In this thesis an algorithm for the noncentral linear density and cumulative distribution function of Wilks' likelihood ratio criterion in MANOVA is derived and it is shown how this algorithm, with modifications, can be used to find the distributions of a number of test criteria for different hypotheses. At the same time previous results regarding percentiles and powers of these criteria are examined and discussed.

Using a geometric approach to best linear unbiased estimation in the general linear model, the additional sum of squares principle, used to generate decompositions, can be generalized allowing for an efficient treatment of augmented linear models. The notion of the admissibility of a new variable is useful in augmenting models. Best linear unbiased estimation and tests of hypotheses can be performed through transformations and reparametrizations of the general linear model. The theory of outliers and influential observations can be generalized so as to be applicable for the general univariate linear model, where three types of outlier and influence may be distinguished. The adjusted models, adjusted parameter estimates, and test statistics corresponding to each type of outlier are obtained, and data adjustments can be effected. Relationships to missing data problems are exhibited. A unified approach to outliers in the general linear model is developed. The concept of recursive residuals admits generalization. The typification of outliers and influential observations in the general linear model can be extended to normal multivariate models. When the outliers in a multivariate regression model follow a nested pattern, maximum likelihood estimation of the parameters in the model adjusted for the different types of outlier can be performed in closed form, and the corresponding likelihood ratio test statistic is obtained in closed form. For an arbitrary outlier pattern, and for the problem of outliers in the generalized multivariate regression model, three versions of the EM-algorithm corresponding to three types of outlier are used to obtain maximum likelihood estimates iteratively. A fundamental principle is the comparison of observations with a choice of distribution appropriate to the presumed type of outlier present. Applications are not necessarily restricted to multivariate normality.

Includes bibliography. / This 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.

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This thesis focuses on modelling the distributions of loss in consumer credit arrangements, both at an individual level and at a portfolio level, and how these might be influenced by loan-specific factors and economic factors. The thesis primarily aims to examine how these factors can be incorporated into a credit risk model through logistic regression models and threshold regression models. Considering the fact that the specification of a credit risk model is influenced by its purpose, the thesis considers the IFRS 7 and IFRS 9 accounting requirements for impairment disclosure as well as Basel II regulatory prescriptions for capital requirements. The thesis presents a critique of the unexpected loss calculation under Basel II by considering the different ways in which loans can correlate within a portfolio. Two distributions of portfolio losses are derived. The Vašíček distribution, which is the assumed in Basel II requirements, was originally derived for corporate loans and was never adapted for application in consumer credit. This makes it difficult to interpret and validate the correlation parameters prescribed under Basel II. The thesis re-derives the Vašíček distribution under a threshold regression model that is specific to consumer credit risk, thus providing a way to estimate the model parameters from observed experience. The thesis also discusses how, if the probability of default is modelled through logistic regression, the portfolio loss distribution can be modelled as a log-log-normal distribution.

Includes bibliography. / This thesis describes the results of a study to establish whether climate variables could be usefully modelled on a daily basis. Three stochastic models are considered for the description of daily climate sequences, which can then be used to generate artificial sequences. The climate variables under consideration are rainfall, maximum and minimum temperature, evaporation, sunshine duration, windrun and maximum and minimum humidity. A simple Markov chain-Weibull model is proposed to model rainfall. Three multivariate models (one proposed by Richardson (1981), two new) are suggested for modelling the remaining climate variables. The model parameters are allowed to vary seasonally, while the error term is assumed to follow an autoregressive process. The models were validated and their general performance·was found to be satisfactory. Some weaknesses were identified and are discussed. The. main conclusion of this study is that daily climate sequences can indeed be usefully described by means of stochastic models.

Includes abstract. Includes bibliographical references.

Kriging is a widely used group of techniques for predicting unobserved responses at specified locations using a set of observations obtained from known locations. Kriging predictors are best linear unbiased predictors (BLUPs) and the precision of predictions obtained from them are assessed by the mean squared prediction error (MSPE), commonly termed the kriging variance.

In the past two decades Mathematical Programming has come to occupy a place of importance in Economic Studies and in Operations Research. Roughly speaking, Mathematical Programming is the analysis of problems of the type: "Find the maximum of a function, when the variables are subject to inequality and equality constraints". The term "Linear Programming" corresponds to the case where, the function to be maximized (the so called objective function) and the equality and inequality constraints are linear. The term "Non-Linear Programming" should then become self-defined. With the introduction of Dantzig's Simplex Method, Linear Programming has become an everyday technique. The same, we regret to say, is not true for Nonlinear Programming because this subject is broader and much more difficult to unify than that of Linear Programming. In fact at present there does exist any unifying theory for Nonlinear Programming. However, we feel that research on this field is gathering tremendous momentum and that in the not too distant future Nonlinear Programming will become both a practical and fundamental tool in many spheres of Science. One of the subject matters of Nonlinear Programming is what we came to call "Quadratic Programming". This name is restricted to the specific problem of maximizing or minimizing a quadratic objective function f(X) = CX + X'DX, where CX is a linear form and X'DX a quadratic form, subject to linear constraints. Historically, Quadratic Programming was the first venture into the theory of Nonlinear Programming. More specifically it is the purpose of this thesis to: (i) Present a unified and simple treatment of the Theory of Concave (Convex) Quadratic Programming (in no way will mathematical rigour be sacrificed for simplicity). (ii) Present a collection of "Simplicial Methods" for solving quadratic programming problems, which are but extensions of the Simplex Method ( for Linear Programming, whose "accuracy" and "convergence" make them completely self-sufficient for the solution of any type of concave (convex) quadratic programming problems.