MTH4260 · Statistics of stochastic processes

Overview

Many practical experiments involve repeated measurements made over a period of time, where the individuals or systems being observed are evolving during the study period. Examples of this kind of data arise in signal processing, financial modelling and mathematical biology. For experiments of this kind, standard statistical methods that assume data points are independent and identically distributed (iid) are of limited value, due to dependencies among measurements. This unit will introduce statistical methods for such processes. Topics: Review of fundamental statistics: their distributions, properties and limitations; Stochastic processes: Markov, ARMA, Stationary and diffusion processes; Likelihood models, Graphical models, Bayesian models; Decision theory, Likelihood ratio tests, Bayesian model comparison; Sufficient statistics, Maximum likelihood estimation, Bayesian estimation; Exponential families; Convergence of random variables and measures; Properties of estimators: bias, consistency, efficiency; Laws of large numbers and ergodic theorems, Central limit theorems; Statistics for stationary processes; Statistics for ARMA processes; Statistics for diffusion processes

Areas of study: Applied mathematics Financial and insurance mathematics Mathematical statistics Mathematics Pure mathematics

Offerings

Assessment

The Handbook lists an examination for this unit.

Assessment details may change. Please refer to the assessment information in Moodle closer to the start of the teaching period.

Requisites

The Handbook lists no prerequisite, corequisite or prohibition for this unit.

Learning outcomes

1. Apply likelihood-based methods to construct, estimate, and compare models for stochastic processes, including maximum likelihood and Bayesian approaches;
2. Analyse and evaluate the properties of estimators, including bias, consistency, efficiency, and asymptotic behaviour, with applications to stationary, ARMA, and diffusion processes;
3. Interpret and apply statistical results from stochastic process models to real-world problems in areas such as signal processing, finance, and mathematical biology;
4. Communicate statistical reasoning and results effectively, both orally and in writing, and collaborate in small groups to solve problems in the statistics of stochastic processes;
5. Extend and deepen understanding of statistical methods for stochastic processes through advanced model synthesis, rigorous analysis, and independent application to complex or novel datasets.

Workload

Three 1-hour seminars One 2-hour applied class (in weeks 2-12) and 7 hours of independent study per week

Student experience, not official rules. Nothing here changes what the Handbook says.