the dependence of the parameter estimate on the percentage of censoring.