Aims - This module aims to provide a grounding in the theoretical foundations of statistical inference and, in particular, to introduce the theory underlying statistical estimation and hypothesis testing. It is primarily intended for third and fourth year undergraduate students and taught postgraduate students registered on the degree programmes offered by the Department of Statistical Science, or jointly with other departments. It also serves as an optional module for students taking the mathematics and statistics stream of the Natural Sciences degree (with prerequisite: STAT0005).

Objectives - On successful completion of the module, a student should be able to: describe the principal features of, and differences between, frequentist, likelihood and Bayesian inference; define and derive a likelihood function based on data from a parametric statistical model, and describe its role in various forms of inference; define a sufficient statistic; describe and apply methods to identify a sufficient statistic; define, derive and apply frequentist criteria for evaluating and comparing estimators; describe, derive and apply lower bounds for the variance of an unbiased estimator; define and derive a maximum likelihood estimate, and observed and expected Fisher information; describe and derive the asymptotic distributions of a maximum likelihood estimator and related quantities; conduct Bayesian analyses of simple problems using conjugate prior distributions, and asymptotic Bayesian analyses of more general problems; define, derive and determine error probabilities of a test of two simple hypotheses; define and conduct a likelihood ratio test; state, prove and apply the Neyman-Pearson lemma.

Applications - The theory of statistical inference underpins statistical design, estimation and hypothesis testing. As such, it has fundamental applications to all fields in which statistical investigations are planned or data are analysed. Important areas include engineering, physical sciences and industry, medicine and biology, economics and finance, psychology and the social sciences.

Content - Frequentist and Bayesian approaches to statistical inference. Summary statistics, sampling distributions. Sufficiency, likelihood, and Fisher information. Asymptotic properties of estimators. Bayesian inference. Hypothesis testing. Likelihood ratio tests, application to linear models.

Key texts - Available from ReadingLists@UCL.