Inference for the extremal index using threshold interexceedance times

\(K\)-gaps and \(D\)-gaps models

The extremal index has several interpretations and leading to different models/methods by which inferences about \(\theta\) can be made. Here we consider a model based on the behaviour of occurrences of exceedances of a high threshold. The \(K\)-gaps model of Süveges and Davison (2010) extends the model of Ferro and Segers (2003) by incorporating a run length parameter \(K\). Under this model threshold inter-exceedance times not larger than \(K\) are part of the same cluster and other inter-exceedance times have an exponential distribution with rate parameter \(\theta\). Thus, \(\theta\) has dual role as the probability that a process leaves one cluster of threshold exceedances and as the reciprocal of the mean time until the process enters the next cluster. For details see Süveges and Davison (2010).

A related approach (Holesovsky and Fusek 2020), which we will call \(D\)-gaps, involves a censoring parameter \(D\). This estimator is similar to the \(K\)-gaps estimator, but the treatment of small inter-exceedance times is different. Threshold inter-exceedances times that are not larger than units are left-censored and contribute to a log-likelihood only the information that they are \(\leq D\).

The exdex package (Northrop and Christodoulides 2022) packages provides functions for performing maximum likelihood about \(\theta\) under the \(K\)-gaps and \(D\)-gaps models.

Bayesian Inference

We use the newlyn dataset, which is analysed in Fawcett and Walshaw (2012). For the sake of illustration we use the default setting, \(K = 1\), which may not be appropriate for these data. See Süveges and Davison (2010) for discussion of this issue and for methodology to inform the choice of \(K\).

The function kgaps_post simulates a random sample from the posterior distribution of \(\theta\) based on a Beta(\(\alpha, \beta\)) prior. The user can choose the values of \(\alpha\) and \(\beta\). The default setting is \(\alpha = \beta = 1\), that is, a U(0,1) prior for \(\theta\). See Attalides (2015) for further information and for a methods for selecting the value of the threshold in this situation. The plot produced below is is histogram of the sample from the posterior with the posterior density superimposed.

library(revdbayes)
# Set a threshold at the 90% quantile
thresh <- quantile(newlyn, probs = 0.90)
postsim <- kgaps_post(newlyn, thresh, k = 1)
plot(postsim, xlab = expression(theta))

The function dgaps_post has the same functionality as kgaps-post, except that the argument k is replaced by an argument D.

References