The University of Southampton

# MATH6153 Statistical Theory and Linear Models

## Module Overview

This module provides an intensive introduction to, or revision of, essential ideas in developing statistical theory. Firstly, basic statistical models are reviewed along with their properties. Then the transformation method for random variables is introduced to derive standard statistical distributions. The middle part of the module concerns the theory for making statistical inference, including methods such as maximum likelihood estimation and likelihood ratio tests, and an introduction to Bayesian methods. Finally, the theory of linear models is introduced, which will include multiple linear regression models and mixed effects models as examples.

### Aims and Objectives

#### Aim

Having successfully completed this module, you will be able to:

• Standard univariate statistical models and their properties.
• Theory of estimation and significance testing, and know when particular tests should be applied.
• Bayesian inference methods for conjugate priors, prediction method and marginal likelihood.
• Theory of linear statistical models.

### Syllabus

• Univariate distributions: Common standard distributions and their properties.
• Estimation: Unbiasedness, Method of Moments,
• Likelihood - score functions, information, maximum likelihood estimators, Cramer-Rao Inequality.
• Confidence intervals: Asymptotic methods and interpretations.
• Hypothesis testing: Neyman-Pearson Lemma and the Generalised likelihood ratio tests.
• Bayesian methods for conjugate priors, prediction method and marginal likelihood.
• Theory of linear models:
• Simple and multiple linear regression,
• The principle of least squares and least squares estimators,
• Linear hypothesis testing,
• Properties of least squares estimators,
• Model selection.

### Learning and Teaching

#### Teaching and learning methods

36 Lectures and 12 Tutorials

Introduction to the Theory of Statistical Inference by Hannelore Liero and Silvelyn Zwanzig (CRC Press, 2012)

Probability and Statistics by MH DeGroot & MJ Schervish (Addison-Wesley, 2001)

Statistical Inference by G Casella & RL Berger (Duxbury, 1990)

### Assessment

#### Assessment methods

Assessment Method Hours % contribution to final mark Feedback
Coursework 6 pieces   30% Marked weekly, coursework will be returned in the week after hand-in, for instant feedback. Feedback for the final examination will be given in terms of an individual mark and a generic email detailing the general nature of student performance.
Other Exam   70%

Referral Method: By examination

### Costs

#### Costs associated with this course

Students are responsible for meeting the cost of essential textbooks, and of producing such essays, assignments, laboratory reports and dissertations as are required to fulfil the academic requirements for each programme of study.

Please also ensure you read the section on additional costs in the University’s Fees, Charges and Expenses Regulations in the University Calendar available at www.calendar.soton.ac.uk.

Course texts are provided by the library and there are no additional compulsory costs associated with the module.

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