Newcastle University

Aims

To develop a knowledge and understanding of basic ideas of statistical inference. To introduce the method of maximum likelihood as a general approach to estimation.

Module Summary

This module develops further the ideas of statistical inference for which the foundations were laid in MAS1341 and MAS1342 (or MAS2341 and MAS2342). Ideas of estimation and hypothesis tests will be reviewed. General properties of maximum likelihood estimators are explored. Many of the important properties apply asymptotically, that is in large samples, and this will be shown by simulation using R. The asymptotic properties allow the construction of approximate (large sample) hypothesis tests and confidence intervals, and this will be shown. In certain important cases, such as the Normal distribution, exact (small sample) results can be obtained and these give tests and confidence intervals for the means and variances of the relevant populations. In situations where no distributional results can be used, an alternative non-parametric approach will be demonstrated for one specific case. The use of the chi-squared, t and F distributions in this will be explained. Exact inference about proportions using the binomial distribution will also be explained.

Outline Of Syllabus

Review of single parameter likelihood inference. Parameter estimation: bias, consistency, mean square error. Sufficiency. Score and information. Asymptotic distribution of maximum likelihood estimator; verification by simulation. Method of moments as an alternative estimation technique. Hypothesis tests: power, effect of sample size. Neyman Pearson Lemma. Inference for one and two normal random samples (using chi-squared, t and F distributions, with definitions by functions of random variables); inference for one and two sample (binomial) proportions. Standard error, confidence intervals and hypothesis testing (Z tests and t-tests) based on asymptotic normality. The Mann Whitney test as a non-parametric alternative to the t-test. Small sample inference based on exact methods. Goodness of fit tests and the use of normal probability plots. Use of R.

Teaching Rationale And Relationship

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. Drop-ins are used to identify and resolve specific queries raised by students and to allow students to receive individual feedback on marked work. Practicals are used both for solution of problems and work requiring extensive computation and for simulation to give insight into the ideas/methods studied. Office hours provide an opportunity for more direct contact between individual students and the lecturer.

Assessment Rationale And Relationship

A substantial formal unseen examination is appropriate for the assessment of the material in this module. Coursework assignments (approximately 4 assignments of approximately equal weight) and a class test allow the students to develop their problem solving techniques, to practise the methods learnt in the module, to assess their progress and to receive feedback; this is thus formative as well as summative assessment. The coursework assignments may be written assignments, computer based assessments or a combination of the two, and in the case of combined assessments the deadlines for the two parts will not necessarily be the same.