## MATH3723 Statistical Theory

### 15 creditsClass Size: 100

Module manager: Dr R. G. Aykroyd
Email: robert@maths.leeds.ac.uk

Taught: Semester 2 (Jan to Jun) View Timetable

Year running 2008/09

### Pre-requisites

 MATH2715 Statistical Methods

This module is approved as an Elective

### 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

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

 Delivery type Number Length hours Student hours Example Class 7 1.00 7.00 Lecture 26 1.00 26.00 Practical 3 1.00 3.00 Private study hours 114.00 Total Contact hours 36.00 Total hours (100hr per 10 credits) 150.00

### Methods of assessment

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

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

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

Exams
 Exam type Exam duration % of formal assessment Standard exam (closed essays, MCQs etc) 3 hr 80.00 Total percentage (Assessment Exams) 80.00

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

The reading list is available from the Library website

Last updated: 19/07/2010

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