The main results in this thesis are the minimax theorems for operators in Schatten ideals of compact operators acting on separable Hilbert spaces, generalized Clarkson-McCarthy inequalities for vector lq-spaces lq (Sp) of operators from Schatten ideals Sp, inequalities for partitioned operators and for Cartesian decomposition of operators. All Clarkson-McCarthy type inequalities are in fact some estimates on the norms of operators acting on the spaces lq (Sp) or from one such space into another.

new class of univariate time series models is developed, the Generalized Structural (GEST) time series model. The GEST model extends Gaussian structural time series models by allowing the distribution of the dependent variable to come from any parametric distribution, including highly skew and=or kurtotic distributions. Furthermore, the GEST model expands the systematic part of time series models to allow the explicit modelling of any or all of the distribution parameters as structural terms and (smoothed) functions of independent variables. The proposed GEST model primarily addresses the difficulty in modelling time-varying skewness and kurtosis (beyond location and dispersion time series models). The originality of the thesis starts from Chapter 6 and in particular Chapter 7 and Chapter 8, with applications of the GEST model in Chapter 9. Chapters 2 and 3 contain the literature review of non-Gaussian time series models, Chapter 4 is a reproduction of Chapter 17 in Pawitan (2001), which contains an alternative method for estimating the hyperparameters instead of using the Kalman filter, and Chapter 5 is an application of Chapter 4 to smoothing Gaussian structural time series models.

The thesis deals with some of the anomalies,that affect the predictive performance of univariate time series. This project should help to improve the forecasts made and should also assist those engaged in time series forecasting in real life situations in industry,government and elsewhere. The problem of testing a set of data for outliers is not new in statistics,methods having been proposed for the general linear model. However, there are very few papers on testing time series data for outliers. The greater part of the thesis is concerned with the effects of outliers on forecasts, statistical methods of detection of outliers and the comparison of these methods. Applications of these methods in real life situations are also considered. A subsidiary part of the thesis is concerned with the shift in the level of the series type of anomaly. Very few papers are published. These papers are reviewed. Tests of detection of this type of anomaly are proposed. The final section considers the contribution made, the findings of the work and areas for further research.

This thesis is concerned with a class of linear functional differential equations of mixed type.

This thesis is concerned with some extensions to and applications of generalised linear models and their implementation in a statistical package. The principal extension considered is the inclusion of extra parameters in the link function of the model in order to create a family of parametric link functions. This technique is applied to standard link functions as well as to the family of composite link functions. The applications of such models are illustrated by reference to several examples. The techniques presented enable complicated models to be fitted in a unified and consistent manner, without the need for specialist software or algorithms. A two-stage algorithm for fitting parametric link functions is presented and a diagnostic procedure applied to this class of extended models. The applications of such models include the analysis of grouped and multivariate data. It is shown that grouped data arising from a truncated or mixture distribution can be represented as a parametric composite link function and the technique applied to extend the analysis of some previously published data sets. Following a transformation, it is shown that certain time series models may modelled using parametric composite link functions. An algorithm is presented for the fitting of such models in which the variance function of the observations may be a quite general function of the mean. A generalisation of the multivariate logistic distribution is introduced with application to the analysis of repeated measurements data. Finally, the results of an investigation into the possible development of a statistical programming language, with particular reference to the fitting of generalised linear models, are considered. An implementation of such a language is reported and some features of the language illustrated.

This thesis investigates how adult learners continuously negotiate their relationship with schoolroom mathematics through discourses akin to being ‘more’ or ‘less’ able to ‘do’ and ‘be’ mathematical. It argues that mathematical identities are politically and socially constructed, and that available forms of knowledge inscribe particular mathematical practices on the individual in the classroom. By paying attention to the precarious and contradictory productions of the self, and investigating the allure of undergoing a transformation of the self, I contribute to critical understandings of the psychic costs of re-engaging with learning mathematics as an adult learner. This analysis is a critical narrative inquiry of stories of adults (not)taming mathematics. As an iterative study into identity formation it puts theory to work in unusual ways. In bringing together internal and external processes (and the intersection of biography, aspiration and discursive practice), I unmask how participants underwent what Mendick (2005) calls “identity work”. Working with a Lacanian psychoanalytical through a Foucauldian tradition, I navigate the construction of selfhood during processes of reinvention as (non)mathematical subjects, experiencing ‘success’ (and alienation) through models of collaborative learning, in the contemporary mathematical classroom. The study examines the lived experiences of 11 adult learners using a range of qualitative methods. I actively seek the complexities within various types of provision (including adult education, further education, work-based learning, and community outreach programs) and the multiple forms of knowledge available (or not) through authoritarian discourses of education. Engaging a mobile epistemology, this thesis connects subject positions, techniques of power, psychic costs of reinventing the self, and how the processes of visceral embodiment of mathematics affects learning in the classroom. It argues that mathematical identities are discursively constructed, and the relationship between selfhood and ‘being’ and ‘doing’ mathematical-ness is told as much through narratives characterised by affection as by fear. Rather than provide answers or ‘best practice’ for the collaborative classroom, I conclude with an explanation of why I question common sense assumptions, such as that adult learners want to be placed in a hierarchical positions and judged as independent mathematical thinkers in class, and the practical implications for this in the classroom.

This research uses a case study approach to examine the learning experiences of mature non-specialist first year undergraduate university students studying mathematics as an ancillary subject. The challenges faced by such students taking mathematics as a subsidiary subject within their main degree have not been adequately addressed in the literature: this study seeks to address this gap. The research took place in a UK inner-city post-1992 university which has a very diverse student intake. A qualitative data set was generated from in-depth and focus group interviews of 22 mature students, the majority of whom were non-specialists taking mathematics as a required ancillary subject. An additional quantitative data set was derived from a questionnaire distributed to 250 students taking first year mathematics modules, either as an ancillary or as a specialism subject. A small number of mature students specialising in mathematics in both the interviews and the survey were included in order to compare the experiences and views of the both specialist and non-specialist groups. The Mixed Methods Research Design adopted combined results from the qualitative and quantitative analyses, and was accompanied by a post-structuralist theoretical framework which examines the discursive practices students were exposed to in relation to their construction of mathematics as a subject and their experiences of learning mathematics. The study shows that the major perceived factors that affect mature non-specialist students learning of mathematics include the pedagogical model that is used; the attitudes and beliefs of the learners; the support available to aid learning; and the prevalent discourses about the learning and perceptions of mathematics. These findings have a number of important implications for policy and practice for teaching mathematics to such students, for our understanding of student identities and for widening participation. The evidence from this study suggest that there should be a shift of government policy on access and financing for mature students; a review of mechanism of financial support for mature students; changes in the organisation and resourcing of small classes; a review of curriculum and pedagogy to fit the diverse background of learners; and the development of mathematics support provisions that are embedded in courses that require mathematical skills.

This study traces the professional learning of student teachers who have lived and studied outside the UK, and successfully applied to follow a Post Graduate Certificate in Education (PGCE) course in London to become teachers of mathematics in English schools. It draws upon Bourdieu’s theory of habitus and field to discuss how these student teachers adapt their capitals, as described in migration studies by Erel (2010) and Nowicka (2015) and how, during initial teacher training (ITT), they develop professional capitals for the teaching of mathematics (Nolan, 2012). Recent migration flows have led to a growth of diversity, as measured by countries of origin, in London and other cities around the world, resulting in what Vertovec (2006) has called superdiversity. Through a series of semi-structured interviews with 16 PGCE student teachers hailing from 13 different countries, this study explores the implications of superdiversity for the practices of training teachers. The focus of the research is on the complications of ‘bring[ing] off’ (MacLure, 2003:55) the embodied performance of becoming a teacher, and on how student teachers develop ‘enough’ (Blommaert and Varis, 2011:5) professional capital to pass the course. This leads to a reassessment of the category ‘highly skilled migrant’, which is used to define those who have academic qualifications for teaching from outside the UK. The study uses instead the term ‘highly qualified migrant’, to argue that a mathematical degree needs to be complemented by knowledge of the national mathematics curriculum, national pedagogies and local communicative resources. It shows how London can become an ‘escalator region’ (Fielding, 1992:1), as the student teachers achieve a working life that matches their academic qualifications, and also their own aspirations and those of their families, in the UK and elsewhere. In so doing, they become part of a teaching workforce that reflects the growing superdiversity of the region’s school pupils.

This thesis introduces two general classes of models for analyzing proportion response variable when the response variable Y can take values between zero and one, inclusive of zero and/or one. The models are inflated GAMLSS model and generalized Tobit GAMLSS model. The inflated GAMLSS model extends the flexibility of beta inflated models by allowing the distribution on (0,1) of the continuous component of the dependent variable to come from any explicit or transformed (i.e. logit or truncated) distribution on (0,1) including highly skewed and/or kurtotic or bimodal distributions. The second proposed general class of model is the generalized Tobit GAMLSS model. The generalized Tobit GAMLSS model relaxes the underlying normal distribution assumption of the latent variable in the Tobit model to a very general class of distribution on the real line. The thesis also provides likelihood inference and diagnostic and model selection tools for these classes of models. Applications of both the models are conducted using different sets of data to check the robustness of the proposed models. The originality of the thesis starts from chapter 4 and in particular chapter 5, 6 and 7 with applications of models in chapter 8, 9 and 10.