MATH2761: Statistical Inference II

• 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)

• None

• None

• 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.

• 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.

Subject-specific Knowledge:

• 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.

Subject-specific Skills:

• 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.

Key Skills:

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

• 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.

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