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# Masters Degrees (Statistics and Actuarial Science)

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### Browsing Masters Degrees (Statistics and Actuarial Science) by Subject "Bayesian statistical decision theory"

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- ItemA 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.
Show more 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.Show more - ItemProjected naive bayes(Stellenbosch : Stellenbosch University, 2020-03) Melonas, Michail C.; Hofmeyr, David; Stellenbosch University. Faculty of Economic and Management Sciences. Dept. of Statistics and Actuarial Science.
Show more ENGLISH SUMMARY : Naïve Bayes is a well-known statistical model that is recognised by the Institute of Electrical and Electronics Engineers (IEEE) as being among the top ten data mining algorithms. It performs classification by making the strong assumption of class conditional mutual statistical independence. Although this assumption is unlikely to be an accurate representation of the true statistical dependencies, naïve Bayes nevertheless delivers accurate classification in many domains. This success can be related to that of linear regression providing reliable estimation in problems where exact linearity is not realistic. There is a rich body of literature on the topic of improving naïve Bayes. This dissertation is concerned with doing so via a projection matrix that provides an alternative representation for the data of interest. We introduce Projected Gaussian naïve Bayes and Projected Kernel naïve Bayes as naïve-Bayes-type classifiers that respectively relies on Gaussianity and kernel density estimation. The proposed method extends the flexibility of the standard naïve Bayes. The approach maintains the simplicity and efficiency of naïve Bayes while improving its accuracy. Our method is shown to be competitive with several popular classifiers on real-world data. In particular, our method’s classification accuracy is compared to that of linear- and quadratic discriminant analysis, the support vector machine and the random forest. There is a close connection between our proposal and the application of naïve Bayes to a class conditionally conducted independent component analysis. In addition to a classification accuracy improvement, the proposed method also provides a tool for visually representing data in low-dimensional space. This visualisation aspect of our method is discussed with respect to the connection to independent component analysis. Our method is shown to give a better visual representation than does linear discriminant analysis on a number of real-world data-sets.Show more - ItemRisk and admissibility for a Weibull class of distributions(Stellenbosch : Stellenbosch University, 2004-12) Negash, Efrem Ocubamicael; Mostert, Paul J.; Stellenbosch University. Faculty of Economy and Management Sciences. Department of Statistics and Actuarial Science.
Show more ENGLISH ABSTRACT: The Bayesian approach to decision-making is considered in this thesis for reliability/survival models pertaining to a Weibull class of distributions. A generalised right censored sampling scheme has been assumed and implemented. The Jeffreys' prior for the inverse mean lifetime and the survival function of the exponential model were derived. The consequent posterior distributions of these two parameters were obtained using this non-informative prior. In addition to the Jeffreys' prior, the natural conjugate prior was considered as a prior for the parameter of the exponential model and the consequent posterior distribution was derived. In many reliability problems, overestimating a certain parameter of interest is more detrimental than underestimating it and hence, the LINEX loss function was used to estimate the parameters and their consequent risk measures. Moreover, the same analogous derivations have been carried out relative to the commonly-used symmetrical squared error loss function. The risk function, the posterior risk and the integrated risk of the estimators were obtained and are regarded in this thesis as the risk measures. The performance of the estimators have been compared relative to these risk measures. For the Jeffreys' prior under the squared error loss function, the comparison resulted in crossing-over risk functions and hence, none of these estimators are completely admissible. However, relative to the LINEX loss function, it was found that a correct Bayesian estimator outperforms an incorrectly chosen alternative. On the other hand for the conjugate prior, crossing-over of the risk functions of the estimators were evident as a result. In comparing the performance of the Bayesian estimators, whenever closed-form expressions of the risk measures do not exist, numerical techniques such as Monte Carlo procedures were used. In similar fashion were the posterior risks and integrated risks used in the performance compansons. The Weibull pdf, with its scale and shape parameter, was also considered as a reliability model. The Jeffreys' prior and the consequent posterior distribution of the scale parameter of the Weibull model have also been derived when the shape parameter is known. In this case, the estimation process of the scale parameter is analogous to the exponential model. For the case when both parameters of the Weibull model are unknown, the Jeffreys' and the reference priors have been derived and the computational difficulty of the posterior analysis has been outlined. The Jeffreys' prior for the survival function of the Weibull model has also been derived, when the shape parameter is known. In all cases, two forms of the scalar estimation error have been t:. used to compare as much risk measures as possible. The performance of the estimators were compared for acceptability in a decision-making framework. This can be seen as a type of procedure that addresses robustness of an estimator relative to a chosen loss function.Show more