---
title: "Inference for the extremal index using threshold interexceedance times"
author: "Paul J. Northrop"
date: "`r Sys.Date()`"
output: 
  rmarkdown::html_vignette:
    toc: true
vignette: >
  %\VignetteIndexEntry{Inference for the extremal index using threshold interexceedance times}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
bibliography: revdbayes.bib
csl: taylor-and-francis-chicago-author-date.csl
---

```{r, include = FALSE}
knitr::opts_chunk$set(comment = "#>", collapse = TRUE)
```

The models considered in the [Introducing revdbayes](revdbayes-a-vignette.html) 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 @CDD2012.

## $K$-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 @SD2010 extends the model of @FS2003 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 @SD2010. 

A related approach [@HF2020], 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 \code{D} units are left-censored and contribute to a log-likelihood only the information that they are $\leq D$. 

The **exdex** package [@exdex] 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 @FW2012.  For the sake of illustration we use the default setting, $K = 1$, which may not be appropriate for these data.  See @SD2010 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 @Attalides2015 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.

```{r, fig.width = 5, fig.align='center'}
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

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