General description
The Linear Statistical Model encompasses all elementary statistical techniques such as one and two mean procedures, straight line fitting, etc, and much more besides. This module sets such models in a matrix formulation and shows the neatness and breadth of application of such modelling, at the same time illustrating some illuminating applications of matrices.

Aims
To introduce the scope and breadth of linear matrix modelling.

Learning outcomes
On completion of this module, a student should be able to:

• describe the properties of the multivariate Normal distribution;
• find and identify the distributions of linear and quadratic forms in Normal variates;
• formulate a given situaton as a (matrix) linear model;
• analyse data from experiments modelled in this way;
• construct confidence intervals/regions for linear combinations of parameters and for ratios of two parameters;
• construct prediction intervals for future observations.

Syllabus
1. DISTRIBUTION THEORY: Random vectors. Multivariate Normal Distribution. Linear Forms. Quadratic forms. Independence.
2. GENERAL LINEAR MODEL OF FULL RANK: Formulation. Least squares and the normal equations. Properties of their solution. Effect of independent homoscedastic errors. The Gauss-Markov Theorem.
3. INFERENCE IN THE FULL RANK CASE: Confidence statements. Confidence regions. Prediction intervals. Confidence limits for ratios.