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# Masters Degrees (Statistics and Actuarial Science)

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- ItemA brief introduction to basic multivariate economic statistical process control(Stellenbosch : Stellenbosch University, 2012-12) Mudavanhu, Precious; Van Deventer, P. J. U.; Stellenbosch University. Faculty of Economic and Management Sciences. Dept. of Statistics and Actuarial Science.
Show more ENGLISH ABSTRACT: Statistical process control (SPC) plays a very important role in monitoring and improving industrial processes to ensure that products produced or shipped to the customer meet the required specifications. The main tool that is used in SPC is the statistical control chart. The traditional way of statistical control chart design assumed that a process is described by a single quality characteristic. However, according to Montgomery and Klatt (1972) industrial processes and products can have more than one quality characteristic and their joint effect describes product quality. Process monitoring in which several related variables are of interest is referred to as multivariate statistical process control (MSPC). The most vital and commonly used tool in MSPC is the statistical control chart as in the case of the SPC. The design of a control chart requires the user to select three parameters which are: sample size, n , sampling interval, h and control limits, k.Several authors have developed control charts based on more than one quality characteristic, among them was Hotelling (1947) who pioneered the use of the multivariate process control techniques through the development of a 2 T -control chart which is well known as Hotelling 2 T -control chart. Since the introduction of the control chart technique, the most common and widely used method of control chart design was the statistical design. However, according to Montgomery (2005), the design of control has economic implications. There are costs that are incurred during the design of a control chart and these are: costs of sampling and testing, costs associated with investigating an out-of-control signal and possible correction of any assignable cause found, costs associated with the production of nonconforming products, etc. The paper is about giving an overview of the different methods or techniques that have been employed to develop the different economic statistical models for MSPC. The first multivariate economic model presented in this paper is the economic design of the Hotelling‟s 2 T -control chart to maintain current control of a process developed by Montgomery and Klatt (1972). This is followed by the work done by Kapur and Chao (1996) in which the concept of creating a specification region for the multiple quality characteristics together with the use of a multivariate quality loss function is implemented to minimize total loss to both the producer and the customer. Another approach by Chou et al (2002) is also presented in which a procedure is developed that simultaneously monitor the process mean and covariance matrix through the use of a quality loss function. The procedure is based on the test statistic 2ln L and the cost model is based on Montgomery and Klatt (1972) as well as Kapur and Chao‟s (1996) ideas. One example of the use of the variable sample size technique on the economic and economic statistical design of the control chart will also be presented. Specifically, an economic and economic statistical design of the 2 T -control chart with two adaptive sample sizes (Farazet al, 2010) will be presented. Farazet al (2010) developed a cost model of a variable sampling size 2 T -control chart for the economic and economic statistical design using Lorenzen and Vance‟s (1986) model. There are several other approaches to the multivariate economic statistical process control (MESPC) problem, but in this project the focus is on the cases based on the phase II stadium of the process where the mean vector, and the covariance matrix, have been fairly well established and can be taken as known, but both are subject to assignable causes. This latter aspect is often ignored by researchers. Nevertheless, the article by Farazet al (2010) is included to give more insight into how more sophisticated approaches may fit in with MESPC, even if the mean vector, only may be subject to assignable cause. Keywords: control chart; statistical process control; multivariate statistical process control; multivariate economic statistical process control; multivariate control chart; loss function.Show more - ItemConfidence 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.
Show more 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.Show more - ItemThe implementation of noise addition partial least squares(Stellenbosch : University of Stellenbosch, 2009-03) Moller, Jurgen Johann; Kidd, M.; University of Stellenbosch. Faculty of Economic and Management Sciences. Dept. of Statistics and Actuarial Science.
Show more When determining the chemical composition of a specimen, traditional laboratory techniques are often both expensive and time consuming. It is therefore preferable to employ more cost effective spectroscopic techniques such as near infrared (NIR). Traditionally, the calibration problem has been solved by means of multiple linear regression to specify the model between X and Y. Traditional regression techniques, however, quickly fail when using spectroscopic data, as the number of wavelengths can easily be several hundred, often exceeding the number of chemical samples. This scenario, together with the high level of collinearity between wavelengths, will necessarily lead to singularity problems when calculating the regression coefficients. Ways of dealing with the collinearity problem include principal component regression (PCR), ridge regression (RR) and PLS regression. Both PCR and RR require a significant amount of computation when the number of variables is large. PLS overcomes the collinearity problem in a similar way as PCR, by modelling both the chemical and spectral data as functions of common latent variables. The quality of the employed reference method greatly impacts the coefficients of the regression model and therefore, the quality of its predictions. With both X and Y subject to random error, the quality the predictions of Y will be reduced with an increase in the level of noise. Previously conducted research focussed mainly on the effects of noise in X. This paper focuses on a method proposed by Dardenne and Fernández Pierna, called Noise Addition Partial Least Squares (NAPLS) that attempts to deal with the problem of poor reference values. Some aspects of the theory behind PCR, PLS and model selection is discussed. This is then followed by a discussion of the NAPLS algorithm. Both PLS and NAPLS are implemented on various datasets that arise in practice, in order to determine cases where NAPLS will be beneficial over conventional PLS. For each dataset, specific attention is given to the analysis of outliers, influential values and the linearity between X and Y, using graphical techniques. Lastly, the performance of the NAPLS algorithm is evaluated for variousShow more - ItemInterest rate model theory with reference to the South African market(Stellenbosch : University of Stellenbosch, 2006-03) Van Wijck, Tjaart; Conradie, W. J.; University of Stellenbosch. Faculty of Economic and Management Sciences. Dept. of Statistics and Actuarial Science.
Show more An overview of modern and historical interest rate model theory is given with the specific aim of derivative pricing. A variety of stochastic interest rate models are discussed within a South African market context. The various models are compared with respect to characteristics such as mean reversion, positivity of interest rates, the volatility structures they can represent, the yield curve shapes they can represent and weather analytical bond and derivative prices can be found. The distribution of the interest rates implied by some of these models is also found under various measures. The calibration of these models also receives attention with respect to instruments available in the South African market. Problems associated with the calibration of the modern models are also discussed.Show more - ItemModelling market risk with SAS Risk Dimensions : a step by step implementation(Stellenbosch : University of Stellenbosch, 2005-03) Du Toit, Carl; Conradie, W. J.; University of Stellenbosch. Faculty of Economic and Management Sciences. Dept. of Statistics and Actuarial Science.
Show more Financial institutions invest in financial securities like equities, options and government bonds. Two measures, namely return and risk, are associated with each investment position. Return is a measure of the profit or loss of the investment, whilst risk is defined as the uncertainty about return. A financial institution that holds a portfolio of securities is exposed to different types of risk. The most well-known types are market, credit, liquidity, operational and legal risk. An institution has the need to quantify for each type of risk, the extent of its exposure. Currently, standard risk measures that aim to quantify risk only exist for market and credit risk. Extensive calculations are usually required to obtain values for risk measures. The investments positions that form the portfolio, as well as the market information that are used in the risk measure calculations, change during each trading day. Hence, the financial institution needs a business tool that has the ability to calculate various standard risk measures for dynamic market and position data at the end of each trading day. SAS Risk Dimensions is a software package that provides a solution to the calculation problem. A risk management system is created with this package and is used to calculate all the relevant risk measures on a daily basis. The purpose of this document is to explain and illustrate all the steps that should be followed to create a suitable risk management system with SAS Risk Dimensions.Show more - Item'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.
Show more 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.Show more - ItemOptimal asset allocation for South African pension funds under the revised Regulation 28(Stellenbosch : Stellenbosch University, 2012-03) Koegelenberg, Frederik Johannes; Van Heerden, J. D.; Stellenbosch University. Faculty of Economic and Management Sciences. Dept. of Statistics and Actuarial Science.
Show more ENGLISH ABSTRACT: On 1 July 2011 the revised version of Regulation 28, which governs the South African pension fund industry with regard to investments, took effect. The new version allows for pension funds to invest up to 25 percent compared to 20 percent, in the previous version, of its total investment in foreign assets. The aim of this study is to determine whether it would be optimal for a South African pension fund to invest the full 25 percent of its portfolio in foreign assets. Seven different optimization models are evaluated in this study to determine the optimal asset mix. The optimization models were selected through an extensive literature study in order to address key optimization issues, e.g. which risk measure to use, whether parametric or non parametric optimization should be used and if the Mean Variance model for optimization defined by Markowitz, which has been the benchmark with regard to asset allocation, is the best model to determine the long term asset allocation strategies. The results obtained from the different models were used to recommend the optimal long term asset allocation for a South African pension fund and also compared to determine which optimization model proved to be the most efficient. The study found that when using only the past ten years of data to construct the portfolios, it would have been optimal to invest in only South African asset classes with statistical differences with regard to returns in some cases. Using the past 20-years of data to construct the optimal portfolios provided mixed results, while the 30-year period were more in favour of an international portfolio with the full 25% invested in foreign asset classes. A comparison of the different models provided a clear winner with regard to a probability of out performance. The Historical Resampled Mean Variance optimization provided the highest probability of out performing the benchmark. From the study it also became evident that a 20-year data period is the optimal period when considering the historical data that should be used to construct the optimal portfolio.Show more