## Compulsory Courses

### M5MS01 Probability for Statistics

Probability and statistics are separate disciplines, although intricately linked. This course covers many of the fundamental ideas in probability theory that are crucial to statisticians. These include:

Review of axiomatic probability theory: probability spaces, distributions and their characteristics [including generating functions], conditional distributions.

Asymptotic theorems and convergence. Convergence modes and stochastic orders, convergence of transformations, laws of large numbers, central limit theorem.

Multivariate normal distribution.

Markov chains, classi cation of chains, stationary distributions, continuous-time Markov chains, (compound) Poisson processes, Brownian motion.

### M5MS02 Fundamentals of Statistical Inference

(Prof Young) Statistical inference is concerned with drawing conclusions about populations and scientific truths from data. This course gives a rigorous treatment of the fundamentals of statistical inference. Topics include:

Approaches to inference: Bayesian, Fisherian, frequentist.
Decision theory: risk, criteria for a decision rule, minimax and Bayes rules, finite decision problems.
Bayesian methods: fundamental elements, choice of prior, general form of Bayes rules. Empirical Bayes, hierarchical modelling. Predictive distributions, shrinkage and James-Stein estimation.
Data reduction and special models. Exponential families, transformation models. Sufficiency and completeness. Conditionality and ancillarity.
Key elements of frequentist theory. Hypothesis testing: Neyman-Pearson, uniformly most powerful tests, two-sided tests, conditional inference and similarity. Optimal point estimation. Confidence sets.
Introduction to likelihood theory. Asymptotic properties of maximum likelihood estimators, testing procedures. Multiparameter problems.

### M5MS03 Applied Statistics

(Dr N Kantas) The course focuses on statistical modelling when applied to realistic problems and real data. We will cover the following topics:

The Normal Linear model (estimation, residuals, residual sum of squares, goodness of fit, hypothesis testing, ANOVA, model comparison). Improving Designs and Explanatory Variables (categorical variables and multi-level regression, experimental design, random and mixed effects models). Diagnostics and Model Selection and Revision (outliers, leverage, misfit, exploratory and criterion based model selection, Box-Cox transformations, weighted regression), Generalised Linear Models (exponential family of distributions, iteratively re-weighted least squares, model selection and diagnostics).

Advanced Topics (Penalised Regression, Time series, Classification, State Space models).

### M5MS04 Computational Statistics

(Prof A Gandy) This course covers a number of computational methods that are key in modern statistics. Topics include:

Statistical Computing: R programming: data structures, programming constructs, object system, graphics.
Numerical methods: root finding, numerical integration, optimisation methods such as EM-type algorithms.
Simulation: generating random variates, Monte Carlo integration.
Simulation approaches in inference: randomisation and permutation procedures, bootstrap, MCMC, Sequential Monte Carlo/particle filtering.