The models considered in the Introducing revdbayes vignette are
based on the assumption that observations of a (univariate) quantity of
interest can be treated as independent and identically distributed (iid)
variates. In many instances these assumptions are unrealistic. In this
vignette we consider the situation when it is not reasonable to make the
former assumption, that is, temporal dependence is present. In this
circumstance a key issue is the strength of dependence between extreme
events. Under conditions that preclude dependence between extreme events
that occur far apart in time, the effect of dependence is local in time,
resulting in a tendency for extreme to a occur in clusters. The most
common measure of the strength of local extremal dependence is the
*extremal index* \(\theta\). For
a review of theory and methods for time series extremes see Chavez-Demoulin and Davison (2012).

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.

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`

.

Attalides, N. 2015. “Threshold-Based Extreme Value
Modelling.” PhD thesis, University College London.

Chavez-Demoulin, V., and A. C. Davison. 2012. “Modelling Time
Series Extremes.” *REVSTAT-Statistical Journal* 10 (1):
109–133.

Fawcett, L., and D. Walshaw. 2012. “Estimating Return Levels from
Serially Dependent Extremes.” *Environmetrics* 23 (3):
272–283. doi:10.1002/env.2133.

Ferro, C. A. T., and J. Segers. 2003. “Inference for Clusters of
Extreme Values.” *Journal of the Royal Statistical Society:
Series B (Statistical Methodology)* 65 (2). Blackwell Publishing:
545–556. doi:10.1111/1467-9868.00401.

Holesovsky, J. P., and M. Fusek. 2020. “Estimation of the Extremal
Index Using Censored Distributions.” *Extremes* 23:
197–213. doi:10.1007/s10687-020-00374-3.

Northrop, P. J., and C. Christodoulides. 2022. *exdex: Estimation of the Extremal Index*. https://CRAN.R-project.org/package=exdex.

Süveges, M., and A. C. Davison. 2010. “Model Misspecification in
Peaks over Threshold Analysis.” *The Annals of Applied
Statistics* 4 (1): 203–221. doi:10.1214/09-AOAS292.