Module MATH2761: Statistical Inference II

Department: Mathematical Sciences

MATH2761: Statistical Inference II

Prerequisites

• One of: Calculus I (Maths Hons) (MATH1081) OR Calculus I (MATH1061)
• AND
• one of: Linear Algebra I (Maths Hons) (MATH1091) OR Linear Algebra I (MATH1071)
• AND:
• Probability I (MATH1597)
• AND:
• Statistics I (MATH1617)

Corequisites

• None

Excluded Combination of Modules

• None

Aims

• To introduce the main concepts underlying statistical inference and methods.
• To develop the statistical and mathematical foundations underlying classical statistical techniques, and develop the basis for the Bayesian approach to statistics.
• To investigate and explore the frequentist and Bayesian approaches to statistical inference.

Content

• Distribution theory: random vectors, change of variables, the multivariate normal distribution, summary statistics for random vectors.
• Estimation: point estimation, bias, standard error, sampling distributions, pivotal statistics, confidence intervals, application to Normal inferences (t, chi square).
• Likelihood: multivariate likelihood, information, large-sample approximations, delta method.
• Frequentist hypothesis testing: significance and power, standard tests, optimal hypothesis testing, likelihood ratio tests, uniformly most powerful test, large-sample results.
• Exponential family: role in likelihood and Bayesian inference.
• Bayesian statistics: conjugacy, standard conjugate pairs (e.g. gamma-Poisson), Jeffreys prior.
• Bayesian inference: predictive distributions, credible intervals, estimation, large-sample approximations.
• Bayesian hypothesis testing: odds ratios, Bayes factors, interpretation of Bayes factors, model comparison.

Learning Outcomes

• Having studied the module students will have developed knowledge and understanding of the essential theory of frequentist and Bayesian statistical analysis and have an ability to use the theory to solve standard problems and make appropriate statistical inferences.
• They will have an awareness of the principles of statistical inference in both frequentist and Bayesian frameworks, including the role of probability, likelihood, and prior information.
• They will have developed an understanding of estimators, sampling distributions, confidence/credible intervals, and hypothesis tests, including likelihood ratios and optimal tests.
• They will be able to recognise and apply key distributional results, including common conjugate priors, large-sample approximations, and the exponential family.
• They will be able to interpret and compare results from frequentist and Bayesian methods, including estimation, prediction, and model comparison.
• They will be able to apply estimation and testing methods to real-world problems and interpret the results appropriately.

• By the end of the course, students will be able to formulate and analyse statistical models using both frequentist and Bayesian methods.
• They will be able to derive and compute estimates, intervals, and test statistics, and assess their properties.
• They will be able to apply likelihood-based, optimal, and Bayesian procedures to real data.
• They will have enhanced their skills at communicating statistical reasoning, results, and limitations.
• They will be able to solve a range of predictable and unpredictable problems in statistical inference.
• They will have developed foundational skills in data analysis and statistical computing using R.

• Analytical and logical reasoning in structuring and solving theoretical and quantitative problems.
• Numeracy and problem-solving in interpreting and applying quantitative methods.
• Communication of technical results and reasoning in various forms.
• Data handling and computational skills for statistical analysis, simulation, and visualisation.

Modes of Teaching, Learning and Assessment and how these contribute to the learning outcomes of the module

• Lectures demonstrate what is required to be learned and the application of the theory to practical examples.
• Problem classes show how to solve example problems in an ideal way, revealing also the thought processes behind such solutions.
• Tutorials provide active problem-solving engagement and immediate feedback to the learning process.
• Practicals develop statistical computing and practical data analysis skills.
• Summative assignments test mastery of foundational topics and provide feedback to students about their mastery of the topics.
• Formative assessments provide feedback to guide students in the correct development of their knowledge and skills in preparation for the summative assessment.
• The end-of-year examination assesses the knowledge acquired and the ability to solve predictable and unpredictable problems.

Teaching Methods and Learning Hours

Summative Assessment

Formative Assessment:

Fortnightly assignments in Weeks 2, 4, 6,8.

■ Students who do not attend monitored activities shown under Teaching Methods and Learning Hours, or who fail to complete the summative or formative assessment(s) specified above, may be subject to the Academic Progress procedures defined in the University's General Regulation V, and may be required to leave the University.