Designing protection mechanisms to safeguard against ever-changing environmental processes requires the accurate estimation of future extreme hazards combined with reliable measures of their uncertainty. This thesis uses the peaks-over-threshold (POT) framework of extreme value modelling to provide high-quality tail inferences. The fundamental problem with POT analyses is selecting the threshold to characterise extreme values, with inferences sensitive to the choice. We develop improved methodology for threshold selection and for quantifying the uncertainty of this selection in tail inferences.

Even for independent and identically distributed data, threshold selection is a difficult task. Existing approaches can be subjective, sensitive to tuning parameters, or rely on asymptotics, resulting in suboptimal performance in practice. We develop a novel, objective, and effective methodology to automate threshold selection and propagate its uncertainty through to high quantile inference. We extend these approaches to handle non-identically distributed data with smooth generalised additive model formulations for the threshold and excess distribution parameters.

We adapt the methodology to address requirements for important applications. For coastal flooding, we focus the goodness-of-fit metric on the upper tail to ensure that the selected thresholds lead to accurate fitting to the most extreme observations. For modelling induced earthquakes, we use geophysical covariates regarding the measurement network and stresses induced by gas extraction, to form spatio-temporal threshold and excess distribution parameter functions. We develop a powerful estimator for a key quantity in seismicity modelling, the magnitude of completion. This estimator reduces the uncertainty and provides stronger evidence of a finite upper-endpoint than in previous research. We expand our uncertainty algorithms to account for the unknown model-covariate formulation and incorporate this uncertainty in inference for future endpoint summaries and quantile estimates relevant for design standards. Our methods have much wider applicability for inference for other induced seismicity contexts and wider environmental hazards.