## 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

Your final mark F in the paper will be calculated according to this formula:

**F = max(E, 0.6E + 0.15A + 0.25T)**

where:

- E is the Exam mark
- A is the Assignments mark
- T is the Tests mark

and all quantities are expressed as percentages.

### Students must abide by the University’s Academic Integrity Policy

**Academic integrity** means being honest in your studying and assessments. It is the basis for ethical decision-making and behaviour in an academic context. Academic integrity is informed by the values of honesty, trust, responsibility, fairness, respect and courage.

**Academic misconduct** is seeking to gain for yourself, or assisting another person to gain, an academic advantage by deception or other unfair means. The most common form of academic misconduct is plagiarism.

Academic misconduct in relation to work submitted for assessment (including all course work, tests and examinations) is taken very seriously at the University of Otago.

All students have a responsibility to understand the requirements that apply to particular assessments and also to be aware of acceptable academic practice regarding the use of material prepared by others. Therefore it is important to be familiar with the rules surrounding academic misconduct at the University of Otago; they may be different from the rules in your previous place of study.

Any student involved in academic misconduct, whether intentional or arising through failure to take reasonable care, will be subject to the University’s Student Academic Misconduct Procedures which contain a range of penalties.

If you are ever in doubt concerning what may be acceptable academic practice in relation to assessment, you should clarify the situation with your lecturer before submitting the work or taking the test or examination involved.

Types of academic misconduct are as follows:

**Plagiarism**

The University makes a distinction between unintentional plagiarism (Level One) and intentional plagiarism (Level Two).

- Although not intended,
*unintentional plagiarism*is covered by the Student Academic Misconduct Procedures. It is usually due to lack of care, naivety, and/or to a lack to understanding of acceptable academic behaviour. This kind of plagiarism can be easily avoided. *Intentional plagiarism*is gaining academic advantage by copying or paraphrasing someone elses work and presenting it as your own, or helping someone else copy your work and present it as their own. It also includes self-plagiarism which is when you use your own work in a different paper or programme without indicating the source. Intentional plagiarism is treated very seriously by the University.

**Unauthorised Collaboration**

Unauthorised Collaboration occurs when you work with, or share work with, others on an assessment which is designed as a task for individuals and in which individual answers are required. This form does not include assessment tasks where students are required or permitted to present their results as collaborative work. Nor does it preclude collaborative effort in research or study for assignments, tests or examinations; but unless it is explicitly stated otherwise, each students answers should be in their own words. If you are not sure if collaboration is allowed, check with your lecturer..

**Impersonation**

Impersonation is getting someone else to participate in any assessment on your behalf, including having someone else sit any test or examination on your behalf.

**Falsiﬁcation**

Falsiﬁcation is to falsify the results of your research; presenting as true or accurate material that you know to be false or inaccurate.

**Use of Unauthorised Materials**

Unless expressly permitted, notes, books, calculators, computers or any other material and equipment are not permitted into a test or examination. Make sure you read the examination rules carefully. If you are still not sure what you are allowed to take in, check with your lecturer.

**Assisting Others to Commit Academic Misconduct**

This includes impersonating another student in a test or examination; writing an assignment for another student; giving answers to another student in a test or examination by any direct or indirect means; and allowing another student to copy answers in a test, examination or any other assessment.

**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.