The clustering of events can have a large impact on society. The extremal index $\theta$ tells how much extreme events cluster. We will compare different types of estimators in this project. First, we review the extremes of different sequences which have different values of $\theta$. We have found significant differences between the extremes. Then, 2 different types of estimators are introduced which both use different ways to divide the data, using disjoint blocks and using sliding blocks. The optimal block lengths are simulated for all those estimators. Using those block lengths, $\theta$ is simulated with all the estimators. From the simulations we conclude that the estimators using sliding blocks perform better. The best-performed estimator that we found from the simulations is used to estimate $\theta$ on data from the KNMI, comparing wind gusts and precipitation at weather stations De Bilt and Vlissingen.

The primary goal of this report is to provide a general overview of offline change-point literature as it is known today. Change-point methods are important statistical problems, where we are interested in determining whenever a certain data-set changes in structure. Furthermore, the term "off-line" is meant to indicate that the data itself is already known, whereas on the other hand we have "on-line" methods which deal with situations where new data is yet being received during localisation of the change-points. In this report we mainly consider off-line methods, as we feel off-line methods provide a more friendly introduction into change-point analysis and on-line methods are in principle just an extension of their off-line counterparts. First off, these off-line change-point methods are considered under different assumptions (parametric, non-parametric). In each case, we treat a solution to the change-point problem under different models(normal and gamma model, mean or varianche change etc.). Eventually we shall also treat some widely used algorithms, meant to extend the problem into the localisation of multiple change-points.indent Aside from theoretical considerations, an equally important part of this report will be focused on empirical results. Both for the statistics as algorithms will there be a performance study where the different methods will be empirically assessed and compared under different models, namely the robustness against "outliers"(extreme data-values) will be investigated. So to complement the primary goal, we will also focus on the following two subgoals:
1) Assessment and comparison of different change-point models
2) Evaluating and improving robustness against outliers

The most common way to evaluate traffic safety is investigating the occurrence and severity of crashes using historical data. This approach however has a number of limitations, the most important of which is probably its reactive nature. An alternative method using non-crash events has gained a lot of attention recently, especially thanks to the rapid improvement of sensing technologies. By gathering trajectory data and calculating various Surrogate Measures of Safety it has become possible to analyse safety without waiting for accidents to happen. Using these indicators combined with Extreme Value Theory (EVT) one can estimate the probability of crashes as extreme (unobserved) events. The primary goal of this thesis is to contribute to the research that has been done so far on the application of Extreme Value Theory to Surrogate Measures for traffic safety analysis. Research questions seek for answers to what we can learn from applying univariate EVT using indicators describing collision course and crossing course interactions, and how we can predict nearness to collision and severity using bivariate EVT models.

This thesis gathers, develops and evaluates several characterizations of multivariate tail dependence. It is established that the stable tail dependence function (STDF) is a suitable copula-based dependence function that fully captures the multivariate extremal dependence structure in all dimensions d≥2 and can be used to visualize the tail dependence structure for bivariate and trivariate problems. Based on the STDF, we propose a multivariate tail dependence coefficient (TDC) as an extension of the well-known bivariate TDC. Importantly, we show that the proposed measure can identify tail independence in all dimensions d≥2, similar to its bivariate variant. The performance of nonparametric estimators for the STDF and, inherently, the multivariate TDC, is assessed with an extensive simulation study, including smoothed and bias-corrected versions of the empirical STDF. Based on the estimators for the STDF and the multivariate TDC, test statistics under the null hypothesis of tail independence are developed and evaluated in another simulation study. The STDF-based estimation and testing procedures are applied to foreign exchange (FX) data to characterize the tail dependence structure between three European FX rates and five worldwide FX rates.

Three interest rate models are researched: Displaced Exponential-Vasicek, Hull-White one factor and Hull-White two factors with time-dependent volatility parameters. The motivation for this is two-fold: firstly, we would like to understand how the capital calculations would be impacted when yield curves are modelled under the three different models. This is done by looking at both magnitude and stability of the risk profiles and scalar risk-measures for three counterparties, which are highly representative for the bank. Secondly, we investigate the benefits and drawbacks of using one model and its corresponding calibration method over the others, with a special attention to the impact on yields correlations.

The first model, calibrated to historical data, is used as a nine-factors model for forward rates and is currently being used within the bank for PFE profiles and CVA regulatory capital. Historical backtest has proven the current model to perform reasonably well on real data and therefore it is used as a benchmark against which the other two models are tested. The two Hull-White models, used as short rate models, are calibrated to the risk-neutral measure (namely, to European swap- tions). Precisely, a two-steps calibration procedure suited for piece-wise constant volatility functions is implemented for both. The stability analysis reveals that the variation of Exposure at Defaults is significant, which might be undesired. On the other side, the two short rate models retain the correlation structure of interest rates better than the current model. This in turn translates into higher capital impact.

In deze scriptie is onderzocht of regressie, een postprocessing methode, een goede toevoeging is aan het model dat het Europees Centrum voor Weersverwachtingen op Middellange Termijn (ECMWF) ontwikkeld heeft en waarvan het KNMI de uitvoer gebruikt, specifiek om te bepalen of er minstens 5 mm neerslag op een dagdeel (00-12 UTC en 12-00 UTC) zal vallen. Hiervoor is gebruik gemaakt van logistische regressie. Voor het verifiëren van de resultaten is er gekeken naar de Brier-Scores en de reliability diagrammen van de regressiemodellen. Deze zijn vervolgens vergeleken met de resultaten van het ECMWF-model. De resultaten zijn vrijwel gelijk, maar de Brier-Scores van het originele model zijn net iets beter dan die van de regressiemodellen. De conclusie die na het onderzoek wordt getrokken is dat het gemaakte regressiemodel geen toevoeging heeft op het ECMWF-model.

Intraday liquidity risk is a subject that applies to all banks, and arises whenever there is a timing mismatch between incoming and outgoing payments within a business day. In case such a mismatch occurs, the bank is exposed to the risk that it is unable to meet its payment obligations at the time expected. A liquidity buffer could help to mitigate this risk.

This thesis presents a framework for intraday liquidity risk management within ABN AMRO Bank, while taking different priorities of transactions into account. We examine the use of extreme value theory (EVT) and propose two metrics to capture the risk: the univariate and multivariate risk metric. The univariate risk metric represents the size of the liquidity buffer for each priority group separately and provides granular view. Making use of a Monte Carlo simulation algorithm in combination with univariate EVT, we
are able to estimate the size of the liquidity buffer for a specified time interval within a business day. We forecast the buffer size 30 days out-of-sample and test the violations against the conditional coverage (CC) hypothesis. Satisfactory results are obtained for the
groups with high and moderate priority when the highest confidence levels are considered: $\alpha$ = 0.1 and 0.05. For the group with low priority, the risk metric performs well for the lowest confidence levels: $\alpha$ = 0.025 and 0.01. The multivariate risk metrics aggregates
the size of the liquidity buffer, while taking the diversification of the priority groups into account. We define a failure set and investigate the use of multivariate EVT.

@en