Masters Degrees (Statistics and Actuarial Science)

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An analysis of income and poverty in South Africa
(Stellenbosch : University of Stellenbosch, 2007-03) Malherbe, Jeanine Elizabeth; De Wet, Tertius; Viljoen, H.; Neethling, Ariane; University of Stellenbosch. Faculty of Economic and Management Sciences. Dept. of Statistics and Actuarial Science.
The aim of this study is to assess the welfare of South Africa in terms of poverty and inequality. This is done using the Income and Expenditure Survey (IES) of 2000, released by Statistics South Africa, and reviewing the distribution of income in the country. A brief literature review of similar studies is given along with a broad de nition of poverty and inequality. A detailed description of the dataset used is given together with aspects of concern surrounding the dataset. An analysis of poverty and income inequality is made using datasets containing the continuous income variable, as well as a created grouped income variable. Results from these datasets are compared and conclusions made on the use of continuous or grouped income variables. Covariate analysis is also applied in the form of biplots. A brief overview of biplots is given and it is then used to obtain a graphical description of the data and identify any patterns. Lastly, the conclusions made in this study are put forward and some future research is mentioned.
Aspects of copulas and goodness-of-fit
(Stellenbosch : Stellenbosch University, 2008-12) Kpanzou, Tchilabalo Abozou; De Wet, Tertius; Stellenbosch University. Faculty of Economic and Management Sciences. Dept. of Statistics and Actuarial Science.
The goodness-of- t of a statistical model describes how well it ts a set of observations. Measures of goodness-of- t typically summarize the discrepancy between observed values and the values expected under the model in question. Such measures can be used in statistical hypothesis testing, for example to test for normality, to test whether two samples are drawn from identical distributions, or whether outcome frequencies follow a speci ed distribution. Goodness-of- t for copulas is a special case of the more general problem of testing multivariate models, but is complicated due to the di culty of specifying marginal distributions. In this thesis, the goodness-of- t test statistics for general distributions and the tests for copulas are investigated, but prior to that an understanding of copulas and their properties is developed. In fact copulas are useful tools for understanding relationships among multivariate variables, and are important tools for describing the dependence structure between random variables. Several univariate, bivariate and multivariate test statistics are investigated, the emphasis being on tests for normality. Among goodness-of- t tests for copulas, tests based on the probability integral transform, Rosenblatt's transformation, as well as some dimension reduction techniques are considered. Bootstrap procedures are also described. Simulation studies are conducted to rst compare the power of rejection of the null hypothesis of the Clayton copula by four di erent test statistics under the alternative of the Gumbel-Hougaard copula, and also to compare the power of rejection of the null hypothesis of the Gumbel-Hougaard copula under the alternative of the Clayton copula. An application of the described techniques is made to a practical data set.
Comparison of methods to calculate measures of inequality based on interval data
(Stellenbosch : Stellenbosch University, 2015-12) Neethling, Willem Francois; De Wet, Tertius; Neethling, Ariane; Stellenbosch University. Faculty of Economic and Management Sciences. Dept. of Statistics and Actuarial Science
ENGLISH ABSTRACT: In recent decades, economists and sociologists have taken an increasing interest in the study of income attainment and income inequality. Many of these studies have used census data, but social surveys have also increasingly been utilised as sources for these analyses. In these surveys, respondents’ incomes are most often not measured in true amounts, but in categories of which the last category is open-ended. The reason is that income is seen as sensitive data and/or is sometimes difficult to reveal. Continuous data divided into categories is often more difficult to work with than ungrouped data. In this study, we compare different methods to convert grouped data to data where each observation has a specific value or point. For some methods, all the observations in an interval receive the same value; an example is the midpoint method, where all the observations in an interval are assigned the midpoint. Other methods include random methods, where each observation receives a random point between the lower and upper bound of the interval. For some methods, random and non-random, a distribution is fitted to the data and a value is calculated according to the distribution. The non-random methods that we use are the midpoint-, Pareto means- and lognormal means methods; the random methods are the random midpoint-, random Pareto- and random lognormal methods. Since our focus falls on income data, which usually follows a heavy-tailed distribution, we use the Pareto and lognormal distributions in our methods. The above-mentioned methods are applied to simulated and real datasets. The raw values of these datasets are known, and are categorised into intervals. These methods are then applied to the interval data to reconvert the interval data to point data. To test the effectiveness of these methods, we calculate some measures of inequality. The measures considered are the Gini coefficient, quintile share ratio (QSR), the Theil measure and the Atkinson measure. The estimated measures of inequality, calculated from each dataset obtained through these methods, are then compared to the true measures of inequality.
Confidence intervals for estimators of welfare indices under complex sampling
(Stellenbosch : University of Stellenbosch, 2010-03) Kirchoff, Retha; De Wet, Tertius; Neethling, Ariane; University of Stellenbosch. Faculty of Economic and Management Sciences. Dept. of Statistics and Actuarial Science.
ENGLISH ABSTRACT: The aim of this study is to obtain estimates and confidence intervals for welfare indices under complex sampling. It begins by looking at sampling in general with specific focus on complex sampling and weighting. For the estimation of the welfare indices, two resampling techniques, viz. jackknife and bootstrap, are discussed. They are used for the estimation of bias and standard error under simple random sampling and complex sampling. Three con dence intervals are discussed, viz. standard (asymptotic), percentile and bootstrap-t. An overview of welfare indices and their estimation is given. The indices are categorized into measures of poverty and measures of inequality. Two Laeken indices, viz. at-risk-of-poverty and quintile share ratio, are included in the discussion. The study considers two poverty lines, namely an absolute poverty line based on percy (ratio of total household income to household size) and a relative poverty line based on equivalized income (ratio of total household income to equivalized household size). The data set used as surrogate population for the study is the Income and Expenditure survey 2005/2006 conducted by Statistics South Africa and details of it are provided and discussed. An analysis of simulation data from the surrogate population was carried out using techniques mentioned above and the results were graphed, tabulated and discussed. Two issues were considered, namely whether the design of the survey should be considered and whether resampling techniques provide reliable results, especially for con dence intervals. The results were a mixed bag . Overall, however, it was found that weighting showed promise in many cases, especially in the improvement of the coverage probabilities of the con dence intervals. It was also found that the bootstrap resampling technique was reliable (by looking at standard errors). Further research options are mentioned as possible solutions towards the mixed results.
Extreme value-based novelty detection
(Stellenbosch : Stellenbosch University, 2017-12) Steyn, Matthys Lucas; De Wet, Tertius; Stellenbosch University. Faculty of Economic and Management Sciences. Dept. of Statistics and Actuarial Science.
ENGLISH SUMMARY : This dissertation investigates extreme value-based novelty detection. An in-depth review of the theoretical proofs and an analytical investigation of current novelty detection methods are given. It is concluded that the use of extreme value theory for novelty detection leads to superior results. The first part of this dissertation provides an overview of novelty detection and the various methods available to construct a novelty detection algorithm. Four broad approaches are discussed, with this dissertation focusing on probabilistic novelty detection. A summary of the applications of novelty detection and the properties of an efficient novelty detection algorithm are also provided. The theory of extremes plays a vital role in this work. Therefore, a comprehensive description of the main theorems and modelling approaches of extreme value theory is given. These results are used to construct various novelty detection algorithms based on extreme value theory. The first extreme value-based novelty detection algorithm is termed the Winner-Takes-All method. The model’s strong theoretical underpinning as well as its disadvantages are discussed. The second method reformulates extreme value theory in terms of extreme probability density. This definition is utilised to derive a closed-form expression of the probability distribution of a Gaussian probability density. It is shown that this distribution is in the minimum domain of attraction of the extremal Weibull distribution. Two other methods to perform novelty detection with extreme value theory are explored, namely the numerical approach and the approach based on modern extreme value theory. Both these methods approximate the distribution of the extreme probability density values under the assumption of a Gaussian mixture model. In turn, novelty detection can be performed in complex settings using extreme value theory. To demonstrate an application of the discussed methods a banknote authentication dataset is analysed. It is clearly shown that extreme value-based novelty detection methods are extremely efficient in detecting forged banknotes. This demonstrates the practicality of the different approaches. The concluding chapter compares the theoretical justification, predictive power and efficiency of the different approaches. Proposals for future research are also discussed.
Levy processes and quantum mechanics : an investigation into the distribution of log returns
(Stellenbosch : Stellenbosch University, 2021-03) Le Roux, Christiaan Hugo; De Wet, Tertius; Stellenbosch University. Faculty of Economic and Management Sciences. Dept. of Statistics and Actuarial Science.
ENGLISH SUMMARY : It is well known that log returns on stocks do not follow a normal distribution as is assumed under the Black-Scholes pricing formula. This study investigates alternatives to Brownian Motion which are better suited to capture the stylized facts of asset returns. Lévy processes and models based on Quantum Mechanical theory are described and t to daily log returns for various JSE Indices. Maximum likelihood estimation is used to estimate the parameters of the Lévy processes and the Cramer-von Mises goodness of t statistic is minimized to estimate the parameters of the Quantum Mechanical models. Q-Q plots and the Kolmogorov-Smirnov t statistic is presented to assess the fit of the various models. The results show that the Lévy processes, specically the Normal Inverse Gaussian process, are the best among the processes considered. The performance of the Quantum Mechanical models could be improved if more eigenstates are considered in the approximation, however the computational expense of these models makes them impractical.
'n Ondersoek na die eindige steekproefgedrag van inferensiemetodes in ekstreemwaarde-teorie
(Stellenbosch : University of Stellenbosch, 2005-03) Van Deventer, Dewald; De Wet, Tertius; University of Stellenbosch. Faculty of Economic and Management Sciences. Dept. of Statistics and Actuarial Science.
Extremes are unusual or rare events. However, when such events – for example earthquakes, tidal waves and market crashes - do take place, they typically cause enormous losses, both in terms of human lives and monetary value. For this reason, it is of critical importance to accurately model extremal events. Extreme value theory entails the development of statistical models and techniques in order to describe and model such rare observations. In this document we discuss aspects of extreme value theory. This theory consists of two approaches: The classical maxima method, based on the properties of the maximum of a sample and the more popular threshold theory, based upon the properties of exceedances of a specified threshold value. This document provides the practitioner with the theoretical and practical tools for both these approaches. This will enable him/her to perform extreme value analyses with confidence. Extreme value theory – for both approaches - is based upon asymptotic arguments. For finite samples, the limiting result for the sample maximum holds approximately only. Similarly, for finite choices of the threshold, the limiting distribution for exceedances of that threshold holds only approximately. In this document we investigate the quality of extreme value based inferences with regard to the unknown underlying distribution when the sample size or threshold is finite. Estimation of extreme tail quantiles of the underlying distribution, as well as the calculation of confidence intervals, are typically the most important objectives of an extreme analysis. For that reason, we evaluate the accuracy of extreme based inferences in terms of these estimates. This investigation was carried out using a simulation study, performed with the software package S-Plus.
The saddle-point method and its application to the hill estimator
(Stellenbosch : Stellenbosch University, 2016-12) Buitendag, Sven; De Wet, Tertius; Stellenbosch University. Faculty of Economic and Management Sciences. Dept. of Statistics & Actuarial Science.
ENGLISH SUMMARY : The saddle-point approximation is a highly accurate approximation of the distribution of a random variable. It was originally derived as an approximation in situations where a parameter takes on large values. However, due to its high accuracy and good behaviour in a variety of applications not involving such a parameter, it has been generalized and applied to the distribution of any random variable with a well-behaved cumulant generating function. In this thesis the theory underlying the saddle-point approximation will be discussed and illustrated with an application to approximate the distribution of the Hill estimator in extreme value theory.
South African security market imperfections
(Stellenbosch : University of Stellenbosch, 2006-03) Jooste, Dirk; De Wet, Tertius; University of Stellenbosch. Faculty of Economic and Management Sciences. Dept. of Statistics and Actuarial Science.
In recent times many theories have surfaced posing challenging threats to the Efficient Market Hypothesis. We are entering an exciting era of financial economics fueled by the urge to have a better understanding of the intricate workings of financial markets. Many studies are emerging that investigate the relationship between stock market predictability and efficiency. This paper studies the existence of calendar-based patterns in equity returns, price momentum and earnings momentum in the South African securities market. These phenomena are commonly referred to in the literature as security market imperfections, financial market puzzles and market anomalies. We provide evidence that suggests that they do exist in the South African context, which is consistent with findings in various international markets. A vast number of papers on the subject exist in the international arena. However, very few empirical studies on the South African market can be found in the public domain. We aim to contribute to the literature by investigating the South African case.
Value at risk and expected shortfall : traditional measures and extreme value theory enhancements with a South African market application
(Stellenbosch : Stellenbosch University, 2013-12) Dicks, Anelda; Conradie, W. J.; De Wet, Tertius; Stellenbosch University. Faculty of Economic and Management Sciences. Dept. of Statistics and Actuarial Science.
ENGLISH ABSTRACT: Accurate estimation of Value at Risk (VaR) and Expected Shortfall (ES) is critical in the management of extreme market risks. These risks occur with small probability, but the financial impacts could be large. Traditional models to estimate VaR and ES are investigated. Following usual practice, 99% 10 day VaR and ES measures are calculated. A comprehensive theoretical background is first provided and then the models are applied to the Africa Financials Index from 29/01/1996 to 30/04/2013. The models considered include independent, identically distributed (i.i.d.) models and Generalized Autoregressive Conditional Heteroscedasticity (GARCH) stochastic volatility models. Extreme Value Theory (EVT) models that focus especially on extreme market returns are also investigated. For this, the Peaks Over Threshold (POT) approach to EVT is followed. For the calculation of VaR, various scaling methods from one day to ten days are considered and their performance evaluated. The GARCH models fail to converge during periods of extreme returns. During these periods, EVT forecast results may be used. As a novel approach, this study considers the augmentation of the GARCH models with EVT forecasts. The two-step procedure of pre-filtering with a GARCH model and then applying EVT, as suggested by McNeil (1999), is also investigated. This study identifies some of the practical issues in model fitting. It is shown that no single forecasting model is universally optimal and the choice will depend on the nature of the data. For this data series, the best approach was to augment the GARCH stochastic volatility models with EVT forecasts during periods where the first do not converge. Model performance is judged by the actual number of VaR and ES violations compared to the expected number. The expected number is taken as the number of return observations over the entire sample period, multiplied by 0.01 for 99% VaR and ES calculations.

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