Module Catalogue 2026/27

Pre Requisite Comment

N/A

Co Requisite Comment

N/A

Aims

This module aims to provide a mathematically rigorous treatment of the fundamental principles of
statistical inference, building upon the foundations of probability and statistics encountered
previously. Students will learn how core inferential procedures—estimation, hypothesis testing, and
interval estimation—arise from general statistical models, and will explore their theoretical properties
and asymptotic behaviour. The module emphasises the logical structure and proofs underlying
statistical methods, preparing students for advanced study or research in statistics, econometrics,
data science, or related mathematical fields.

Outline Of Syllabus

Statistical models: sufficiency, completeness, exponential families, chi-square, t-distributions and Fdistributions. Unbiased estimation: Rao–Blackwell theorem, uniformly minimum variance unbiased
estimators, Lehmann–Scheffé theorem, Cramér–Rao lower bound, Gauss-Markov theorem.
Hypothesis testing: Neyman–Pearson lemma, uniformly most powerful tests, likelihood-ratio test.
Confidence sets. Large sample theory: asymptotics of estimators, and related confidence sets and
hypothesis tests. Covariance estimation.

Intended Knowledge Outcomes

By the completion of the module, students should be able to:
- define and interpret formal statistical models, including sufficiency and completeness;
- derive and evaluate properties of estimators using unbiasedness, variance, and efficiency criteria;
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- explain and apply fundamental optimality results such as the Rao–Blackwell and Lehmann–Scheffé
theorems and the Cramér–Rao bound;
- derive and interpret classical tests (likelihood-ratio, Wald, score) and their asymptotic properties;
- construct and analyse interval estimates and confidence regions from both finite-sample and largesample perspectives

Intended Skill Outcomes

Students completing the module will be able to:
- formulate and solve statistical inference problems using rigorous mathematical reasoning;
- derive and justify properties of estimators and tests from first principles;
- apply asymptotic methods to approximate and analyse inferential procedures;
- critically evaluate the efficiency and optimality of competing statistical methods;
- translate theoretical results into practical inferential strategies in applied settings;
- communicate precise mathematical arguments clearly, using correct statistical notation and logic;
- use computational tools to illustrate asymptotic behaviour empirically.
Students will develop skills across the cognitive domain (Bloom's taxonomy, 2001 revised edition):
remember, understand, apply, analyse, evaluate and create

Teaching Rationale And Relationship

The teaching methods are appropriate to allow students to develop a wide range of skills, from understanding basic concepts and facts to higher-order thinking.

Lectures are used for the delivery of theory and explanation of methods, illustrated with examples, and for giving general feedback on marked work. Problem Classes are used to help develop the students’ abilities at applying the theory to solving problems.

Assessment Rationale And Relationship

A substantial formal unseen examination is appropriate for the assessment of the material in this module. The format of the examination will enable students to reliably demonstrate their own knowledge, understanding and application of learning outcomes. The assurance of academic integrity forms a necessary part of programme accreditation.

Exam problems may require a synthesis of concepts and strategies from different sections, while they may have more than one ways for solution. The examination time allows the students to test different strategies, work out examples and gather evidence for deciding on an effective strategy, while carefully articulating their ideas and explicitly citing the theory they are using.

The in course assessment allows students to develop their problem solving techniques, to practise the methods learnt in the module, to assess their progress and to receive feedback; the assessment therefore has a secondary formative purpose as well as a primary summative purpose.

General Notes

N/A