MMath Mathematics with Financial Mathematics

This module provides students with the methodological foundations in model-based statistical learning, in particular likelihood estimation and inference and simple linear regression models. The theoretical and methodological discussions are complemented by practical computer application.

This module thus links the core level 1 module (Introduction to Statistics) and the optional theoretical and applied level 3 modules (Statistical Modelling, Statistical Inference , Extreme Values and Financial Risk, Time Series, Multivariate Statistics and Machine Learning, Medical Statistics).

Desirable
• Basic knowledge of the R statistical programming language

• to introduce the general principles of likelihood-based inference and testing for general models (i.e. for both discrete and continuous distributions),
• to provide an introduction to linear regression models,
• to offer a first overview of Bayesian statistical inference, and
• to demonstrate corresponding computational procedures in R.

On successful completion of the course students will be able to:

• apply model-based approaches in statistical data analysis;
• derive maximum likelihood estimates and compute corresponding confidence intervals;
• perform statistical testing from a likelihood perspective;
• analyse and fit linear regression models;
• address simple inference problems from a Bayesian point of view;
• use R to apply these techniques on actual data.

• Likelihood-based inference: likelihood function, score function, maximum likelihood estimators (MLE), Fisher information, likelihood intervals, invariance principle, relationship to ordinary least-squares estimation (OLS). [6]
• Generalised likelihood ratio tests: one and two sample problems, most powerful tests (Neyman Pearson lemma). [4]
• Linear regression: standard linear regression model, OLS/MLE estimation of regression coefficients and their variances, coefficient of determination, prediction intervals, testing of regression coefficients, variable selection. [6]
• Bayesian learning: Bayes‘ theorem, prior and posterior probabilities, information update, credible intervals, properties of Bayes‘ estimators, shrinkage effect, Bayes factor. [6]

• Coursework (1 in-class exam): weighting 20%
• End of semester examination: weighting 80%

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding. The in-class test also provides an opportunity for students to receive feedback. Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

• Faraway, J. J. 2015. Linear Models with R (second edition). Chapman and Hall/CRC. (recommended)

  Held, L, and Bove, D.S. 2014.  Applied Statistical Inference. Springer. (recommended)

  Hoff, P. 2009. A first course in Bayesian Statistics.  Springer. (recommended)

The independent study hours will normally comprise the following. During each week of the taught part of the semester:

·         You will normally have approximately 60-75 minutes of video content. Normally you would spend approximately 2-2.5 hrs per week studying this content independently

·         You will normally have exercise or problem sheets, on which you might spend approximately 1.5hrs per week

·         There may be other tasks assigned to you on Blackboard, for example short quizzes or short-answer formative exercises

·         In some weeks you may be preparing coursework or revising for mid-semester tests 

Together with the timetabled classes, you should be spending approximately 6 hours per week on this course unit.

The remaining independent study time comprises revision for and taking the end-of-semester assessment.