MA3501: Elements of Mathematical Statistics

School Cardiff School of Mathematics
Department Code MATHS
Module Code MA3501
External Subject Code 100406
Number of Credits 10
Level L6
Language of Delivery English
Module Leader Professor Jonathan Gillard
Semester Autumn Semester
Academic Year 2014/5

Outline Description of Module

Is the arithmetic average the best way of estimating the mean of a probability distribution? Is Student's t-test the best way of testing null hypotheses about the mean? The answers to these questions are assumed to be yes in elementary statistics, this module shows that there is a firm mathematical basis for this assumption.

The first part of the module is a study of methods of estimation of parameters of probability distributions.Brief comparisons are made of maximum likelihood estimation, the method of moments approach and Bayesian inference. The properties that are desirable in estimators are identified, and by using a series of results it is shown that under fairly general conditions maximum likelihood estimators have optimal properties. It is even possible, for some statistical estimation problems, to identify estimators that are unbiased (correct on average, in the long run) and have a smaller theoretical variance than any other unbiased estimator.

The second part of the module covers the testing of hypotheses in statistics. It is shown how optimal statistical tests can be devised and a link is made with maximum likelihood.

The module presents a coherent view of estimation and hypothesis testing in a firm theoretical framework.

Prerequisite Modules: MA2500 Foundations of Probability and Statistics

On completion of the module a student should be able to

  • Understand the theoretical framework that underpins classical statistical theory
  • Demonstrate a sound knowledge of mathematical manipulation in statistical theory
  • Find optimum parameter estimators for a range of common probability models
  • Identify optimum test procedures both for simple and composite hypotheses

How the module will be delivered

27 - 50 minute lectures

Some handouts will be provided in hard copy or via Learning Central, but students will be expected to take notes of lectures.

Students are also expected to undertake at least 50 hours private study including preparation of solutions to given exercises.

Skills that will be practised and developed

Skills:

An ability to understand the salient features of estimation and hypothesis testing in mathematical statistics.

Demonstrate an ability to perform mathematical manipulations and use the results in statistical problems.

Transferable Skills:

An ability to understand a logical presentation of argument, and put this into the context of justifying the classical approach to statistical analysis.

How the module will be assessed

Formative assessment is carried out by means of coursework assignments, generally taking the form of a class test and coursework.  Feedback to students on solutions to selected exercises is also provided. This, together with feedback on performance in the class test indicates to students their progress towards achieving the learning outcomes of the module.  

The class test is also used for summative assessment. It allows students to demonstrate their progress in attaining a level of achievement appropriate at that stage in the module.

The major component of summative assessment is the written examination at the end of the module.  This gives students the opportunity to demonstrate their overall achievement of learning outcomes.  It also allows them to give evidence of the higher levels of knowledge and understanding required for above average marks.

The examination paper has a choice of three from four equally weighted questions.

Assessment Breakdown

Type % Title Duration(hrs)
Exam - Autumn Semester 85 Elements Of Mathematical Statistics 2
Class Test 15 Class Test N/A

Syllabus content

  • maximum likelihood estimation and the method of moments
  • minimum variance unbiased estimation and the Cramer-Rao inequality
  • sufficient statistics and the factorization theorem
  • the Rao-Blackwell theorem
  • completeness and the exponential family of distributions
  • Simple and composite hypothesis testing, critical region, significance level and power
  • the Neyman-Pearson lemma
  • uniformly most powerful tests
  • likelihood ratio tests

Essential Reading and Resource List

Probability and Statistics, DeGroot, M. H., Addison-Wesley

Statistical Inference, Casella, G., & Berger, R. L., Duxbury

Background Reading and Resource List

Not applicable.


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