Asymptotic theory for the inference of the latent trawl model for extreme values

Abstract

This article develops statistical inference methods and their asymptotic theory for the latent trawl model for extremes, which captures serial dependence in the time series of exceedances above a threshold. We review two methods based on pairwise likelihood and show that they underestimate the serial dependence in the extremes. We propose two generalized method of moments procedures based on auto-covariance matching to overcome this shortcoming. Out of those four inference approaches, two are single-stage strategies while the others have two stages, and we provide central limit theorems in the sense of weakly approaching sequences of distributions for all of them. This additional flexibility ensures good behavior between the estimators and estimates of the limiting distribution.In an empirical illustration using London air pollution data, we find that the two-stage auto-covariance matching scheme yields a high-quality inference. It comprises two interpretable steps and correctly captures the serial dependence structure of extremes while performing on par with other methods in terms of marginal fit.

Limit Theorems for Trawl Processes

“…In a more applied realm, Noven et al [18] develop a latent trawl process model for extreme values and apply it to environmental time series. This work is partially extended by Courgeau and Veraart [9], who derive an asymptotic theory for inference on the latent trawl model for extreme values. Further work in the direction of extreme values has been done by Bacro et al [1], who propose hierarchical space-time modelling of asymptotically independent exceedances based on a space-time extension of the trawl process and apply their model to precipitation data.…”

“…With regards to estimation methodology for trawl processes, in addition to the aforementioned works [1,5,9,18,26], Doukhan [12] introduces spectral estimation for non-linear long range dependent discrete time trawl processes, and Shephard and Yang [25] develop likelihood inference for exponential-trawl processes.…”