# 2022/23 Undergraduate Module Catalogue

## MATH3723 Statistical Theory

### 15 creditsClass Size: 80

Module manager: Dr Leonid Bogachev
Email: L.V.Bogachev@leeds.ac.uk

Taught: Semester 2 (Jan to Jun) View Timetable

Year running 2022/23

### Pre-requisites

 MATH2715 Statistical Methods

This module is not approved as a discovery module

### Module summary

This module gives a general unified theory of the problems of estimation and hypotheses testing. It covers Bayesian inference, making comparisons with classical inference.

### Objectives

- To give a general unified theory of the problems of estimation and hypotheses testing.
- To study Bayesian inference, making comparisons with classical inference.

By the end of this module, students should be able to:
a) write down the likelihood given a parametric model, produce different estimators, and be able to assess their efficiency;
b) improve, if possible, upon these estimators via the Rao-Blackwell Theorem;
c) evaluate the best critical region for a given parametric model and evaluate most powerful tests of simple hypotheses;
d) perform likelihood ratio tests for hypotheses on several parameters;
e) discuss the use of prior distributions and Bayes Theorem;
f) discuss Bayesian analogues of classical inferential procedures - point and interval estimation, prediction and hypothesis tests;
g) obtain the Bayes solution to some basic decision problems.

### Syllabus

You have met the problems of estimation and hypothesis testing in your first and second years. Most of the work has been intuitive so far, but the key questions are not unresolved, such as which average should we use; mean, mode or median? Assuming underlying models, the module will give a general theory to answer such questions in a unified way. Also, in testing of hypotheses, one might wonder if there is an optimal way to obtain the tests so that Type I and Type II errors are controlled in some way. The Bayesian approach to inference is strikingly different from the classical view covered in the first part of the course. Bayesian inference essentially incorporates additional and usually subjective assumptions about the underlying population into the inference procedure, over and above the sample data. Over the past 50 years it has sparked off fierce controversy between its proponents and the classical statisticians. We shall examine the controversy and compare the two schools of thought.

Topics included:
- Point Estimation.
- Unbiased estimation.
- Unbiased linear estimation, best unbiased linear estimator.
- Maximum likelihood estimators: small sample properties, large sample properties.
- Newton-Raphson, best unbiased estimators.
- Cramer-Rao inequality, sufficiency, completeness, Neyman-Fisher Criterion, Rao-Blackwell Theorem, minimum variance unbiased estimators, mean square error.
- Hypotheses testing: Two types of error, Neyman-Pearson Lemma and its applications, similar tests, uniformly most powerful tests, likelihood ratio test and its applications.
- Wilks Theorem.
- Bayesian analysis: prior distributions, conjugate priors, interpretations of priors.
- Bayesian analogues of classical procedures - point and interval estimates, prediction, tests.
- Bayesian decision theory.
- Comparison of classical and Bayesian inference.

### Teaching methods

 Delivery type Number Length hours Student hours Lecture 33 1.00 33.00 Practical 1 2.00 2.00 Private study hours 115.00 Total Contact hours 35.00 Total hours (100hr per 10 credits) 150.00

### Private study

Studying and revising of course material.
Completing of assignments and assessments.

### Opportunities for Formative Feedback

Regular problem solving assignments

### Methods of assessment

Coursework
 Assessment type Notes % of formal assessment Practical . 20.00 Total percentage (Assessment Coursework) 20.00

There is no resit available for the coursework component of this module. If the module is failed, the coursework mark will be carried forward and added to the resit exam mark with the same weighting as listed above.

Exams
 Exam type Exam duration % of formal assessment Open Book exam 2 hr 30 mins 80.00 Total percentage (Assessment Exams) 80.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated