This thesis is focused on the models based on extreme value theory and their practical applications. Specifically are described the block maxima models and the models based on threshold exceedances. Both of these methods are described in thesis theoretically. Apart from theoretical description there are also practical calculations based on simulated or real data. The applications of block maxima models are focused on choice of block size, suitability of the models for specific data and possibilities of extreme data analysis. The applications of models based on threshold exceedances are focused on choice of threshold and on suitability of the models. There is an example of the model used for calculations of reinsurance premium for extreme claims in the case of nonproportional reinsurance.

In the present thesis we will firstly familiarize ourselves with the term of operational risk, it's definition presented in the directives Basel II and Solvency II, and afterwards with the methods of calculation Capital Requirements for Operational Risk, set by these directives. In the second part of the thesis we will concentrate on the methods of modelling operational loss data. We will introduce the Extreme Value Theory which describes possible approaches to modelling data with significant values that occur infrequently; the typical characteristic of operational risk data. We will mainly focus on the model for threshold exceedances which utilizes Generalized Pareto Distribution to model the distribution of those excesses. The teoretical knowledge of this theory and the appropriate modelling will be applied on simulated loss data. Finally we will test the ability of presented methods to model loss data distributions.

[pt] Nesta dissertação avalia-se o grau de associação entre pares de excessos de retornos, simultâneos e defasados no tempo, usando-se o conceito de cópulas. Cópulas assimétricas são ajustadas aos pares de distribuições de retornos e coeficientes de dependência de cauda, as medidas de interdependência e contágio baseadas nessas cópulas, são calculados para 10 pares de índices de mercados. Tais coeficientes balizam a escolha do par de ativos com melhor desempenho em períodos de estresse. Se excessos defasados são incluídos, então estes coeficientes também indicam a direção e intensidade de propagação das crises (contágio). Os resultados encontrados na nossa investigação mostram que a técnica utilizada é eficaz na montagem de carteiras em que se pretende aproveitar os ganhos extremos conjuntos dos ativos e, ao mesmo tempo, evitar perdas extremas conjuntas. O uso de retornos defasados, porém, foi um artifício pouco producente, refletindo possivelmente o contágio quase instantâneo entre os mercados financeiros mundiais, nos dias de hoje. / [en] In this dissertation we evaluate the degree of association between pairs of excess of returns, simultaneous and lagged, using the concept of copulas. Asymmetric copulas are fitted to 10 pairs of distributions of returns of world markets índices. From these copulas coefficients of tail dependence are obtained for the right and left tails. Isong those coefficients as measures of cross dependence and contagion between markets one can pick the pair of returns that show the best performance in periods of stress. If lagged excess of returns are included, then these coefficients provide information on the direction and intensity of the contagion spread. Our results have shown that such technique isd efficent in constructing a portfolio in which one wants to take advantage of joint extreme gains of pairs of returns and, simultaneously, avoid losses associated with the occurrence of joint negative extremes. The use of lagged returns in this context hás shown no extra gain, maybe reflecting the fact that, nowadays, the spread of contagion between world financial markets is almost instantaneous.

Urbanisation alters the hydrologic response of a catchment, resulting in increased runoff rates and volumes, and loss of infiltration and base flow. Quantification of uncertainties is important in hydrologic designs of urban infrastructure. Major sources of uncertainty in the Intensity Duration Frequency (IDF) relationships are due to insufficient quantity and quality of data leading to parameter uncertainty and, in the case of projections of future IDF relationships under climate change, uncertainty arising from use of multiple General Circulation Models (GCMs) and scenarios. The work presented in the thesis presents methodologies to quantify the uncertainties arising from parameters of the distribution fitted to data and the multiple GCMs using a Bayesian approach. High uncertainties in GEV parameters and return levels are observed at shorter durations for Bangalore City. Twenty six GCMs from the CMIP5 datasets, along with four RCP scenarios are considered for studying the effects of climate change. It is observed that the uncertainty in short duration rainfall return levels is high when compared to the longer durations. Further it is observed that parameter uncertainty is large compared to the model uncertainty. Disaggregation of precipitation extremes from larger time scales to smaller time scales when the extremes are modeled with the GPD is burdened with difficulties arising from varying thresholds for different durations. In this study, the scale invariance theory is used to develop a disaggregation model for precipitation extremes exceeding specified thresholds. A scaling relationship is developed for a range of thresholds obtained from a set of quantiles of non-zero precipitation of different durations. The disaggregation model is applied to precipitation datasets of Berlin City, Germany and Bangalore City, India. From both the applications, it is observed that the uncertainty in the scaling exponent has a considerable effect on uncertainty in scaled parameters and return levels of shorter durations. A Bayesian hierarchical model is used to obtain spatial distribution of return levels of precipitation extremes in urban areas and quantify the associated uncertainty. Applicability of the methodology is demonstrated with data from 19 telemetric rain gauge stations in Bangalore City, India. For this case study, it is inferred that the elevation and mean monsoon precipitation are the predominant covariates for annual maximum precipitation. For the monsoon maximum precipitation, it is observed that the geographic covariates dominate while for the summer maximum precipitation, elevation and mean summer precipitation are the predominant covariates. In this work, variation in the dependence structure of extreme precipitation within an urban area and its surrounding non-urban areas at various durations is studied. The Berlin City, Germany, with surrounding non-urban area is considered to demonstrate the methodology. For this case study, the hourly precipitation shows independence within the city even at small distances, whereas the daily precipitation shows a high degree of dependence. This dependence structure of the daily precipitation gets masked as more and more surrounding non-urban areas are included in the analysis. The geographical covariates are seen to be predominant within the city and the climatological covariates prevail when non-urban areas are added. These results suggest the importance of quantification of dependence structure of spatial precipitation at the sub-daily timescales, as well as the need to more precisely model spatial extremes within the urban areas. The work presented in this thesis thus contributes to quantification of uncertainty in precipitation extremes through developing methodologies for generating probabilistic future IDF relationships under climate change, spatial mapping of probabilistic return levels and modeling dependence structure of extreme precipitation in urban areas at fine resolutions.

Made available in DSpace on 2014-12-17T15:26:38Z (GMT). No. of bitstreams: 1 DanielMC_DISSERT.pdf: 1549569 bytes, checksum: 5ad46f43cc6bf2e74f6fc1e20e5e2dc5 (MD5) Previous issue date: 2010-05-31 / Conselho Nacional de Desenvolvimento Cient?fico e Tecnol?gico / Present day weather forecast models usually cannot provide realistic descriptions of local and particulary extreme weather conditions. However, for lead times of about a small number of days, they provide reliable forecast of the atmospheric circulation that encompasses the subscale processes leading to extremes. Hence, forecasts of extreme events can only be achieved through a combination of dynamical and statistical analysis methods, where a stable and significant statistical model based on prior physical reasoning establishes posterior statistical-dynamical model between the local extremes and the large scale circulation. Here we present the development and application of such a statistical model calibration on the besis of extreme value theory, in order to derive probabilistic forecast for extreme local temperature. The dowscaling applies to NCEP/NCAR re-analysis, in order to derive estimates of daily temperature at Brazilian northeastern region weather stations / Os dados de rean?lise de temperatura do ar e precipita??o do NCEP National Centers for Environmental Predictions ser?o refinados para a produ??o dos n?veis de retorno para eventos extremos nas 8 capitais do Nordeste Brasileiro - NB: S?o Luis, Teresina, Fortaleza, Natal, Jo?o Pessoa, Recife, Macei?, Aracaju e Salvador. A grade do Ncep possui resolu??o espacial de 2.5? x 2.5? disponibilizando s?ries hist?ricas de 1948 a atualidade. Com esta resolu??o a grade envolve o NB utilizando 72 localiza??es (s?ries). A primeira etapa consiste em ajustar os modelos da Distribui??o Generalizada de Valores Extremos (GEV) e da Distribui??o Generalizada de Pareto (GPD) para cada ponto da grade. Utilizando o m?todo Geoestat?stico denominado Krigagem, os par?metros da GEV e GPD ser?o interpolados espacialmente. Considerando a interpola??o espacial dos par?metros, os n?veis de retorno para extremos de temperatura do ar e precipita??o poder?o ser obtidos aonde o NCEP n?o fornece informa??o relevante. Visando validar os resultados desta proposta, ser?o ajustados os modelos GEV e GPD as s?ries observacionais di?rias de temperatura e precipita??o de cada capital nordestina, e assim comparar com os resultados obtidos a partir da interpola??o espacial. Por fim o m?todo de Regress?o Quant?lica ser? utilizado como m?todo mais tradicional com a finalidade de compara??o de m?todos.

The thesis is focused on extreme value theory and its applications. Initially, extreme value distribution is introduced and its properties are discussed. At this basis are described two models mostly used for an extreme value analysis, i.e. the block maxima model and the Pareto-distribution threshold model. The first one takes advantage in its robustness, however recently the threshold model is mostly preferred. Although the threshold choice strongly affects estimation quality of the model, an optimal threshold selection still belongs to unsolved issues of this approach. Therefore, the thesis is focused on techniques for proper threshold identification, mainly on adaptive methods suitable for the use in practice. For this purpose a simulation study was performed and acquired knowledge was applied for analysis of precipitation records from South-Moravian region. Further on, the thesis also deals with extreme value estimation within a stationary series framework. Usually, an observed time series needs to be separated to obtain approximately independent observations. The use of the advanced theory for stationary series allows to avoid the entire separation procedure. In this context the commonly applied separation techniques turn out to be quite inappropriate in most cases and the estimates based on theory of stationary series are obtained with better precision.

The US sub-prime mortgage crisis greatly affected not only the US economy but also other countries in the world. This thesis employs the extreme value theory and Value at Risk (VaR) analysis to assess the impact of the US sub-prime mortgage crisis on various stock markets of the MSCI indexes, including 10 countries and 7 areas. It is reasonable to guess that VaR value should increase after the crisis. The empirical analyses on these indexes conclude that (1) the American market indexes not only do not agree with the guess after the crisis but four American indexes are identical; (2) not all the Asia market indexes consist with the guess; (3) the European market indexes agree with the guess; (4) MSCI AC PACIFIC, NEW ZEALAND, and AUSTRALIA consist with the guess; (5) the behavior for the positive log returns is different from that for the negative returns in some MSCI indexes. Over speaking, the impacts of US sub-prime mortgage crisis on those countries are not the same.

Les précipitations et les débits des cours d'eau constituent les deux variables hydrométéorologiques les plus importantes pour l'analyse des bassins versants. Ils fournissent des informations fondamentales pour la gestion intégrée des ressources en eau, telles que l’approvisionnement en eau potable, l'hydroélectricité, les prévisions d'inondations ou de sécheresses ou les systèmes d'irrigation.Dans cette thèse de doctorat sont abordés deux problèmes distincts. Le premier prend sa source dans l’étude des débits des cours d’eau. Dans le but de bien caractériser le comportement global d'un bassin versant, de longues séries temporelles de débit couvrant plusieurs dizaines d'années sont nécessaires. Cependant les données manquantes constatées dans les séries représentent une perte d'information et de fiabilité, et peuvent entraîner une interprétation erronée des caractéristiques statistiques des données. La méthode que nous proposons pour aborder le problème de l'imputation des débits se base sur des modèles de régression dynamique (DRM), plus spécifiquement, une régression linéaire multiple couplée à une modélisation des résidus de type ARIMA. Contrairement aux études antérieures portant sur l'inclusion de variables explicatives multiples ou la modélisation des résidus à partir d'une régression linéaire simple, l'utilisation des DRMs permet de prendre en compte les deux aspects. Nous appliquons cette méthode pour reconstruire les données journalières de débit à huit stations situées dans le bassin versant de la Durance (France), sur une période de 107 ans. En appliquant la méthode proposée, nous parvenons à reconstituer les débits sans utiliser d'autres variables explicatives. Nous comparons les résultats de notre modèle avec ceux obtenus à partir d'un modèle complexe basé sur les analogues et la modélisation hydrologique et d'une approche basée sur le plus proche voisin. Dans la majorité des cas, les DRMs montrent une meilleure performance lors de la reconstitution de périodes de données manquantes de tailles différentes, dans certains cas pouvant allant jusqu'à 20 ans.Le deuxième problème que nous considérons dans cette thèse concerne la modélisation statistique des quantités de précipitations. La recherche dans ce domaine est actuellement très active car la distribution des précipitations exhibe une queue supérieure lourde et, au début de cette thèse, il n'existait aucune méthode satisfaisante permettant de modéliser toute la gamme des précipitations. Récemment, une nouvelle classe de distribution paramétrique, appelée distribution généralisée de Pareto étendue (EGPD), a été développée dans ce but. Cette distribution exhibe une meilleure performance, mais elle manque de flexibilité pour modéliser la partie centrale de la distribution. Dans le but d’améliorer la flexibilité, nous développons, deux nouveaux modèles reposant sur des méthodes semiparamétriques.Le premier estimateur développé transforme d'abord les données avec la distribution cumulative EGPD puis estime la densité des données transformées en appliquant un estimateur nonparamétrique par noyau. Nous comparons les résultats de la méthode proposée avec ceux obtenus en appliquant la distribution EGPD paramétrique sur plusieurs simulations, ainsi que sur deux séries de précipitations au sud-est de la France. Les résultats montrent que la méthode proposée se comporte mieux que l'EGPD, l’erreur absolue moyenne intégrée (MIAE) de la densité étant dans tous les cas presque deux fois inférieure.Le deuxième modèle considère une distribution EGPD semiparamétrique basée sur les polynômes de Bernstein. Plus précisément, nous utilisons un mélange creuse de densités béta. De même, nous comparons nos résultats avec ceux obtenus par la distribution EGPD paramétrique sur des jeux de données simulés et réels. Comme précédemment, le MIAE de la densité est considérablement réduit, cet effet étant encore plus évident à mesure que la taille de l'échantillon augmente. / Precipitation and streamflow are the two most important meteorological and hydrological variables when analyzing river watersheds. They provide fundamental insights for water resources management, design, or planning, such as urban water supplies, hydropower, forecast of flood or droughts events, or irrigation systems for agriculture.In this PhD thesis we approach two different problems. The first one originates from the study of observed streamflow data. In order to properly characterize the overall behavior of a watershed, long datasets spanning tens of years are needed. However, the quality of the measurement dataset decreases the further we go back in time, and blocks of data of different lengths are missing from the dataset. These missing intervals represent a loss of information and can cause erroneous summary data interpretation or unreliable scientific analysis.The method that we propose for approaching the problem of streamflow imputation is based on dynamic regression models (DRMs), more specifically, a multiple linear regression with ARIMA residual modeling. Unlike previous studies that address either the inclusion of multiple explanatory variables or the modeling of the residuals from a simple linear regression, the use of DRMs allows to take into account both aspects. We apply this method for reconstructing the data of eight stations situated in the Durance watershed in the south-east of France, each containing daily streamflow measurements over a period of 107 years. By applying the proposed method, we manage to reconstruct the data without making use of additional variables, like other models require. We compare the results of our model with the ones obtained from a complex approach based on analogs coupled to a hydrological model and a nearest-neighbor approach, respectively. In the majority of cases, DRMs show an increased performance when reconstructing missing values blocks of various lengths, in some of the cases ranging up to 20 years.The second problem that we approach in this PhD thesis addresses the statistical modeling of precipitation amounts. The research area regarding this topic is currently very active as the distribution of precipitation is a heavy-tailed one, and at the moment, there is no general method for modeling the entire range of data with high performance. Recently, in order to propose a method that models the full-range precipitation amounts, a new class of distribution called extended generalized Pareto distribution (EGPD) was introduced, specifically with focus on the EGPD models based on parametric families. These models provide an improved performance when compared to previously proposed distributions, however, they lack flexibility in modeling the bulk of the distribution. We want to improve, through, this aspect by proposing in the second part of the thesis, two new models relying on semiparametric methods.The first method that we develop is the transformed kernel estimator based on the EGPD transformation. That is, we propose an estimator obtained by, first, transforming the data with the EGPD cdf, and then, estimating the density of the transformed data by applying a nonparametric kernel density estimator. We compare the results of the proposed method with the ones obtained by applying EGPD on several simulated scenarios, as well as on two precipitation datasets from south-east of France. The results show that the proposed method behaves better than parametric EGPD, the MIAE of the density being in all the cases almost twice as small.A second approach consists of a new model from the general EGPD class, i.e., we consider a semiparametric EGPD based on Bernstein polynomials, more specifically, we use a sparse mixture of beta densities. Once again, we compare our results with the ones obtained by EGPD on both simulated and real datasets. As before, the MIAE of the density is considerably reduced, this effect being even more obvious as the sample size increases.

Made available in DSpace on 2017-05-12T14:46:53Z (GMT). No. of bitstreams: 1 Wagner.pdf: 5111902 bytes, checksum: b4edf3498cca6f9c7e2a9dbde6e62e18 (MD5) Previous issue date: 2013-12-07 / The purpose of hydrologic frequency analysis is to relate magnitude of events with their occurrence frequency based on probability distribution. The generalized probability distributions can be used on the study concerning extreme hydrological events: extreme events, logistics and Pareto. There are several methodologies to estimate probability distributions parameters, however, L-moments are often used due to computational easiness. Reliability of quantiles with high return period can be increased by LH-moments or high orders L-moments. L-moments have been widely studied; however, there is little information about LH-moments on literature, thus, there is a great research requirement on such area. Therefore, in this study, LH-moments were studied under two approaches commonly used in hydrology: (i) local frequency analysis (LFA) and (ii) regional frequency analysis (RFA). Moreover, a database with 227 rainfall stations was set (daily maximum annual), in Paraná State, from 1976 to 2006. LFA was subdivided into two steps: (i) Monte Carlo simulations and (ii) application of results to database. The main result of Monte Carlo simulations was that LH-moments make 0.99 and 0.995 quantiles less biased. Besides, simulations helped on creating an algorithm to perform LFA by generalized distributions. The algorithm was applied to database and enabled an adjustment of 227 studied series. In RFA, the 227stations have been divided into 11 groups and regional growth curves were obtained; while local quantiles were obtained from the regional growth curves. The difference between local quantiles obtained by RFA was quantified with those obtained via LFA. The differences may be approximately 33 mm for return periods of 100 years. / O objetivo da análise de frequência das variáveis hidrológicas é relacionar a magnitude dos eventos com sua frequência de ocorrência por meio do uso de uma distribuição de probabilidade. No estudo de eventos hidrológicos extremos, podem ser usadas as distribuições de probabilidade generalizadas: de eventos extremos, logística e Pareto. Existem diversas metodologias para a estimativa dos parâmetros das distribuições de probabilidade, no entanto, devido às facilidades computacionais, utilizam-se frequentemente os momentos-L. A confiabilidade dos quantis com alto período de retorno pode ser aumentada utilizando os momentos-LH ou momentos-L de altas ordens. Os momentos-L foram amplamente estudados, todavia, os momentos-LH apresentam literatura reduzida, logo, mais pesquisas são necessárias. Portanto, neste estudo, os momentos-LH foram estudados sob duas abordagens comumente utilizadas na hidrologia: (i) Análise de frequência local (AFL) e (ii) Análise de frequência regional (AFR). Além disso, foi montado um banco de dados com 227 estações pluviométricas (máximas diárias anuais), localizadas no Estado do Paraná, no período de 1976 a 2006. A AFL subdividiu-se em duas etapas: (i) Simulações de Monte Carlo e (ii) Aplicação dos resultados ao banco de dados. O principal resultado das simulações de Monte Carlo foi que os momentos-LH tornam os quantis 0,99 e 0,995 menos enviesados. Além disso, as simulações viabilizaram a criação de um algoritmo para realizar a AFL utilizando as distribuições generalizadas. O algoritmo foi aplicado ao banco de dados e possibilitou ajuste das 227 séries estudadas. Na AFR, as 227 estações foram dividas em 11 grupos e foram obtidas as curvas de crescimento regional. Os quantis locais foram obtidos a partir das curvas de crescimento regional. Foi quantificada a diferença entre os quantis locais obtidos via AFL com aqueles obtidos via AFR. As diferenças podem ser de aproximadamente 33 mm para períodos de retorno de 100 anos.

許多觀察到的時間序列資料，多呈現高峰厚尾（leptokurtic）的現象，本文引用時間序列資料為Paretian分配之假設，估計各個國家股價指數報酬率於不同頻率資料下之最大級數動差，以觀察其厚尾程度。實證結果發現，各個國家指數報酬率於不同頻率資料下之四級以上動差大部分存在，且不隨資料之頻率不同，而有不同的表現。由此可推論，各個國家股價指數報酬率之歷史分配，其離群值之活動並不嚴重。接著，利用樣本分割預測檢定（Sample Split Prediction Test）來檢定所觀察各個國家股價指數報酬率於同一樣本期間內，其左右尾之厚尾程度是否一致，及檢定所觀察各個國家指數報酬率於跨期間左尾或右尾之厚尾程度是否穩定。在同一樣本期間，檢定時間序列之左右尾之厚尾程度是否一致之檢定中，發現各個國家指數報酬率在所觀察樣本期間內，其左右尾之厚尾程度大致相同；而在跨期間之樣本分割預測檢定中，發現各個國家指數報酬率在像是1987年10月美國股市大崩盤、1990年至1991年間之波斯灣戰爭、1997年亞洲金融風暴等事件前後，其左（右）尾之厚尾程度有顯著差異。最後提出Cusum of Squares檢定，係用於檢定一時間序列資料在所觀察之樣本期間內，其非條件變異數是否為一常數。 Cusum of Squares檢定之檢定結果顯示，本文之各個國家指數報酬率在所觀察之樣本期間內，其非條件變異數並非為一常數。進一步觀察各個國家指數報酬率之Cusum of Squares圖，並綜合前述跨期間樣本分割預測檢定之結果，可推論在處理較長樣本期間之時間序列資料可能遇到結構性變動之情況時，跨期間之樣本分割預測檢定及Cusum of Squares檢定可提供結構性變動可能發生之時點。