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 endeavours at the University of Otago are built upon an essential commitment to academic integrity.
The two most common forms of academic misconduct are plagiarism and unauthorised collaboration.
Academic misconduct: Plagiarism
Plagiarism is defined as:
- Copying or paraphrasing another person’s work and presenting it as your own.
- Being party to someone else’s plagiarism by letting them copy your work or helping them to copy the work of someone else without acknowledgement.
- Using your own work in another situation, such as for the assessment of a different paper or program, without indicating the source.
- Plagiarism can be unintentional or intentional. Even if it is unintentional, it is still considered to be plagiarism.
All students have a responsibility to be aware of acceptable academic practice in relation to the use of material prepared by others and are expected to take all steps reasonably necessary to ensure no breach of acceptable academic practice occurs. You should also be aware that plagiarism is easy to detect and the University has policies in place to deal with it.
Academic misconduct: 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 student’s answers should be in their own words. If you are not sure if collaboration is allowed, check with your lecturer.
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.
