STAT3001- Mathematical Statistics

Aims of course

To provide a grounding in the theoretical foundations of statistical inference and, in particular, to introduce the theory underlying statistical estimation and hypothesis testing, and to provide the theory underlying the methods taught in the first and second years of degree courses offered by the Department of Statistical Science or jointly with other Departments.

Objectives of course

On successful completion of the course, a student should be able to

describe the principal features of, and differences between, frequentist, likelihood and Bayesian inference;

define and derive the likelihood function based on data from a parametric statistical model, and describe its role in various forms of inference;

define a sufficient statistic; describe, calculate and apply methods of identifying 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 the maximum likelihood estimator, and the observed and expected information;

describe, derive and apply the asymptotic distributions of the 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 apply the error probabilities of a test between two simple hypotheses; define and conduct a likelihood ratio test; state 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.