Standard statistical modelling approaches are typically biased in tail behaviour estimation due to the modelling parameters being driven by the main body of the distribution. As such, the area of extreme value theory provides an asymptotically justified approach to model the probabilistic behaviour of rare events. Extreme value methods are used in a wide range of applications, for example they are often used for modelling storm surges in hydrology and heatwaves in medical statistics.

Most of our work has been motivated by the well-established negative impacts of poor air quality, more specifically, extreme episodes of ozone concentrations on human health. Our focus has been on a multivariate ozone dataset, which shows complex temporal and spatial trends. We begin by proposing the use of extreme value theory to validate numerical process-based model forecasts. Next, we examine the temporal dependence structure and propose a new measure to determine the order of an extreme Markov process. Then we present novel applications of the extreme Markov processes, simulating extremal chain behaviour of key health related scenarios and short-lead-time forecasts of extreme events. Finally, we present a spatial model to evaluate the risk of extreme events of ozone across Great Britain.