This online module provides you with the mathematical underpinning for statistical methods in general and – in particular – for other OU statistics modules. You will gain a thorough grounding in mathematical statistics, together with generic skills. You will study distribution theory, leading on to the theory of statistical inference developed under both classical and Bayesian approaches. In the classical case, you will focus on maximum likelihood estimation. You’ll also explore the development of these ideas in the context of linear modelling (regression and extensions). To study this module, you should have a sound knowledge of basic statistical ideas and competence in calculus, algebra and matrices, as provided by the appropriate OU level 1 and 2 study.
What you will study
Other OU statistics modules focus on hands-on practical applications of statistical techniques and interpretation of data and statistical analyses. This module complements these modules by providing the mathematical theory underlying the methods and concepts, including a treatment of both classical and Bayesian statistics. A considerable amount of mathematics is sometimes required for this development.
This module is delivered online, with integrated use of exercises, animations, audio and video segments. You will also be provided with printed versions of the main units and extra exercises.
The module is divided into four blocks of study.
Block 1: Review and distribution theory
The first block comprises a review unit and units introducing distribution theory. The review is mostly of fundamental statistical ideas of the type taught in Analysing data (M248), (see Entry section below for details); there is also a speedy reminder of important relevant methods in mathematics, including calculus and matrices. Two units in this block introduce the theory of continuous distributions. You will learn, for example, how to evaluate moments of distributions and about other properties of some important univariate distributions. The mathematical structure of multivariate distributions will be explored, with some emphasis on the multivariate normal distribution.
Block 2: Classical inference
The second block is about the classical approach to statistical inference. You will learn how to use calculus to obtain maximum likelihood estimators of parameters. You will also learn about the properties of maximum likelihood estimation and of point estimation more generally. The mathematics underlying hypothesis tests and confidence intervals will be explored. There is also a unit on asymptotic (large sample) analysis, giving an insight into how statisticians study properties of statistical procedures by approximate methods.
Block 3: Bayesian statistics
In the third block you’ll consider the Bayesian approach to statistical inference. The emphasis is first on so-called conjugate analysis which constitutes the type of Bayesian analysis most amenable to straightforward mathematical development. You’ll consider prior to posterior analysis first, followed by Bayesian estimation based on decision theory. Markov chain Monte Carlo (MCMC) is a technique often used for tackling Bayesian problems which are not conjugate; you’ll investigate the mathematical ideas leading to the basic methods of MCMC.
Block 4: Linear modelling
The fourth and final block gives some of the mathematical development underlying linear modelling. The material covers linear regression on a single explanatory variable; multiple linear regression where there is more than one explanatory variable; and generalised linear modelling for regression situations where the normal distribution is not a suitable model for variation in the response. Both classical and Bayesian approaches to the analysis of these models are considered.
You will learn
Successful study of this module should enhance your skills in understanding some useful mathematical theory, interpreting mathematical results in a statistical context, constructing logical arguments, and finding solutions to problems.
This module may help you to gain membership of the Institute of Mathematics and its Applications (IMA). For further information, see the IMA website.
This module may also help you to apply for the professional award of Graduate Statistician conferred by The Royal Statistical Society (RSS).
You need no pre-requisites to study Mathematical statistics. However, we recommend that you’re familiar with the following mathematical topics:
- basic differentiation and integration
We recommend you also have previous basic knowledge of statistical science; we’ll include some revision of the following topics:
- normal, Poisson and binomial distributions
- the Central Limit Theorem
- point estimation
- maximum likelihood estimation
- confidence intervals
- hypothesis testing
- simple linear regression
Check you’re ready to study Mathematical statistics with our self-assessed quiz.
Talk to an advisor if you’re still not sure if you’re ready.
Essential mathematics 1 (MST124)
and Essential mathematics 2 (MST125)
are ideal preparation to reach the mathematical competence level we recommend for studying Mathematical statistics
. However, we recommend also completing Mathematical methods, models and modelling (MST210)
or Pure mathematics (M208)
All the study materials (including the Study Guide), activities, assessment and study support are delivered online via the module website. You will also be provided with printed versions of the main units and extra exercises.
You will need
Calculator with basic mathematical functions (exp, log, etc.), but not necessarily with statistical functions.
A computing device with a browser and broadband internet access is required for this module. Any modern browser will be suitable for most computer activities. Functionality may be limited on mobile devices.
Any additional software will be provided, or is generally freely available. However, some activities may have more specific requirements. For this reason, you will need to be able to install and run additional software on a device that meets the requirements below.
A desktop or laptop computer with either:
- Windows 7 or higher
- macOS 10.7 or higher
The screen of the device must have a resolution of at least 1024 pixels horizontally and 768 pixels vertically.
To participate in our online-discussion area you will need both a microphone and speakers/headphones.
Our Skills for OU study website has further information including computing skills for study, computer security, acquiring a computer and Microsoft software offers for students.