Given a linear regression model y = Xβ + e, where e has a multivariate normal distribution N(0, Σ) consequences of the erroneous assumption that e is distributed as N(0, I) are considered. For a general linear hypothesis concerning the parameters β, in a general model the distribution of the statistic to test the hypothesis, derived under the erroneous assumption is studied. Particular linear hypotheses concerning particular linear models are investigated so as to describe the effects of various patterns of serial correlation on the test statistics arising from these hypotheses. Attention is specially paid to the models of one- and two- way analysis of variance.

Linear algebra is one of the first advanced mathematics courses that students encounter at university level. The transfer from a primarily procedural or algorithmic school approach to an abstract and formal presentation of concepts through concrete definitions, seems to be creating difficulty for many students who are barely coping with procedural aspects of the subject. This research proposes applying APOS theory, in conjunction with Tall’s three worlds of embodied, symbolic and formal mathematics, to create a framework in order to examine the learning of a variety of linear algebra concepts by groups of first and second year university students. The aim is to investigate the difficulties in understanding some linear algebra concepts and to propose potential paths for preventing them. As part of this research project several case studies were conducted where groups of first and second year students were exposed to teaching and learning some introductory linear algebra concepts based on the framework and expressed their thinking through their involvements in tests, interviews and concept maps. The results suggest that the students had limited understanding of the concepts, they struggled to recognise the concepts in different registers, and their lack of ability in linking the major concepts became apparent. However, they also revealed that those with more representational diversity had more overall understanding of the concepts. In particular the embodied introduction of the concept proved a valuable adjunct to their thinking. Since difficulties with learning linear algebra by average students are universally acknowledged, it is anticipated that this study may provide suggestions with the potential for widespread positive consequences for learning.

Automatic Interaction Detection (A.I.D.) is the name of a computer program, first used in the social sciences, to find the interaction between a set of predictor variables and a single dependent variable. The program proceeds in stages, and at each stage the categories of a predictor variable induce a split of the dependent variable into two groups, so that the between groups sum of squares ( BSS ) is a maximum. In this way, the optimum split defines the interaction between predictor and dependent variable, and the criterion BSS is taken as a measure of the explanatory power of the split. One of the strengths of A.I.D. is that this interaction is established without any reference to a specific model, and for this reason it is widely used in practice. However this strength is also its weakness; with no model there is no measure of its significance. Barnard (1974) has said: “… nowadays with more and more apparently sophisticated computer programs for social science, failure to take account of possible sampling fluctuations is leading to a glut of unsound analyses … I have in mind procedures such as A.I.D., the automatic interaction detector, which guarantees to get significance out of any data whatsoever. Methods of this kind require validation …” The aim of this thesis is to supply Part of that validation by investigating the null distribution of the optimum BSS for a single predictor at a single stage of A.I.D., so that the significance of any particular split can be judged. The problem of the overall significance of a complete A.I.D. analysis, combining many stages, still remains to be solved. In Chapter 1 the A.I.D. method is described in more detail and an example is presented to illustrate its use. A null hypothesis that the dependent variable observations have independent and identical normal distributions is proposed as a model for no interaction. In Chapters 2 and 3 the null distributions of the optimum BSS for a single predictor are derived and tables of percentage points are given. In Chapter 4 the normal assumption is dropped and non-parametric A.I.D. criteria, based on ranks, are proposed. Tables of percentage points, found by direct enumeration and by Monte Carlo methods, are given. In Chapter 5 the example presented in Chapter 1 is used to illustrate the application of the theory and tables in Chapters 2, 3 and 4 and some final conclusions are drawn.

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Under the democratic systems of government instilled in many sovereign states, the party in government maintains a constitutional right to call an early election. While the constitution states that there is a maximum period between elections, early elections are frequently called. This right to call an early election gives the government a control to maximize its remaining life in power. The optimal control for the government is found by locating an exercise boundary that indicates whether or not a premature election should be called. This problem draws upon the body of literature on optimal stopping problems and stochastic control. Morgan Poll’s two-party-preferred data are used to model the behaviour of the poll process and a mean reverting Stochastic Differential Equation (SDE) is fitted to these data. Parameters of this SDE are estimated using the Maximum Likelihood Estimation (MLE) Method. Analytic analysis of the SDE for the poll process is given and it will be proven that there is a unique solution to the SDE subject to some conditions. In the first layer, a discrete time model is developed by considering a binary control for the government, viz. calling an early election or not. A comparison between a three-year and a four-year maximum term is also given. A condition when the early exercise option is removed, which leads to a fixed term government such as in the USA is also considered. In the next layer, the possibility for the government to use some control tools that are termed as ‘boosts’ to induce shocks to the opinion polls by making timely policy announcements or economic actions is also considered. These actions will improve the government’s popularity and will have some impacts upon the early-election exercise boundary. An extension is also given by allowing the government to choose the size of its ‘boosts’ to maximize its expected remaining life in power. In the next layer, a continuous time model for this election timing is developed by using a martingale approach and Ito’s Lemma which leads to a problem of solving a partial differential equation (PDE) along with some boundary conditions. Another condition considered is when the government can only call an election and the opposition can apply ‘boosts’ to raise its popularity or just to pull government’s popularity down. The ultimate case analysed is when both the government and the opposition can use ‘boosts’ and the government still has option to call an early election. In these two cases a game theory approach is employed and results are given in terms of the expected remaining life in power and the probability of calling and using ‘boosts’ at every time step and at certain level of popularity.

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