Statistics: Theory and Practice

BUEM003S6 (30 credits)

Aims

This module will provide you with an overview of the main theoretical ideas that fundamentally underpin practices in day-to-day routine, or innovative, uses of the theory of statistics, and its applications. It follows on from the “Probability and Statistics” module to give a more in-depth understanding.

For those focusing on statistics, the module will provide some of the necessary pre-requisite knowledge for modules found on final year undergraduate and Masters programmes in Statistics. It can also serve as a more complete ‘stopping-off’ point for mathematicians who wish to push their statistical knowledge beyond an introductory level.

On the computational side, by studying this module you will gain a working knowledge of a high-level statistical programming language, such as S-PLUS.

Teaching and Assessment

Teaching for this module will take place throughout the year, with eight evenings in each of the Autumn and Spring Terms and two evenings of revision and consolidation in the Summer Term.

Of the final course mark, 80% is based on a three-hour exam in June and the other 20% is from assessed coursework.

Coursework will consist of short, problem based assignments. You will have around three weeks to complete each assignment. Some statistical computing may be involved.

The examination in the Summer Term has two sections. Section A (worth 40%) consists of compulsory short questions. Section B (also worth 40%) contains several longer questions of which you must answer two.

Syllabus

• Introduction to S-PLUS;
• Design and analysis of simple experiments: to include (but not necessarily restricted to) one and two-way completely randomized designs;
• Joint distribution of several variables and likelihood functions: with special emphasis on the effects of the variables being i) mutually independent, or ii) drawn from the same distribution, or both; multivariate normal distribution, with particular attention to the bivariate normal;
• Further distribution theory: sums of independent Chi-squared random variables, F-distributions, and how they relate to analysis of variance techniques;
• Introduction to the theory of statistical inference: likelihood, sufficiency, estimation, hypothesis testing;
• Simple and multiple linear regression.

Learning Outcomes

On successful completion of this module you will be expected to be able to:

• set-up and carry out (or supervise the statistical implementation of) a simple designed experiment which allows for the testing of the influence of certain factors using ANOVA techniques;
• collate and analyze data arising from a simple designed experiment within a package (like S-PLUS), and draw appropriate conclusions;
• specify and recognise the joint distribution of several random variables given appropriate assumptions on the marginal distributions and their dependence structure;
• specify and recognise the multivariate normal distribution, and some of its important properties: particularly in relation to specific graphical properties of the bivariate normal distribution;
• derive key results pertaining to the Chi-squared and Fisher distributions, and relate these to  the theoretical basis for the ANOVA technique;
• formulate and derive maximum likelihood estimators (and appreciate how these differ from those based on the method of moments);
• determine whether a statistic is sufficient for a given parameter;
• appreciate the theoretical underpinning behind hypothesis testing and have an acknowledgment of how hypothesis tests are carried out across several different paradigms;
• determine whether a given data set is amenable to analysis using multiple linear regression;
• import or enter data into a statistical package, like S-PLUS, and perform multiple linear regression by principally using command line functions (rather than menu-driven GUI operations);
• interpret and draw conclusions from a statistical analysis, and present these conclusions so that they can  either i) be well understood by a statistician, or ii) be accessible (in a non-misleading way) to the intelligent lay-person/non-statistician (who may be involved in policy development).