Introductory description

To provide a systematic introduction to major ideas of statistical inference, with an emphasis on likelihood methods of modelling, estimation, and testing.

Pre-requisites:

• ST228 Mathematical Methods for Statistics and Probability, and
• ST229 Probability for Mathematical Statistics.

This module is core for students with their home department in Statistics.

It is not available to other students, for whom ST232/ST233 (Introduction to Mathematical Statistics) is provided as an alternative.

Leads To: many ST3 and ST4 modules.

Module web page

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

This module continues the systematic study of the theory of mathematical statistics.

1. The notion of a parametrized statistical model for data.
2. The definition of likelihood and examples of using it to compare possible parameter values.
3. Parameter estimates and in particular maximum likelihood estimates. Examples including estimated means and variances for Gaussian variables.
4. The notion of estimator and its sampling distribution; interval estimation. Examples of calculating sampling distributions.
5. Frequentist approaches to model comparison and selection; hypothesis testing and p-values.
6. The Bayesian approach to statistical inference; concepts of prior, posterior, conjugacy.
7. Bayesian estimators and credible intervals, posterior predictive checking.
8. Bayesian model selection.

Learning outcomes

By the end of the module, students should be able to:

• describe the main notions of statistical inference including a (parametrized) statistical model, an estimator and its sampling distribution, and hypothesis tests; and to understand their uses and limitations.
• calculate maximum likelihood estimators in a variety of examples.
• use likelihood ratios to compare models and to design hypothesis tests in a variety of examples.
• derive properties of sampling distributions of estimators in a variety of examples.
• compare models using methods of model selection in both a frequentist and Bayesian setting.
• communicate solutions to problems accurately with structured and coherent arguments.

Indicative reading list

Main reference book for this module:

1. Statistical Inference, G. Casella and R. L. Berger.
2. Bayesian Data Analysis, A. Gelman et al.

Further possibilities for reference:
3. Probability and statistics by example: 1: Basic probability and statistics, Y. M. Suhov, M. Kelbert.
4. Introductory Statistics, S.M. Ross.
5. Introduction to Probability and Statistics for Engineers and Scientists, S. M. Ross.

View reading list on Talis Aspire

Subject specific skills

Select and apply appropriate mathematical and/or statistical techniques.

Create structured and coherent arguments communicating them in written form.

Construct and develop logical mathematical arguments with clear identification of assumptions and conclusions.

Transferable skills

Written communication skills: Students complete written assessments that require precise and unambiguous communication in the manner and style expected in mathematical sciences.

Verbal communication skills: Students are encouraged to discuss and debate formative assessment and lecture material within small-group tutorials sessions. Students can continually discuss specific aspects of the module with the module leader. This is facilitated by statistics staff office hours.

Problem-solving skills: The module requires students to solve problems with complex solutions and this requirement is embedded in the module’s assessment.