**Course co-ordinator(s):** Dr Marcelo Pereyra (Edinburgh), Marjan Qazvini (Malaysia).

**Aims:**

To describe and compare the main approaches to statistical inference: including classical and Bayesian, and to develop students’ skills in practical, computer- based estimation and inference. This module also aims to develop students’ independent research skills, and their report writing skills.

**Summary:**

This module will consist of a mixture of lectures, tutorials, computer practicals and project work. First and second year courses have discussed how to draw conclusions from data, and introduced some basic methods in an informal way. In this course we take a more fundamental approach to estimation and quantifying the accuracy of estimates.

In lectures we introduce the principles of classical and Bayesian inference discussing their different philosophical bases, and comparing the different solutions that each method gives to various problems of inference. The properties and fundamental importance of the likelihood are described, along with some important results on the sampling properties of estimators. The course will emphasise worked examples and there will be project work based on the computer implementation of the theory taught in lectures and tutorials. Projects will require students to apply statistical methods covered in the lectures, to implement these methods using the **R** programming language and to present their results.

## Detailed Information

**Pre-requisite course(s):** F78PA Probability and Statistics A & F78PB Probability and Statistics B .

**Location: **Edinburgh.

**Semester: **1.

**Syllabus:**

- Introduction
- Inference and decision-making
- Background material: distributions and random variables

- Parameter estimation
- Notation and definitions
- The method of moments
- Interval estimation
- The method of pivots

- Likelihood
- Definition and example
- The likelihood principle and sufficiency
- Point and interval estimation
- Invariance under transformations

- Classical meets Likelihood: the CramИr-Rao Theorem and after
- A lower bound for the variance of an unbiased estimator
- Attaining the CramИr-Rao lower bound
- Minimum variance unbiased estimators (MVUEs)
- Classic properties of MLEs
- Extension to multi-parameter models

- Classical inference in practice
- Practical sampling problems
- Case studies
- Computing exercises
- Project work and report writing

- Introduction to Bayesian inference
- Introduction and prior information
- Prior and posterior distributions
- Conjugate priors
- Bayesian estimation
- Choice of prior distribution
- Link to decision theory
- Utility and the loss function

- Hypothesis testing
- Revision of classical approach – N-P lemma up to GLRT’s
- Bayesian approach to testing

**Learning Outcomes: Subject Mastery**

After studying this module, students should be able to:

- understand the main approaches to statistical inference and the relations between them, and to apply appropriate statistical methods in a practical application.

**Reading list:**

- Garthwaite, PH, Jolliffe, IT, and Jones, B (2002)
*Statistical Inference (2nd ed.)*, Oxford UP.

*[A good reference for the course, though – like all the following books – it goes well beyond what we will cover.]* - Pawitan, Y (2001)
*In all likelihood*, Oxford UP.

*[A possible alternative to Garthwaite et al.]* - Barnett, V (1982)
*Comparative Statistical Inference*(2nd Edn), Wiley.

*[A good general reference.]* - Berger, JO (1985)
*Statistical Decision Theory and Bayesian Analysis*, Springer.

*[Difficult reading, but comprehensive.]* - Casella, G & Berger RL (1990)
*Statistical Inference*, Duxbury Press.

*[Very comprehensive particularly on classical methods.]*

**Assessment Methods:**

There is a 2-hour exam on the lecture material in December (60%) and project work (40%).

The module is synoptically linked with F79MB Statistical Models B.

**SCQF Level: **9.

**Credits:** 15.

## Other Information

**Help:** If you have any problems or questions
regarding the course, you are encouraged to contact the lecturer

**VISION:** further information and course materials
are available on VISION