This thesis aims to improve the current risk classification for (company) car insurance at Achmea, focusing on WAM and ARD coverages. By using cluster analysis, specifically K-prototypes and spectral clustering, policyholders are grouped based on zip codes and license plates to enhance the current claim frequency models (and thus premium pricing models). The application of spectral clustering (with U-SPEC as the observation reduction technique) led to significant improvement of the current claim frequency GLM for the ARD coverage. This thesis highlights the potential of cluster analysis in actuarial science, offering new methods for mixed data types and high-dimensional clustering, thus providing a foundation for more accurate claim frequency models. The time stabilities and stabilities with respect to the number of observations of the clusters are also investigated.

Automated Market Makers (AMMs) are a novel type of market makers that eliminate the need for a coun-terparty in a trade. This thesis analyses the properties of several types of AMMs, and in particular the con-centrated liquidity market maker. An axiomatic definition of AMMs is provided, and two types of constant function market makers (CFMMs), called the constant sum market maker (CSMM) and constant product market maker (CPMM), are explored. The concentrated liquidity market maker, which improves liquidity provision compared to the CPMM, is thoroughly analyzed, and it is shown that it’s trading function can be formulated as a composition of functions. This thesis also conjectures that the concentrated liquidity trading function can be approximated by taking an infinite number of compositions. Additionally, a simulation study is conducted using transaction mocking. The simulation study supports the conjecture, and brings several other noteworthy properties of the concentrated liquidity market maker to light.

This thesis concerns modeling residential real estate selling prices in a hedonic price model framework on a small spatial-temporal granularity. The research addresses the challenge of sparse spatial-temporal real estate data, i.e. many combinations of location and time with few or no transactions, by employing spatial dynamic factor models (SDFMs). Two types of SDFMs are employed: an SDFM with a 1D spatial structure based on the spatial random walk and an SDFM with a 2D spatial structure based on the Gaussian random field. To capture the information on the property characteristics, spatial dynamic factor models are combined with two different data-driven models, namely a neural network (NN) and an interpretable version of an NN, the local generalized linear model network (LGLMN). Both a Bayesian approach and an algorithmic approach are employed to estimate the models on both a PC and a high-performance computer (HPC). A simulation study is conducted to demonstrate the ability of an NN to capture linear and non-linear structures when combined with an SDFM and to show the ability of the LGLMN to replicate a linear structure. Furthermore, the models are evaluated on real transaction data from the municipality of Rotterdam. The findings demonstrate that the algorithmically estimated NN-adjusted SDFM based on the spatial random walk (NN-SRW-DFM) outperforms the other models in terms of accuracy with an out-of-sample MAPE of 0.128. Moreover, the results highlight a trade-off between accuracy, speed, and interpretability.

This paper primarily employs Item Response Theory (IRT) to estimate item characteristics and the proficiency levels of students as reflected in the exam results. The process includes the application of algorithms for item characteristic parameter estimation and the utilization of statistical techniques for item selection from an item pool. Additionally, Classical Test Theory (CTT) is also used to gain insights into the item characteristics. The mathematical frameworks behind every algorithm and model will be introduced in detail. Our ultimate objective is to create an item bank that unifies all items from various exam versions onto a common scale. The technique to put items from different versions on the same scale is the test equating technique, and it will also be described in detail.

After a comprehensive analysis to determine the appropriate model within the framework of Item Response Theory (IRT), the Rasch model has been selected for all versions of the exams. One item from version 9 has been removed from the item pool due to its unsatisfactory item fit. Based on the Item-Map plot, the conclusion can be made that all three exam versions appear to be relatively easy for students to answer, as evidenced by the high bar of the exams. Ultimately, the item bank has been successfully established through the application of two test-equating methods, with results indicating its reliability.

American option pricing has been an active research area in financial engineering over the past few decades. Since no analytic closed-form solution exists, various numerical approaches have been developed. Among all proposed methods, the least square Monte Carlo(LSMC) approach is the most successful and popular. The LSMC utilizes linear regression for the estimation of the continuation values for the option. However, the accuracy of the LSMC is dependent on the chosen basis functions, where no objective strategy exists for the basis function selection process. Recently, it has been proposed to use kernel ridge regression (KRR) with a bundling technique for high-dimensional American option pricing to avoid selecting the basis functions.
The Heston model is an example of the stochastic volatility model, where only limited literature can be found concerning multi-dimensional American option pricing under this model. In this thesis, we reproduce the proposed KRR-based methods. Additionally, we extend the KRR-based methods for high-dimensional American options under the Heston model. We also involve the LSMC method for evaluating the pricing efficiency and accuracy of KRR-based methods. As no reliable benchmarks exist for the valuation of American options under the high-dimensional Heston model, we implement the primal-dual approach and use its result as the reference price. Nevertheless, the obtained benchmarks are biased. Therefore, we can only conclude that the KRR-based methods apply to American option pricing under the high-dimensional Heston model. Moreover, the KRR-based techniques are computationally more efficient than the LSMC method.

This thesis is concerned with finding the asymptotic distributions of linear spectral statistics of the nonlinear shrinkage estimator for large covariance matrices derived by Ledoit and Wolf (2012). It provides some new inferential procedures for large-dimensional data and shed some light on the power of the new statistical tests on the structure of the large covariance matrices. After a short review of the relevant theory, two linear spectral statistics are proposed which are deduced from the nonlinear shrinkage estimator where for one of these linear spectral statistics its limiting distribution is derived. This results in a ready to use sphericity test statistic in the large-dimensional framework and is one of the main results of this thesis. The sphericity test corresponding to this new test statistic is called the nonlinear shrinkage test (NLS-ε). The theoretical results are illustrated by means of a simulation study where the new nonlinear shrinkage test is compared with al- ready existing tests, in particular the commonly used corrected likelihood ratio test and the corrected John’s test. It is demonstrated that the new nonlinear shrinkage test is most powerful under a non homogeneous variance alternative where it outperforms well known sphericity tests. Moreover, it is observed that the new nonlinear shrinkage test encounters some problems when different alternatives are combined.

Cardiac complications after surgery are common irrespective of the underlying condition. The postoperative level of troponin T is a good marker for cardiac complications. Little is known on the pathology of the release of troponin T in the blood, while a better understanding might provide the ability to reduce the complications. The goal of the thesis is to find patterns in intraoperative data that are related to the release of troponin T in the blood during surgery. The states resulting from estimating an MSVAR on intraoperative hemodynamic data were interpreted and related to postoperative troponin T measurements. The MSVAR was estimated in two ways: with the EM algorithm and in Bayesian fashion with the Gibbs sampler. Prior distributions were chosen and a Gibbs sampler was developed for estimating the MSVAR with these priors. The differences between the EM algorithm and the Gibbs sampler are mostly fundamental and not practical. Furthermore, the MSVAR is an appropriate model for modelling intraoperative hemodynamic data. The states of the MSVAR were related to various surgery variables, but did not have any prognostic value for predicting postoperative troponin T. The states related to the external shocks continuously given to the patient during surgery rather than the patient's state.

In this paper, we want to create a prediction model for target attainment and a scoring system based on these models. The Random Forest model, Logistic Regression model, and Naive Bayes model are employed to achieve this. Classification trees and predictors from the random forest model are used to create the scoring system. The theory behind each model and scoring system is described in detail. The properties of proper scoring rules were checked when making our scoring systems. We have made three models which can reasonably predict target attainment. The logistic regression model performed the best for internal and external validation, with an AUC of 0.84 and 0.80, respectively, whereas the RF model performed slightly worse. The AUC of the random forest differed by 0.01 in both validation sets. The scoring system performed reasonably with an AUC of 0.74. These models and the scoring system perform reasonably well and will help the medical practitioner assess if an ICU patient will attain their beta-lactam antibiotic target.

In this thesis, a factor model which estimates multivariate time series is extended to include an asymmetric relation between the returns of assets and the volatility of said assets. The model proposed in this thesis uses the classical factor model, with univariate logarithmic volatility equations to model the factors as well as the asset innovations. The volatility equations for the factors are extended to contain an asymmetric relationship with the factor returns of the day before. In this thesis, a method to estimate this asymmetric model is developed, the method of estimation mainly relies upon MCMC methods. A method to estimate the logarithmic likelihood for the model is provided as well. This method uses a particle filter to estimate the distribution of the volatility. Using the logarithmic likelihood, it is shown that the asymmetry in the data is identifiable, by comparing the likelihood of the model to the likelihood of the classical factor model, as well as to the likelihood of a factor model with a jump extension. Finally, the model is tested on a real data set of daily returns where its effectiveness is again compared to the classical model and the jump model.

Measuring variable importance is often a difficult task: among others models can be complex and covariates can interact with each other and can be correlated. This study focuses on two questions: First, what should be the theoretical measure of variable importance under a given data-generating model? And second, what are the best estimates of these theoretical measures? Two theoretical measures and some corresponding estimates are presented of which one is the well-known random forests variable importance measure (Breiman, 2001). A simulation study is done for both linear and nonlinear models to find out what are the best estimates of variable importance measures for given data-generating models. Most measures struggle when covariates are correlated, but make an improvement in performance when the number of split variables is tuned.

The Shapley value method is an explanatory method that describes the feature attribution of Machine Learning models. There are three different definitions of the Shapley values, namely Conditional Expectation Shapley, Marginal Expectation Shapley and Baseline Shapley. A comparison is made between the three definitions and they are applied to one statistical and two Machine Learning models that predict house transaction prices. Most existing methods to approximate Shapley values assume independence, which is in practice often violated. An existing copula-based method that tries to take into account the dependency is extended to apply to problems with continuous and discrete features. This copula-based method approximates the Shapley values more accurately than other methods. The Conditional Expectation Shapley values give unnatural explanations, therefore other definitions of the Shapley values are more suitable. The Baseline Shapley values seem to be the most promising since there is an accurate and fast approximation method and the B Shapley values are the easiest to interpret.

In this thesis we shall consider sample covariance matrices Sn in the case when the dimension of the data increases with the sample size to infinity ,while the ratio approaches a fixed constant. We will derive a new statistic based on the general linear shrinkage estimator by Bodnar et al. (2014)[1] We will show that the new statistic is normally distributed under the null hypothesis that the true covariance matrix is the identity, where we assume the existence of the fourth moment of our data.

Furthermore, we will do simulation study that compares our new statistic to tests from finite dimensional statistics that have been altered to work in high dimensional statistics by Wang and Yao [3]. We will look at three different hypothesis, the equicorrelation case, the auto-regressive case and a fixed ratio case. After that, we will look at the non-linear shrinkage estimator based on the work by Ledoit and Peche (2011) [11], and show that, under the null hypothesis, constructing a test is not directly possible like it is in the linear case.

Time series analysis is used to predict future behaviour of processes and is widely used in the finance sector. In this paper we will analyse the modelling of multivariate time series of financial data using vector autoregressive processes. The goal is that the reader will understand the presented models and could theoretically perform time series analysis by himself. Two specific models will be explained: the Vector Autoregressive model (VAR model) and the Vector Error Correction Model (VECM). We will describe various methods to analyse multivariate time series using these models, such as forecasting the process, variance decomposition of the forecast error, causality analysis and impulse response analysis. Examples of these models and analysis methods will be presented and investigated. Finally, we will perform a time series analysis with these models on Dutch indices and stock data. We conclude that real-world data often does not fit the VAR model and VECM requirements and that further improved models should be considered as well.

The aim of this thesis is to model fully collateralized exposures in the presence of the Margin Period of Risk, i.e., the time between the last successful collateral call to the time where the amount of the loss crystallizes. We start with introducing a closed-form expression to model fully collateralized exposures for fixed versus floating interest rate swaps. Then the Brownian bridge method is introduced and applied, which allows for improvements computational wise. Finally, fully collateralized exposures are modeled in the presence of the Margin Period of Risk, with the Least Squares Monte Carlo method in the case of one European call option under Black-Scholes assumptions.