Asymptotic Theory for the Inference of the Latent Trawl Model for Extreme Values

We extend the study of an established parametric latent model for extreme values which captures serial dependence in the exceedances above a threshold using so-called trawl processes - a family of stationary and infinitely divisible random processes. We prove that the observed process is strongly mixing and we introduce a two-step Generalised Method of Moments (GMM) inference approach which is partly based on auto-covariance matching and a likelihood maximisation leveraging non-extreme values. We prove two novel central limit theorems for both the pairwise likelihood estimators and the above-mentioned GMM method in the sense of weakly approaching sequences of distributions for strongly mixing sequences. We remark that the Kullback-Leibler divergence can be minimised with non-neighbouring parameter values which leads to the introduction of a different parametrisation compared to that of the original model formulation. Finally, we apply those two inference approaches to a London hourly air pollution dataset featuring six pollutants: the GMM approach is superior in matching the autocorrelation function and still provides a good marginal fit whilst being faster by an order of magnitude at obtaining confidence bounds