The thesis intent is to provide a set of statistical methodologies in the field of Extreme Value Theory (EVT) with a particular application to energy losses, in Gigawatt-hours (GWh) experienced by electrical generating units (GUâs). Due to the complexity of the energy market, the thesis focuses on the volume loss only and does not expand into the price, cost or mixes thereof (although the strong relationship between volume and price is acknowledged by some initial work on the energy price [SMP] is provided in Appendix B) Hence, occurrences of excessive unexpected energy losses incurred by these GUâs formulate the problem. Exploratory Data Analysis (EDA) structures the data and attempts at giving an indication on the categorisation of the excessive losses. The size of the GU failure is also investigated from an aggregated perspective to relate to the Generation System. Here the effect of concomitant variables (such as the Load Factor imposed by the market) is emphasised. Cluster Analysis (2-Way Joining) provided an initial categorising technique. EDA highlights the shortfall of a scientific approach to determine the answer to the question at when is a large loss sufficiently large that it affects the System. The usage of EVT shows that the GWh Losses tend to behave as a variable in the Fréchet domain of attraction. The Block Maxima (BM) and Peak-Over-Threshold (POT), the latter as semi and full parametric, methods are investigated. The POT methodologies are both applicable. Of particular interest is the Q-Q plots results on the semiparametric POT method, which yielded results that fit the data satisfactorily (pp 55-56). The Generalised Pareto Distribution (GPD) models well the tail of the GWh Losses above a threshold under the POT full parametric method. Different methodologies were explored in determining the parameters of the GPD. The method of 3-LM (linear combinations of Probability Weighted Moments) is used to arrive at initial estimates of the GPD parameters. A GPD is finally parameterised for the GWh Losses above 766 GWh. The Bayesian philosophy is also utilised in this thesis as it provides a predictive distribution of (high quantiles) the large GWh Losses. Results are found in this part of the thesis in so far that it utilises the ratio of the Mean Excess Function (the expectation of a loss above a certain threshold) over its probability of exceeding the threshold as an indicator to establish the minimum of this ratio. The technique was developed for the GPD by using the Fisher Information Matrix (FIM) and the Delta-Method. Prediction of high quantiles were done by using Markov Chain Monte Carlo (MCMC) and eliciting the GPD Maximal Data Information (MDI) prior. The last EVT methodology investigated in the thesis is the one that uses the Dirichlet process and the method of Negative Differential Entropy (NDE). The thesis also opened new areas of pertinent research.

This thesis focuses on objective Bayesian statistics, by evaluating a number of noninformative priors. Choosing the prior distribution is the key to Bayesian inference. The probability matching prior for the product of different powers of k binomial parameters is derived in Chapter 2. In the case of two and three independently distributed binomial variables, the Jeffreys, uniform and probability matching priors for the product of the parameters are compared. This research is an extension of the work by Kim (2006), who derived the probability matching prior for the product of k independent Poisson rates. In Chapter 3 we derive the probability matching prior for a linear combination of binomial parameters. The construction of Bayesian credible intervals for the difference of two independent binomial parameters is discussed. The probability matching prior for the product of different powers of k Poisson rates is derived in Chapter 4. This is achieved by using the differential equation procedure of Datta & Ghosh (1995). The reference prior for the ratio of two Poisson rates is also obtained. Simulation studies are done to com- pare different methods for constructing Bayesian credible intervals. It seems that if one is interested in making Bayesian inference on the product of different powers of k Poisson rates, the probability matching prior is the best. On the other hand, if we want to obtain point estimates, credibility intervals or do hypothesis testing for the ratio of two Poisson rates, the uniform prior should be used. In Chapter 5 the probability matching prior for a linear contrast of Poisson parameters is derived, this prior is extended in such a way that it is also the probability matching prior for the average of Poisson parameters. This research is an extension of the work done by Stamey & Hamilton (2006). A comparison is made between the confidence intervals obtained by Stamey & Hamilton (2006) and the intervals derived by us when using the Jeffreys and probability matching priors. A weighted Monte Carlo method is used for the computation of the Bayesian credible intervals, in the case of the proba- bility matching prior. In the last section of this chapter hypothesis testing for two means is considered. The power and size of the test, using Bayesian methods, are compared to tests used by Krishnamoorthy & Thomson (2004). For the Bayesian methods the Jeffreys prior, probability matching prior and two other priors are used. Bayesian estimation for binomial rates from pooled samples are considered in Chapter 6, where the Jeffreys prior is used. Bayesian credibility intervals for a single proportion and the difference of two binomial proportions estimated from pooled samples are considered. The results are compared This thesis focuses on objective Bayesian statistics, by evaluating a number of noninformative priors. Choosing the prior distribution is the key to Bayesian inference. The probability matching prior for the product of different powers of k binomial parameters is derived in Chapter 2. In the case of two and three independently distributed binomial variables, the Jeffreys, uniform and probability matching priors for the product of the parameters are compared. This research is an extension of the work by Kim (2006), who derived the probability matching prior for the product of k independent Poisson rates. In Chapter 3 we derive the probability matching prior for a linear combination of binomial parameters. The construction of Bayesian credible intervals for the difference of two independent binomial parameters is discussed. The probability matching prior for the product of different powers of k Poisson rates is derived in Chapter 4. This is achieved by using the differential equation procedure of Datta & Ghosh (1995). The reference prior for the ratio of two Poisson rates is also obtained. Simulation studies are done to com- pare different methods for constructing Bayesian credible intervals. It seems that if one is interested in making Bayesian inference on the product of different powers of k Poisson rates, the probability matching prior is the best. On the other hand, if we want to obtain point estimates, credibility intervals or do hypothesis testing for the ratio of two Poisson rates, the uniform prior should be used. In Chapter 5 the probability matching prior for a linear contrast of Poisson parameters is derived, this prior is extended in such a way that it is also the probability matching prior for the average of Poisson parameters. This research is an extension of the work done by Stamey & Hamilton (2006). A comparison is made between the confidence intervals obtained by Stamey & Hamilton (2006) and the intervals derived by us when using the Jeffreys and probability matching priors. A weighted Monte Carlo method is used for the computation of the Bayesian credible intervals, in the case of the proba- bility matching prior. In the last section of this chapter hypothesis testing for two means is considered. The power and size of the test, using Bayesian methods, are compared to tests used by Krishnamoorthy & Thomson (2004). For the Bayesian methods the Jeffreys prior, probability matching prior and two other priors are used. Bayesian estimation for binomial rates from pooled samples are considered in Chapter 6, where the Jeffreys prior is used. Bayesian credibility intervals for a single proportion and the difference of two binomial proportions estimated from pooled samples are considered. The results are compared

Non-responses in survey data are a prevalent problem. Various techniques for the handling of missing data have been studied and published. The application of a regularised iterative multiple correspondence analysis (RIMCA) algorithm in single imputation (SI) has been suggested for the handling of missing data in survey analysis. Multiple correspondence analysis (MCA) as an imputation procedure is appropriate for survey data, since MCA is concerned with the relationships among the variables in the data. Therefore, missing data can be imputed by exploiting the relationship between observed and missing data. The RIMCA algorithm expresses MCA as a weighted principal component analysis (PCA) of a data triplet ( ), which represents a weighted data matrix, a metric and a diagonal matrix containing row masses, respectively. Performing PCA on a triplet involves the generalised singular value decomposition of the weighted data matrix . Here, standard singular value decomposition (SVD) will not suffice, since constraints are imposed on the rows and columns because of the weighting. The success of this algorithm lies in the fact that all eigenvalues are shrunk and the last components are omitted; thus a âdouble shrinkageâ occurs, which reduces variance and stabilises predictions. RIMCA seems to overcome overfitting and underfitting problems with regard to categorical missing data in surveys. The idea of applying the RIMCA algorithm in MI was appealing, since advantages of MI occur over SI, such as an increase in the accuracy of estimations and the attainment of valid inferences when combining multiple datasets. The aim of this study was to establish the performance of RIMCA in MI. This was achieved by two objectives: to determine whether RIMCA in MI outperforms RIMCA in SI and to determine the accuracy of predictions made from RIMCA in MI as an imputation model. Real and simulated data were used. A simulation protocol was followed creating data drawn from multivariate Normal distributions with both high and low correlation structures. Varying the percentages of missing values in the data and missingness mechanisms (missing completely at random (MCAR) and missing at random (MAR)), as is done by Josse et al. (2012), were created in the data. The first objective was achieved by applying RIMCA in both SI and MI to real data and simulated data. The performance of RIMCA in SI and MI were compared with regard to the obtained mean estimates and confidence intervals. In the case of the real data, the estimates were compared to the mean estimates of the incomplete data, whereas for the simulated data the true mean values and confidence intervals could be compared to the estimates obtained from the imputation procedures. The second objective was achieved by calculating the apparent error rates of predictions made by the RIMCA algorithm in SI and MI in simulated datasets. Along with the apparent error rates, approximate overall success rates were calculated in order to establish the accuracy of imputations made by the SI and MI. The results of this study show that the confidence intervals provided by MI are wider in most of the cases, which confirmed the incorporation of additional variance. It was found that for some of the variables the SI procedures were statistically different from the true confidence intervals, which shows that SI was not suitable in these instances for imputation. Overall the mean estimates provided by MI were closer to the true values, with respect to the simulated and real data. A summary of the bias, mean square errors and coverage for the imputation techniques over a thousand simulations were provided, which also confirmed that RIMCA in MI was a better model than RIMCA in SI in the contexts provided by this research.

Peak electricity demand is an energy policy concern for all countries throughout the world, causing blackouts and increasing electricity tariffs for consumers. This calls for load curtailment strategies to either redistribute or reduce electricity demand during peak periods. This thesis attempts to address this problem by providing relevant information through a frequentist and Bayesian modelling framework for daily peak electricity demand using South African data. The thesis is divided into two parts. The first part deals with modelling of short term daily peak electricity demand. This is done through the investigation of important drivers of electricity demand using (i) piecewise linear regression models, (ii) a multivariate adaptive regression splines (MARS) modelling approach, (iii) a regression with seasonal autoregressive integrated moving average (Reg-SARIMA) model (iv) a Reg-SARIMA model with generalized autoregressive conditional heteroskedastic errors (Reg-SARIMA-GARCH). The second part of the thesis explores the use of extreme value theory in modelling winter peaks, extreme daily positive changes in hourly peak electricity demand and same day of the week increases in peak electricity demand. This is done through fitting the generalized Pareto, generalized single Pareto and the generalized extreme value distributions. One of the major contributions of this thesis is quantification of the amount of electricity which should be shifted to off peak hours. This is achieved through accurate assessment of the level and frequency of future extreme load forecasts. This modelling approach provides a policy framework for load curtailment and determination of the number of critical peak days for power utility companies. This has not been done for electricity demand in the context of South Africa to the best of our knowledge. The thesis further extends the autoregressive moving average-exponential generalized autoregressive conditional heteroskedasticity model to an autoregressive moving average exponential generalized autoregressive conditional heteroskedasticity-generalized single Pareto distribution. The benefit of this hybrid model is in risk modelling of under and over demand predictions of peak electricity demand. Some of the key findings of this thesis are (i) peak electricity demand is influenced by the tails of probability distributions as well as by means or averages, (ii) electricity demand in South Africa rises significantly for average temperature values below 180C and rises slightly for average temperature values above 220C and (iii) modelling under and over demand electricity forecasts provides a basis for risk assessment and quantification of such risk associated with forecasting uncertainty including demand variability.

South African medical schemes spend billions of Rands each year on medical care costs for their beneficiaries near their end of life. Hospi-centric benefit design, fee-for-service reimbursement arrangements and fragmented, silo-based delivery of care result in high, often unnecessary spending near the end of life. Factors including an ageing population, increasing incidence rates of cancer and other non-communicable diseases, and high levels of multi-morbidity among beneficiaries near their end of life further drive end-of-life care costs. Low levels of hospice or palliative care utilisation, a high proportion of deaths in-hospital and chemotherapy use in the last weeks of life point to potentially poor-quality care near the end of life. The usual care pathway for serious illness near the end of life acts like a funnel into private hospitals. This often entails resource intensive care that includes aggressive care interventions right up until death. The result is potentially sub-optimal care and poor healthcare outcomes for many scheme beneficiaries and their surviving relatives. Understanding the complex nature of the end of life, the different care pathways, the available insurance benefits, the interactions between key stakeholders and the multitude of factors that drive end-of-life care costs are vital to setting end-of-life care reform in motion. In order to increase value at the end of life, i.e. to increase quality and/or to reduce costs, benefit design reform, alternative reimbursement strategies, effective communication and multi-stakeholder buy-in is key.

Post the Millennium Development Goal project a significant number of countries are still faced with the challenge of monitoring child mortality. Despite numerous enquiries since 1996 to provide this basic health indicator, South Africa has experienced prolonged periods of uncertainty regarding the level and trend of infant and under-5 mortality. The thesis develops an analytical framework to review all available data sources and methods of analysis and presents the results of the four approaches adopted to measure child mortality trends. Reviewing the demographic indicators produced from seven census and survey enquiries, the overall performance and the strengths and limitations of each approach is evaluated. Poor and extremely poor quality of data for child mortality emerges as a pervasive challenge to census and survey data. The thesis presents the remarkable improvement in the completeness of birth and death registration through South Africa's CRVS system, particularly since 2000, illustrating the possibility of using CRVS data to monitor provincial child mortality in the future and highlighting statistical challenges arising from the movement of children. In conclusion, South Africa should focus on improving CRVS for purposes of monitoring childhood mortality provincially and the comprehensive evaluation of available data is a useful lesson for other upper-middle-income countries.

Banks are key to a well-functioning economy. Periods of economic stress could put banks and therefore the financial system at risk so regulators such as the Prudential Authority in South Africa need to know if banks are resilient to economic stress. A model that forecasts the impact of severe economic stress is developed using publicly available information. The model forecasts the credit losses, deposit volumes and other general equity movements of the biggest five full-service South African banks to assess capital and liquidity strain for any defined macroeconomic stress scenario over the next 3 years. The full-service banks being considered account for more than 90% of all bank lending in deposits in the market and therefore covers the vast majority of banking systemic risk in South Africa. It is shown that different macroeconomic factors affect these banks in different ways due to differences in the type of customers with deposits with each institution and differences in credit risk associated with various loan products. From an overall market perspective economic growth, lending levels, household debt levels and equity markets are the key drivers of deposit volumes. Credit risk in turn is primarily driven by interest rates, inflation and household debt to disposable income. / Dissertation (MSc (Actuarial Science))--University of Pretoria, 2020. / Insurance and Actuarial Science / MSc (Actuarial Science) / Unrestricted

The United Nations Population Division publishes fertility projections for all countries in the World Population Prospects (WPP). These are the most widely used projections for planning and policy implementation. Despite a substantial body of literature that suggests education has a significant impact on fertility, these projections do not incorporate changes in the composition of the population by level of education. We therefore propose and implement a method that incorporates education composition change in projecting fertility. We investigate fertility differentials by level of education, then evaluate how education influences fertility independently; and finally, a model is fitted to project fertility rates by education levels. In both cases, the fertility rates by education level are then weighted by the IIASA educational attainment distributions to get the national fertility rates. These national fertility rates are in turn validated against the WPP fertility rates to evaluate how good the proposed method works. Fertility is high among the less educated relative to educated women. Education proves to be an important driver of fertility decline in Southern Africa. The proposed model is a good fit for countries with sufficient DHS data. However, there are other sources of data that are available, for example, the census data but we could not rely on them since they only give summary information. Validation was done to evaluate how good the model is working. This exercise produced consistent results with the observed fertility estimates. The percentage difference between the projected and WPP fertility estimates varied from 1 to 5 percent in Lesotho, Namibia and Zimbabwe. In conclusion, the model can also be used for other countries. Furthermore, education composition change should be considered when projecting fertility since it has proven to be a significant driver of fertility change. Data quality and availability issues were a major limitation to our study and in future should be improved.

In developing countries, vital registration is the best source of death data that can be used to estimate adult mortality provided they are sufficiently complete. However, they are usually insufficient for estimating mortality sub-nationally due to incomplete registration. This research adapts a method used by Dorrington, Moultrie and Timæus at the provincial level to determine whether it is adequate for estimating adult mortality at the district municipality level in the year prior to the 2001 census. The method uses registration data adjusted for completeness of registration to scale (up or down) the deaths reported by households in the census by age group for each sex. The process of correcting the registered deaths in the year prior to the 2001 census involves estimating intercensal completeness for each population group and each sex between 1996 and 2001 using the average of results from the GGB and the SEG+δ methods. Thereafter, the results are used to estimate the completeness in each of the years within the intercensal period. Thus, an estimate of completeness is obtained in the year prior to the 2001 census for correcting the registered deaths at the population group level. These registered deaths are then used to obtain population group specific adjustment factors to correct the deaths reported by households at the district level, and thereafter to estimate adult mortality rates. Most districts in Kwa-Zulu-Natal have amongst the highest rates of adult mortality, while most districts in the Western Cape have amongst the lowest rates. Results show the Buffalo metropolitan municipality to have higher mortality than that expected for most of the district metropolitan municipalities for both sexes. The same is true for women in Mangaung metropolitan district. It is suspected that HIV prevalence had a significant impact on different levels of adult mortality in the districts, although some adults in the more urban provinces may have died in other provinces. At the provincial level, the method produces marginally higher estimates of adult mortality than the other sources. Provinces that reflect a higher level of mortality appear to deviate more from other research findings than those reflecting lower mortality. In conclusion, the method produces district estimates of ₄₅q₁₅ that are consistent with provincial estimates from other sources and with estimates of HIV prevalence at the district level.

Includes bibliographical references / Over the last few decades, there has been vast interest in the modelling of asset returns using jump diffusion processes. This was in part as a result of the realisation that the standard diffusion processes, which do not allow for jumps, were not able to capture the stylized facts that return distributions are leptokurtic and have heavy tails. Although jump diffusion models have been identified as being useful to capture these stylized facts, there has not been consensus as to how these jump diffusion models should be calibrated. This dissertation tackles this calibration issue by considering the basic jump diffusion model of Merton (197G) applied to South African equity and interest rate market data. As there is little access to frequently updated volatility surfaces and option price data in South Africa, the calibration methods that are used in this dissertation are those that require historical returns data only. The methods used are the standard Maximum Likelihood Estimation (MLE) approach, the likelihood profiling method of Honore (1998), the Method of Moments Estimation (MME) technique and the Expectation Maximisation (EM) algorithm. The calibration methods are applied to both simulated and empirical returns data. The simulation and empirical studies show that the standard MLE approach sometimes produces estimators which are not reliable as they are biased and have wide confidence intervals. This is because the likelihood function required for the implementation of the MLE method is not bounded. In the simulation studies, the MME approach produces results which do not make statistical sense, such as negative variances, and is thus not used in the empirical analysis. The best method for calibrating the jump diffusion model to the empirical data is chosen by comparing the width of the bootstrap confidence intervals of the estimators produced by the methods. The empirical analysis indicates that the best method for calibrating equity returns is the EM approach and the best method for calibrating interest rate returns is the likelihood profiling method of Honore (1998).