## STAT362 Probability and Inference 2

Second Semester |

This course continues the theoretical development begun in STAT 261. The tools developed by statisticians for analysing data have become a major factor in the advancement of scientific knowledge. Why are these tools so useful? The reason is that they are based on an agreed system of mathematical and statistical reasoning. In order to be confident that the methods used by a statistician are reliable we need an understanding of this theory.

### Paper details

In STAT362 we will use some of the skills that you developed in STAT 261 and begin to apply them to important statistical problems. In particular we will start to look at the theoretical basis for the linear models developed in STAT 241, STAT 341 and time series models developed in STAT 352. We will also consider in more detail the theory behind hypothesis testing and estimation, including Bayesian methods.

Much of statistical analysis involves postulating a model for observed data, fitting the model to data and selecting between models, and finally making inference based on the chosen model. The first part of the course will look at matrix methods for the general linear model, an important class of models that is used in regression analysis and ANOVA. It will look at likelihood-based model inference, and conclude with Bayesian methods.

### Potential students

All students majoring in Statistics under the Statistics Theme must take this paper. It is strongly recommended for all statistic students. STAT 261 and STAT 362 should also be considered by students with a strong mathematics background interested in a career in scientific research.

### Main topics

- The general linear model
- The likelihood function
- Bayesian inference
- Maximum likelihood estimation
- Hypothesis testing using the likelihood function
- Model selection using the likelihood function

### Prerequisites

STAT 261, MATH 160, MATH 170

### Required text

A full set of lecture notes is provided.

### Useful references

*Mathematical Statistics with Applications* by Wackerly, Mendenhall and Scheaffer.

*An Introduction to Mathematical Statistics and its Applications* by Larsen and Marx

### Lecturers

Matthew Schofield, room 237

### Lectures

3 lectures per week Tuesday, Thursday and Friday at 10am in room 241.

### Tutorials

Wednesday 3pm-5pm every week

### Internal Assessment

The internal assessment is made up of two parts:

- 3/8 = 37.5% from 6 exercises. Assignments are to be word processed using LaTeX.
- 5/8 = 62.5% from a mid-term test

### Exam format

A three-hour exam with 6 questions worth an equal number of marks. All topics are examined.

### Final mark

**Thomas Bayes**, 1702-1761, who followed his father into the Ministry in England. He was intensely interested in mathematics at a time when calculus was in its infancy. His theory of probability was published in

*Essay towards solving a problem in the doctrine of chances*by the Royal Society of London in 1764 — three years after his death, being submitted by a friend who recognised its value.

### Sir Ronald Aylmer Fisher

(1890-1962) ...is considered one of the founders of modern statistics because of his many important contributions. He studied the design of experiments by introducing the concept of randomisation and the analysis of variance, procedures now used throughout the world.

In 1921 he introduced the concept of likelihood. The likelihood of a parameter is proportional to the probability of the data and it gives a function which usually has a single maximum value, which he called the maximum likelihood.

In 1922 he identified three fundamental problems of statistics:

The contributions Fisher made included the development of methods suitable for small samples, the discovery of the precise distributions of many sample statistics and the invention of analysis of variance. He introduced the term maximum likelihood and studied hypothesis testing.