### Browsing by Author "Troskie, Casper G"

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- ItemOpen AccessA multivariate statistical approach to the assessment of nutrition status(1972) Fellingham, Stephen A; Troskie, Casper GAttention is drawn to the confusion which surrounds the concept of nutrition status and the problem of selecting an optimum subset of variables by which nutrition status can best be assessed is defined. Using a multidisciplinary data set of some 60 variables observed on 1898 school children from four racial groups, the study aims to identify statistically, both those variables which are unrelated to nutrition status and also those which, although related, are so highly correlated that the measurement of all would be an unnecessary extravagance. It is found that, while the somatometric variables provide a reasonably good (but non-specific) estimate of nutrition status, the disciplines form meaningful groups and the variables of the various disciplines tend to supplement rather than replicate each other. Certain variables from most of the disciplines are, therefore, necessary for an optimum and specific estimate of nutrition status. Both the potential and the shortcomings of a number of statistical techniques are demonstrated.
- ItemOpen AccessAccurate estimation of risk when constructing efficient portfolios for the capital asset pricing model(2010) Zwane, Samkelo Sifiso; Clark, Allan; Troskie, Casper GIn this paper, we investigate the behaviour of the efficient frontier and optimal portfolio of the Troskie-Hossain Capital Asset Pricing Model (TrosHos CAPM) and Sharpe Capital Asset Pricing Model (Sharpe CAPM) when the covariance structure of the residuals is correlated under the Markowitz formulation. By building in the dynamic time series models: AR, GARCH and AR/GARCH we were able to model the autocorrelation and heteroskedasticity of the residuals.
- ItemOpen AccessAccurate portfolio risk-return structure modelling(2006) Hossain, Nafees; Troskie, Casper G; Guo, RenkuanMarkowitz's modem portfolio theory has played a vital role in investment portfolio management, which is constantly pushing the development on volatility models. Particularly, the stochastic volatility model which reveals the dynamics of conditional volatility. Financial time series and volatility models has become one of the hot spots in operations research. In this thesis, one of the areas we explore is the theoretical formulation of the optimal portfolio selection problem under Ito calculus framework. Particularly, a stochastic variation calculus problem, i.e., seeking the optimal stochastic volatility diffusion family for facilitating the best portfolio selection identified under the continuous-time stochastic optimal control theoretical settings. One of the properties this study examines is the left-shifting role of the GARCH(1, 1) (General Autoregressive Conditional Heteroskedastic) model's efficient frontier. This study considers many instances where the left shifting superior behaviour of the GARCH(1, 1) is observed. One such instance is when GARCH(1, 1) is compared within the volatility modelling extensions of the GARCH environ in a single index framework. This study will demonstrate the persistence of the superiority of the G ARCH ( 1, 1) frontier within a multiple and single index context of modem portfolio theory. Many portfolio optimization models are investigated, particularly the Markowitz model and the Sharpe Multiple and Single index models. Includes bibliographical references (p. 313-323).
- ItemOpen AccessThe address sort and other computer sorting techniques(1971) Underhill, Leslie G; Troskie, Casper GOriginally this project was to have been a feasibility study of the use of computers in the library. It soon became clear that the logical place in the library at which to start making use of the computer was the catalogue. Once the catalogue was in machine-readable form it would be possible to work backwards to the book ordering and acquisitions system and forwards to the circulation and book issue system. One of the big advantages in using the computer to produce the catalogue would be the elimination of the "skilled drudgery" of filing. Thus vast quantities of data would need to be sorted. And thus the scope of this project was narrowed down from a general feasibility study, firstly to a study of a particular section of the library and secondly to one particularly important aspect of that section - that of sorting with the aid of the computer. I have examined many, but by no means all computer sorting techniques, programmed them in FORTRAN as efficiently as I was able, and compared their performances on the IBM 1130 computer of the University of Cape Town. I have confined myself to internal sorts, i.e. sorts that take place in core. This thesis stops short of applying the best of these techniques to the library. I intend however to do so, and to work back to the original scope of my thesis.
- ItemOpen AccessThe applicability of discriminant analysis techniques on the multivariate normal and non-normal data types in marketing research.(1985) Van Deventer, Petrus Jacobus Uys; Troskie, Casper GThe purpose of the procedures described is to assign “objects” or "observations" in some optimum fashion to one of two or more populations. In routine banking a bank manager may wish to classify clients who wish to make loans as low or high credit risks on the basis of the elements of certain accounting statements. In such a case there are two definite distinct classes. Another investigation may be initiated to determine whether buying habits are different with respect to the categories: urban, sub-urban and rural clients. Note that in the first example the classes are determined before any sample of observations is investigated, i.e. the sample results do not influence the choice of groups. In the latter case one is trespassing on the terrain of cluster analysis.In the first case we have two types of problems, namely that of devising a classification rule from samples of already classified objects and that of imposing the classification scheme on the objects. The term "discrimination" refers to the process of deriving classification rules from samples of classified objects and the term "classification" refers to applying the rules to knew objects of unknown class. Although it is possible to convert raw data into more easily grasped forms like cartoon faces (Chernoff, 1973) this still represents the problem that any grouping or classification based on these diagrams is subjective.
- ItemOpen AccessAspects of multivariate complex quadratic forms(1981) Conradie, Willem Jacobus; Troskie, Casper GIn this study the distributional properties of certain multivariate complex quadratic forms and their characteristic roots are investigated. Multivariate complex distribution theory was originally introduced by Wooding (1956), Turin (1960) and Goodman (1963a) when they derived and studied the multivariate complex normal distribution. The multivariate complex normal distribution is the basis of complex distribution theory and plays an important role in various areas. In the area of multiple time-series the complex distribution theory is found very useful.
- ItemOpen AccessAspects of non-central multivariate t distributions(1973) Juritz, June M; Troskie, Casper G
- ItemOpen AccessContributions to the theory of generalized inverses, the linear model and outliers(1982) Dunne, Timothy Terence; Troskie, Casper GColumn-space conditions are shown to be at the heart of a number of identities linking generalized inverses of rectangular matrices. These identities give some new insights into reparametrizations of the general linear model, and into the imposition of constraints, when the variance-covariance structure is ÏƒÂ².I. Hypothesis-test statistics for non-estimable functions are shown to give no further information than underlying estimable functions. For an arbitrary variance-covariance structure the "sweep-out" method is generalized. The John and Draper model for outliers is extended, and distributional results established. Some diagnostic statistics for outlying or influential observations are considered. A Bayesian formulation of outliers in the general linear model is attempted.
- ItemOpen AccessA detailed investigation of the linear model and some of its underlying assumptions(1977) Coutsourides, Dimitris; Troskie, Casper GThe purpose of this thesis is to provide a study of the linear model. The whole work has been split into 6 chapters. In Chapter 1 we define and examine the two linear models, i.e. the regression and the correlation model. More specifically we show that the regression model is the conditional version of the correlation model. In Chapter 2 we deal with the problem of multicollinearity. We investigate the sources of near singularities, we give some methods of detecting the multicollinearity, and we state briefly methods for overcoming this problem. In Chapter 3 we consider the least squares method with restrictions, and we dispose of some tests for testing the linear restrictions. The theory concerning the sign of least squares estimates is discussed, then we deal with the method for augmenting existing data. Chapter 4 is mainly devoted to ridge regression. We state methods for selecting the best estimate for k. Some extensions are given dealing with the shrinkage estimators and the linear transforms of the least squares. In Chapter 5 we deal with the principal components, and we give methods for selecting the best subset of principal components. Much attention was given to a method called fractional rank and latent root regression analysis. In Chapter 6 comparisons were performed between estimators previously mentioned. Finally the conclusions are stated.
- ItemOpen AccessThe distribution of the complex rectangular co-ordinates and its applications(1967) Dines, Malcolm; Troskie, Casper GGoodman derived the complex Hishart distribution with the aid of characteristic functions and Fourier transforms. From this the distribution of the complex rectangular co-ordinates were derived as an application of the complex Hishart distribution. In the present paper we give a direct and simplified method of deriving the distribution of the complex rectangular co-ordinates. From this distribution the complex Hishart distribution will be derived as an application. Some properties of the distribution of the complex rectangular coordinates will be given. In addition some applications and in particular the application of the distribution in the derivation of the distribution of the "complex" generalized variance will also be given.
- ItemOpen AccessDistributions of certain test statistics in multivariate regression(1980) Coutsourides, Dimitris; Troskie, Casper GThis thesis is principally concerned with test criteria for testing different hypotheses for the multivariate regression. In this preface a brief summary of each of the succeeding chapters is given. In Chapter 1 the problem of testing the equality of two population multiple correlation coefficients in identical regression experiments has been studied. The author's results are extentions to those of Schuman and Bradley. In Chapter 2 the results of Chapter 1 are extended to the multivariate case, in other words, the author has constructed tests in order to test the equality of two population generalized multiple correlation matrices. In Chapter 3 the author shows that the Ridge Regression, Principal Components and Shrunken estimators yield the same central t and F statistics as the ordinary least square estimator. In Chapter 4 using the results of Aitken, simultaneous tests for the Cp-criterion of Mallows are constructed. Some comments on extrapolation and prediction are made. In Chapter 5 the Ridge and Principal components residuals are studied. Their use for detecting outliers, when multi-collinearity is present, is examined.
- ItemOpen AccessDynamic and robust estimation of risk and return in modern portfolio theory(2008) Mupambirei, Rodwel; Troskie, Casper GThe portfolio selection method developed by Markowitz gives a rational investor a way of evaluating different investment options in a portfolio using the expected return and variance of the returns. Sharpe uses the same optimization approach but estimates the mean and covariance in a regression framework using the index models. Sharpe makes a crucial assumption that the residuals from different assets are uncorrelated and that the beta estimates are constant. When the Sharpe model parameters are estimated using ordinary least squares, the regression assumptions are violated when there is significant autocorrelation and heteroskedasticity in the residuals. Furthermore, the presence of outlying observations in the data leads to unreliable estimates when the ordinary least squares method is used. We find significant correlation in the residuals from different shares and thus we use the Troskie-Hossain model which relaxes this assumption and ultimately produces an efficient frontier that is almost identical to the Markowitz model. The combination of the GARCH and AR models to remove both autocorrelation and heteroskedasticity is used on the single index model and it causes the efficient frontier to shift significantly to the left. Using dynamic estimation through the Kalman filter, it is noticed that the beta coefficients are not constant and that the resulting efficient frontiers significantly outperform the Sharpe model. In order to deal with the problem of outlying observations in the data, we propose using the Minimum Covariance Determinant, (MCD) estimator as a robust version of the Markowitz formulation. Robust alternatives to the ordinary lea.st squares estimator are also investigated and they all cause the efficient frontier to shift to the left. Finally, to solve the problem of collinearity in the multiple index framework, we construct orthogonal indices using principal components regression to estimate the efficient frontier.
- ItemOpen AccessIdentifying outliers and influential observations in general linear regression models(2004) Katshunga, Dominique; Troskie, Casper GIdentifying outliers and/or influential observations is a fundamental step in any statistical analysis, since their presence is likely to lead to erroneous results. Numerous measures have been proposed for detecting outliers and assessing the influence of observations on least squares regression results. Since outliers can arise in different ways, the above mentioned measures are based on motivational arguments and they are designed to measure the influence of observations on different aspects of various regression results. In what follows, we investigate how one can combine different test statistics based on residuals and diagnostic plots to identify outliers and influential observations (both in the single and multiple case) in general linear regression models.
- ItemOpen AccessInformation theoretic measure of complexity and stock market analysis : using the JSE as a case study(2010) Oyenubi, Adeola; Troskie, Casper G; Clark, AlanBozdogan [8] [6] [7] developed a new model selection criteria called information measure of complexity (ICOMP) for model selection. In contrast to Akaike's [1] information criterion (AIC) and other AIC type criteria that are traditionally used for regression analysis, ICOMP takes into account the interdependencies of the parameter estimates. This paper is divided into two parts. In the first part we compare and contrast ICOMP with AIC and other AIC type selection criterion for model selection in regression analysis involving stock market securities. While in the second part we apply the definition of information theoretic measure of complexity to portfolio analysis. We compare the complexity of a portfolio of securities with its' measure of diversification (PDI) and examine the similarities and differences between the two quantities as it affects portfolio management.
- ItemOpen AccessModel selection-regression and time series applications(2003) Clark, Allan Ernest; Troskie, Casper GIn any statistical analysis the researcher is often faced with the challenging task of gleaning relevant information from a sample data set in order to answer questions about the area under investigation. Often the exact data generating process that governs any data set is unknown, indicating that we have to estimate the data generating process by using statistical methods. Regression analysis and time series analysis are two statistical techniques that can be used to undertake such an analysis. In practice researcher will propose one model or a group of competing models that attempts to explain the data being investigated. This process is known as model selection. Model selection techniques have been developed to aid researchers in finding a suitable approximation to the true data generating process. Methods have also been developed that attempt to distinguish between different competing models. Many of these techniques entail using an information criterion that estimates the "closeness" of a fitted model to the unknown data generating process. This study investigates the properties of Bozdogan's Information complexity measure (ICOMP) when undertaking time series and regression analysis. Model selection techniques have been developed for both time series and regression analysis. The regression analysis techniques however often provide unsatisfactory results due to poor experimental designs. Poor experimental design could induce collinearities causing parameter estimates to become unstable with large standard errors. Time series analysis utilizes lagged autocorrelation- and lagged partial autocorrelation coefficients in order to specify the lag structure of the model. In certain data sets this process is not informative in determining the order of an ARIMA model. ICOMP guards against collinearity by considering the interaction between the parameters being estimated in a model. This study investigates the properties of ICOMP when undertaking regression and time series analysis by means of a simulation study. Bibliography: pages 250-263.
- ItemOpen AccessModern portfolio optimization using robust estimation techniques(2005) Van Straaten, Conrad; Troskie, Casper GRather than following a normal distribution, share returns and market proxies have been shown to follow skewed distributions, with long tails in some cases. In this dissertation various robust estimation techniques are investigated in an attempt to minimise the influence that outliers may have on the estimation and to better estimate the input parameters for the Markowitz and Sharpe portfolio models. The main goal is to ascertain whether or not the input parameters determined, using the robust procedures, yield better results than the Ordinary Least Squares (OLS) procedure.
- ItemOpen AccessA multivariate statistical approach to the assessment of nutrition status(1972) Fellingham, Stephen Arthur; Troskie, Casper GAttention is drawn to the confusion which surrounds the concept of nutrition status and the problem of selecting an optimum subset of variables by which nutrition status can best be assessed is defined. Using a multidisciplinary data set of some 60 variables observed on 1898 school children from four racial groups, the study aims to identify statistically, both those variables which are unrelated to nutrition status and also those which, although related, are so highly correlated that the measurement of all would be an unnecessary extravagance. It is found that, while the somatometric variables provide a reasonably good (but non-specific) estimate of nutrition status, the disciplines form meaningful groups and the variables of the various disciplines tend to supplement rather than replicate each other. Certain variables from most of the disciplines are, therefore, necessary for an optimum and specific estimate of nutrition status. Both the potential and the shortcomings of a number of statistical techniques are demonstrated.
- ItemOpen AccessOutliers, influential observations and robust estimation in non-linear regression analysis and discriminant analysis(1993) Van Deventer, Petrus Jacobus Uys; Troskie, Casper GIncludes bibliography.
- ItemOpen AccessPortfolio construction using index regression models(2008) Steyn, Dirk; Troskie, Casper GIn this dissertation we review the Sharpe Index Model and an innovation on this model introduced by Hossain, Troskie and Guo (2005b). These models are extended to the multi index framework. We then empirically investigate the impact of the models on portfolio creation over an extensive data set. Next we extend these models by modelling the regression residuals as ARMA and GARCH(l, 1) processes and investigate the effect on the resulting portfolios. We then introduce the topic of bounded influence regression and apply it to financial data by down weighting extreme returns prior to regression. A new weighting function is introduced in this dissertation and the effects on the efficient frontiers and resulting market portfolios for the chosen set of shares are investigated.
- ItemOpen AccessPricing methods for American options(2003) Duvel, Heimo; Abraham, Haim; Troskie, Casper GThis thesis is about the comparison of Pricing models for the valuation of American Options. Three classes of numerical approaches are considered. These are Lattice Methods, Analytic Approximations and Monte Carlo Simulation. Methods will be contrasted in terms of accuracy and speed of the computed American option price. One particular method utilises regression when estimating the American option price. For this approach the impact of outliers and multicollinearity is examined and alternative regression models fitted. Monte Carlo Simulation is implemented to calculate early exercise probabilities of American options in the South African market. Results are compared for both call and put options. A test set of 3550 options is simulated with parameters mirroring the South African economy. On this set, the accuracy of all methods is assessed relative to a benchmark price, which is computed by a convergent lattice approach. Finally, American Symmetry is used to evaluate both put and call options.