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 so-called latent trawl model for extremes, which captures serial dependence in the time series of exceedances above a threshold. We review two inference methods based on the pairwise likelihood (PL) approach, a classical single-stage and a two-stage estimation; in the latter, we first estimate the marginal parameters, then the parameters describing the serial dependence. We derive the asymptotic theory in both cases and find that the asymptotic behaviour and goodness-of-fit are similar to the two-stage method being slightly preferred in our empirical work. However, we also show that PL-based approaches tend to underestimate the serial dependence in the extremes, and we propose a generalised method of moments procedure (both a single-stage and a two-stage procedure) based on autocovariance matching to overcome this shortcoming. We prove the strong mixing property of our model and provide central limit theorems for the four inference approaches. For single-stage approaches, the convergence is shown in the sense of weakly approaching sequences of distributions for strongly mixing sequences. For the two-stage strategies, we provide an asymptotic variance for the serial parameter which takes the estimation error arising from the first stage into account. In an empirical illustration using London air pollution data featuring six pollutants, we find that the two-stage autocovariance matching scheme seems to be performing best.