Module Identifier MA36010
Module Title COMPARATIVE STATISTICAL INFERENCE
Co-ordinator Mr Alan Jones
Semester Semester 1
Pre-Requisite MA26010
Course delivery Lecture   19 x 1 hour lectures
Seminars / Tutorials   3 x 1 hour example classes
Assessment
Assessment TypeAssessment Length/DetailsProportion
Semester Exam2 Hours (written examination)  100%
Supplementary Assessment2 Hours (written examination)  100%

#### Learning outcomes

On completion of this module, a student should be able to:
• construct and interpret Classical confidence intervals for a population mean (Normal) and a probability parameter;
• set up a Bayesian analysis of the same situation;
• interpret prior and posterior distributions for parameters and construct Bayesian confidence intervals;
• explain the differences between Classical and Bayesian analyses;
• extend the ideas to other distributional families;
• explain the bases of other forms of inference such as the fiducial and likelihood approaches.

#### Brief description

This module re-examines the ideas of confidence intervals and hypothesis testing in Classical Inference and considers their interpretation more deeply. An alternative approach known as Bayesian Inference is introduced and developed, and consideration given to the formal description of prior information along with the way this information is modified in the presence of data. The concepts prior, posterior, predictive and preposterior are introduced. Applications to inferences about a (Normal) population mean and a (Binomial) probability parameter are discussed in detail, and extensions to other distributional families indicated. [The meanings and interpretations of the two approaches are discussed at length, along with the philosophical bases of other forms of statistical inference, such as the fiducial and likelihood approaches.]

#### Aims

To introduce the basic ideas and concepts of statistical inference.

#### Content

1. CLASSICAL INFERENCE Basic ideas. Point estimators, bias, mean squared error. Consistency, relative efficiency. Likelihood. The Cramer-Rao Theorem and the minimum variance bound. Efficiency. MVBUE's and their existence. Sufficiency. Rao-Blackwell Theorem and its application.
2. BAYESIAN INFERENCE Bayes' Theorem. Prior and posterior odds. Prior and posterior distributions. Conjugate families. Prior knowledge and prior ignorance. Quantification of knowledge. Predictive distributions. Preposterior distributions. Bayesian point estimation, loss functions.
3. CONFIDENCE STATEMENTS Classical: pivotal functions, confidence intervals, exact confidence intervals for discrete data. Shortest intervals.Bayesian: highest density intervals, predictive intervals.
4. HYPOTHESIS TESTING Classical: null and alternative hypotheses. Neyman Pearson theory. UMP tests. Bayesian: Bayesian decisions, risk, preposterior risk.
5. OVERVIEW Comparisons between Classical and Bayesian approaches. Ideas of fiducial and likelihood approaches.