Masters Degrees (Mathematical Statistics and Actuarial Science)

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Open Access
Assessing the willingness of rural homeowners to insure their homes in South Africa using multilevel modelling
(University of the Free State, 2023) Skenjana, Samkele; Koning, Frans Frederik
There has been an increase in urbanisation in the last decade as more South African seeks better work opportunities in the urban areas. Despite this notable increase, there are individuals who still prefer to build houses and reside in rural areas. There are reasons why people in South Africa have opted to invest in properties in rural areas. Firstly, the process to obtain land in rural areas is through a traditional leader or chief of the village (much easier than in the urban areas). Secondly, the cost of the land in the rural areas is significantly lower than the cost of land in the urban areas. As valid as these reasons are, they have drawbacks such as owners of the not having a title deed for their land and an accurate amount of the value of their land. This makes it difficult to insure these homes. The absence of a rural home insurance in South Africa that focuses on these rural homeowners fitting the description above was the driving force behind the need for this study. Literature thus far has been focused on agricultural and crop insurance in rural areas. This study will explore the challenges in the rural insurance market in South Africa and factors affecting the willingness of these rural residents to insure their rural homes.
Open Access
The association of statistics anxiety, attitude toward statistics and mathematics self-concept with performance in a business statistics course
(University of the Free State, 2018) Mokhele, Moletenyane; Van der Merwe, Linda; Von Maltitz, Michael J.
Statistics anxiety is a pervasive problem in many fields of study. A large proportion of students identify statistics courses as the most anxiety-inducing courses in their curriculum. It is important to investigate students’ anxiety as it can negatively affect students’ performance and their overall psychological and physiological condition. Furthermore, understanding about a student’s level of anxiety may help teachers find ways to reduce the level of anxiety and enhance the learning experienced by students. This empirical study examined the relationship between statistics anxiety, attitude toward statistics, and mathematics self-concept as well as their effect on performance in an introductory business statistics course with 103 students (50 males and 53 females). In addition, the study aimed to determine whether statistics anxiety differs by gender and to investigate the experiences and opinions of students regarding statistics anxiety by means of interviews. Statistics anxiety and attitude toward statistics was measured using the Statistics Anxiety Rating Scale (STARS). Ten questions were added to the STARS to measure mathematics self-concept. Performance measures included two tests and final examination marks. Face-to-face, semi-structured interviews were conducted after the examination was written.
Open Access
The capability approach and measurement: operationalizing capability indicators in higher education
(University of the Free State, 2015-01) Ruswa, Anesu; Walker, Melanie; Chikobvu, Delson
The thesis contributes to work in the field of operational measurement of Human Capabilities. Although a number of studies have examined the challenges posed in the measurement of Human Capabilities, there has not been a focus on the empirical merits of the methods and methodologies followed in identification and measurement of valuable capabilities especially in the Higher Education context. To this end, this study provides insights into the identification of valuable student capabilities through an exposition of the methods which can be followed to create and measure robust indicators of student capabilities. A quantitative inquiry determines which Human capabilities students in Higher Education institutions have reason to value and the results of this process are compared to a theoretical student capabilities literature. The thesis advocates for a human development approach over a human capital approach in evaluating the wellbeing of students. The study is significant in that it aids policy and decision makers in Higher Education to identify what students value and thus be in a position to fashion curricula, programmes and policies in a way which best benefits the subjects. To achieve the above mentioned goal, the thesis draws substantially on the work of Paul Anand, Amartya Sen, Flavio Comim, Enrica Chiappero Martinetti, Ingrid Robeyns, Melanie Walker and Sabina Alkire, among others, who have researched and advanced in the field of operational measurement of human capabilities in the Higher Education environment.
Open Access
Exotic equity derivatives: a comparison of pricing models and methods with both stochastic volatility and interest rates
(University of the Free State, 2017) Scheltema, Jaundre; Venter, Jan-Paul
The traditional Black Scholes methodology for exotic equity option pricing fails to capture the features of latent stochastic volatility and observed stochastic interest rate factors exhibited in financial markets today. The detailed study presented here shows how these shortcomings of the Black Scholes methodology have been addressed in literature by examining some of the developments of stochastic volatility models with constant and stochastic interest rates. A subset of these models, notably with models developed within the last two years, are then compared in a simulated study design against a complex Market Model. Each of the select models were chosen as “best” representatives of their respective model class. The Market Model, which is specified through a system of Stochastic Differential Equations, is taken as a proxy for real world market dynamics. All of the select models are calibrated against the Market Model using a technique known as Differential Evolution, which is a globally convergent stochastic optimiser, and then used to price exotic equity options. The end results show that the Heston-Hull-CIR Model (H2CIR) outperforms the alternative Double Heston and 4/2 Models respectively in producing exotic equity option prices closest to the Market Model. Various other commentaries are also given to assess each of the select models with respect to parameter stability, computational run times and robustness in implementation, with the final conclusions supporting the H2CIR Model in preference over the other models. Additionally a second research question is also investigated that relates to Monte Carlo pricing methods. Here the Monte Carlo pricing schemes used under the Black Scholes and other pricing methodologies is extended to present a semi-exact simulation scheme built on the results from literature. This new scheme is termed the Brownian Motion Reconstruction scheme and is shown to outperform the Euler scheme when pricing exotic equity derivatives with relatively few monitoring or option exercise dates. Finally, a minor result in this study involves a new alternative numerical method to recover transition density functions from their respective characteristic functions and is shown to be competitive against the popular Fast Fourier Transform method. It is hoped that the results in this thesis will assist investment and banking practitioners to obtain better clarity when assessing and vetting different models for use in the industry, and extend the current range of techniques that are used to price options.
Open Access
Regularised iterative multiple correspondence analysis in multiple imputation
(University of the Free State, 2013-07) Nienkemper, Johané; Von Maltitz, M. J.;
English: 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 ‘double shrinkage’ 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.
Open Access
Second-order estimation procedures for complete and incomplete heavy-tailed data
(University of the Free State, 2016) Maribe, Gaonyalelwe; Verster, Andréhette
This thesis investigates the second-order re ned peaks over threshold model called the Extended Pareto Distribution (EPD) introduced by Beirlant et al. (2009). Focus is placed on estimation of the Extreme Value Index (EVI). Firstly we investigate the e ectiveness of the EPD in modelling heavy-tailed distributions and compare it to the Generalized Pareto Distribution (GPD) in terms of the bias, mean squared error and variance of the EVI. This is done through a simulation study and the Maximum Likelihood (ML) method of estimation is used to make the comparison. In practice, data can be tampered by some arbitrary process or study design. We therefore investigate the performance of the EPD in estimating the EVI for heavy-tailed data under the assumption that the data is completely observable and uncontaminated, random right censored and contaminated respectively. We suggest an improved ML numerical procedure in the estimation of EPD parameters under the assumption that data is completely observable and uncontaminated. We further propose a Bayesian EPD estimator of the EVI and show through a simulation study that this estimator leads to much improved results as the ML EPD estimator. A small case study is conducted to assess the performance of the Bayesian EPD estimator and the ML EPD estimator using a real dataset from a Belgian reinsurance rm. We investigate the performance of some well known parametric and semi-parametric estimators of the EVI adapted for censoring by a simulation study and further illustrate their performance by applying them to a real survival dataset. A censored Bayesian EPD estimator for right censored data is then proposed through an altered expression of the posterior density. The censored Bayesian EPD estimator is compared with the censored ML EPD estimator through a simulation study. Behaviour of the minimum density power divergence estimator (MDPDE) is assessed at uncontaminated and contaminated distributions respectively through an exhaustive simulation study including other EPD estimators mentioned in this thesis. The comparison is made in terms of the bias and mean squared error. EVI estimates from the di erent estimators are then used to estimate quantiles, the results are reported concurrently with the EVI estimates. We illustrate the performance of all mentioned estimators on a real dataset from geopedology, in which a few abnormal soil measurements highly in uence the estimates of the EVI and high quantiles.
Open Access
Stochastic ordering with applications to reliability theory
(University of the Free State, 2015-01) Khalema, Tokelo; Finkelstein, Maxim
Abstract not available
Open Access
A survey on participation and attitude to sports among undergraduate students in junior residences at the University of the Free State
(University of the Free State, 2016-01) Mangoejane, Patricia Kekeletso; Chikobvu, Delson
The main objective of this study is to assess and quantify participation in sporting activities by students and to determine the factors influencing students’ intentions to participate or not to participate in sports at the University of the Free State. The data are obtained from interviewing students participating or not participating in various sporting codes available at the University of the Free State (main campus in Bloemfontein, South Africa). A systematic random sampling technique was used as the interviewing team knocked on every fifth door in a given residence to ensure that all corners of each residence were reached. The students found at the residence at that particular time, were asked to fill in the questionnaire. Tables and charts are used for illustration of results. T-tests, F-tests, Principal component analysis, Cluster comparison analysis and Item analysis are also performed for further analysis. Three hundred and eight students (308) (61% females and 39% males) living in junior residences were interviewed for this research. The majority of participants (75%) were non-whites (blacks, coloured, and Asians); this was in line with the University of the Free State enrolment structure of the year 2011 (75% non-whites and 25% whites). The reasons provided by the participants for their participation in sporting activities were indicated as keeping fit (91%), releasing of stress (89.35%), gaining a feeling of wellbeing (83%), increasing in physical abilities (81%) and previous school sports involvement (67%). Students from second academic year upwards mostly raised the positive response that they relied on regular exercise to achieve academic success. The researcher concludes that certain variables, namely gender, age group, race, marital status preferred language of study, faculty of study, academic year of study, previous school sport participation, current sport participation, participated sporting codes, reasons for sport participation and reasons for non-sport participation for students, are the most important variables that the Kovsie Sport and management of sports, should focus on in order to encourage students to participate in sporting activities. Through sports, students are also able to interact with one another and participate in different sporting codes offered by the university.

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