Masters Degrees (Statistics and Actuarial Science)

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Browsing Masters Degrees (Statistics and Actuarial Science) by browse.metadata.advisor "Harvey, Justin"

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    A Bayesian extreme value approach to the optimal reinsurance problem in a multivariate risk setting
    (Stellenbosch : Stellenbosch University, 2023-12) Steenkamp, Shaun Francois; Harvey, Justin; Stellenbosch University. Faculty of Economic and Management Sciences. Dept. of Statistics and Actuarial Science.
    ENGLISH SUMMARY: This thesis investigates a Bayesian extreme value theory approach to analyse the optimal reinsurance problem, more specifically the optimal layer selection of an excess of loss reinsurance contract. This thesis suggests a simulation approach to the optimization of the layer selection. This thesis proposes a multivariate excess of loss (XL) reinsurance structure, referred to as the simultaneous XL reinsurance structure and applies the developed optimization algorithm to this structure in several numerical examples. The approach takes a particular focus on extreme risks, thereby investigating the optimal reinsurance contract that best protects the insurance company from rare large claims. The methodology is explained for a univariate risk case, thereafter the model is extended to the bivariate and the multivariate risk cases. The optimal reinsurance agreement can be investigated using a variety of different models. This thesis develops a risk measure minimization model, with a focus on the conditional tail expectation (CTE) riskmeasure. The model allows for the insurance company’s reinsurance budget as a constraint in the optimization problem. Bayesian techniques are especially useful in problems where data is sparse, therefore this thesis suggests utilizing a Bayesian approach to the optimal reinsurance problem where rare large claims are considered. A Bayesian extreme value theory approach could improve the process of investigating the optimal reinsurance problem by utilising Markov Chain Monte Carlo (MCMC) methods to supplement the information from the data that the insurance company has available. The approach is extended into the bivariate and multivariate risk cases where a fictitious insurer, involved in various lines of business is considered. The dependence structure is modelled using a copula approach. Numerical examples are examined, and the results are interpreted. This thesis takes a focus on the tail of the data, thereby evaluating the optimal excess of loss reinsurance contract for very large claims with very small probabilities. The research suggests an algorithm for evaluating the optimal reinsurance strategy in a multivariate risk environment for insurance companies involved in different lines of business. The analysis will improve understanding and assist decision making on the reinsurance strategy from the insurer’s perspective.
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    Bayesian machine learning : theory and applications
    (Stellenbosch : Stellenbosch University, 2020-12) Payne, Megan Wendy; Harvey, Justin; Stellenbosch University. Faculty of Economic and Management Sciences. Dept. of Statistics and Actuarial Science.
    ENGLISH SUMMARY : Machine learning problems in general are concerned with the ability of different methods and algorithms to extract useful and interpretable information from large datasets, possibly ones which are corrupt due to noisy measurements or errors in data capturing. As the size and complexity of data increases, the demand for efficient and robust machine learning techniques is greater than ever. All statistical techniques can be divided into either a frequentist approach or a Bayesian approach depending on how probability is interpreted and how the unknown parameter set is treated. Bayesian methods have been present for several centuries; however, it was the advent of improved computational power and memory storage that catalysed the use of Bayesian modelling approaches in a wider range of scientific fields. This is largely due to many Bayesian methods requiring the computation of complex integrals, sometimes ones that are analytically intractable to compute in closed form, now being more accessible for use since approximation methods are less time-consuming to execute. This thesis will consider a Bayesian approach to statistical modelling and takes the form of a postgraduate course in Bayesian machine learning. A comprehensive overview of several machine learning topics are covered from a Bayesian perspective and, in many cases, compared with their frequentist counterparts as a means of illustrating some of the benefits that arise when making use of Bayesian modelling. The topics covered are focused on the more popular methods in the machine learning literature. Firstly, Bayesian approaches to classification techniques as well as a fully Bayesian approach to linear regression are discussed. Further, no discussion on machine learning methods would be complete without consideration of variable selection techniques, thus, a range of Bayesian variable selection and sparse Bayesian learning methods are considered and compared. Finally, probabilistic graphical models are presented since these methods form an integral part of Bayesian artificial intelligence. Included with the discussion of each technique is a practical implementation. These examples are all easily reproducible and demonstrate the performance of each method. Where applicable, a comparison of the Bayesian and frequentist methods are provided. The topics covered are by no means exhaustive of the Bayesian machine learning literature but rather provide a comprehensive overview of the most commonly encountered methods.