# Module Information

Module Identifier
MA36510
Module Title
Linear Statistical Models
2021/2022
Co-ordinator
Semester
Semester 1
Pre-Requisite
Other Staff

#### Assessment

Due to Covid-19 students should refer to the module Blackboard pages for assessment details

Assessment Type Assessment length / details Proportion
Semester Exam 2 Hours   (Written Examination)  100%
Supplementary Exam 2 Hours   (Written Examination)  100%

### Learning Outcomes

On successful completion of this module students should be able to:

1. describe the properties of the multivariate Normal distribution;
2. find and identify the distributions of linear and quadratic forms in Normal variates;
3. formulate a given situation as a (matrix) linear model;
4. analyse data from experiments modelled in this way;
5. construct confidence intervals / regions for linear combinations of parameters and for unknown population variance.
6. carry out simple linear hypothesis tests based on ANOVA.
7. calculate leverages and residuals, and use them for outlier detection.

### Brief description

The Linear Statistical Model encompasses a variety of elementary statistical techniques such as linear regression, design models, ANOVA, etc, and much more besides. This module introduces the general matrix formulation of the linear model, and demonstrates the neatness of its systematic application to a wide range of statistical problems.

### Aims

To introduce the scope and breadth of linear matrix modelling.

### Content

1. MULTIVARIATE PROBABILITY THEORY: Random vectors. Multivariate Normal distribution. Affine transformations, quadratic forms, and their distributions. Independence.
2. GENERAL LINEAR MODEL OF FULL RANK: Formulation. Ordinary Least Squares estimator and normal equations. The assumption of independent homoscedastic errors. The BLUE and Gauss-Markov Theorem. Consideration of the design matrix. The hat matrix and leverage.
3. FURTHER INFERENCE IN THE FULL RANK CASE: Estimation of population variance. Confidence regions and intervals. Consideration of outliers: residuals and leverage. Tests of linear hypotheses.

### Notes

This module is at CQFW Level 6