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 compare the frequentist and Bayesian approaches to statistical inference.

Content

• Distribution theory: random vectors, change of variables, the multivariate normal distribution.
• Multivariate statistics and estimation: summary statistics for random vectors, application to Normal inferences (t, chi sq), estimation, confidence intervals.
• Likelihood: multivariate likelihood, information, large-sample approximations, delta method.
• Bayesian statistics: conjugacy, gamma-Poisson, normal-gamma, Jeffreys prior.
• Bayesian inference: predictive distributions, large-sample approximations.
• Exponential family: role in likelihood and Bayesian inference.
• Frequentist hypothesis testing: optimal hypothesis testing, likelihood ratio tests, large-sample results; two sample comparisons; chi-square tests.
• Bayesian hypothesis testing: Bayes factors, model comparison.

Learning Outcomes

• By the end of the module students will:
• Be able to solve a range of predictable and unpredictable problems in statistical inference.
• Have an awareness of the abstract theoretical concepts underlying statistics.
• Have a knowledge and understanding of fundamental theories of these subjects demonstrated through one or more of the following topic areas: statistical inference, frequentist and likelihood methods, Bayesian statistics.

• Students will have the ability to undertake and defend the use of alternative mathematical skills in the following areas with minimal guidance: statistical modelling, statistical analysis of unseen data sets.
• Students will have enhanced mathematical skills in the following areas: Statistical computing with R.

• Students will have basic mathematical skills in the following areas: problem solving, statistical modelling, data analysis, statistical computation.

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 consolidate the studied material, explore theoretical ideas in practice, enhance practical understanding, and develop practical data analysis skills.
• Formative assessments provide feedback to guide students in the correct development of their knowledge and skills in preparation for the summative assessment.
• Summative assignments test achievement of learning outcomes and provide feedback to students about their mastery of the topics.
• 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.

■ Attendance at all activities marked with this symbol will be monitored. Students who fail to attend these activities, or to complete the summative or formative assessment specified above, will be subject to the procedures defined in the University's General Regulation V, and may be required to leave the University