## Estimation of the Extremal Index

### What does exdex do?

The extremal index θ is a measure of the degree of local dependence in the extremes of a stationary process. The exdex package performs frequentist inference about θ using two types of methodology.

One type (Northrop, 2015) is based on a model that relates the distribution of block maxima to the marginal distribution of the data, leading to a semiparametric maxima estimator. Two versions of this type of estimator are provided, following Northrop, 2015 and Berghaus and Bücher, 2018. A slightly modified version of the latter is also provided. Estimates are produced using both disjoint and sliding block maxima, the latter providing greater precision of estimation. A graphical block size diagnostic is provided.

The other type of methodology uses a model for the distribution of threshold inter-exceedance times (Ferro and Segers, 2003). Three versions of this type of approach are provided: the iterated weight least squares approach of Süveges (2007), the K-gaps model of Süveges and Davison (2010) and a similar approach of Holesovsky and Fusek (2020) that we refer to as D-gaps. For the K-gaps and D-gaps models the exdex package allows missing values in the data, can accommodate independent subsets of data, such as monthly or seasonal time series from different years, and can incorporate information from censored inter-exceedance times. Graphical diagnostics for the threshold level and the respective tuning parameters K and D are provided.

### A simple example

The following code estimates the extremal index using the semiparametric maxima estimators, for an example dataset containing a time series of sea surges measured at Newlyn, Cornwall, UK over the period 1971-1976. The block size of 20 was chosen using a graphical diagnostic provided by choose_b().

library(exdex)
theta <- spm(newlyn, 20)
theta
#>
#> Call:
#> spm(data = newlyn, b = 20)
#>
#> Estimates of the extremal index theta:
#>           N2015   BB2018  BB2018b
#> sliding   0.2392  0.3078  0.2578
#> disjoint  0.2350  0.3042  0.2542
summary(theta)
#>
#> Call:
#> spm(data = newlyn, b = 20)
#>
#>                   Estimate Std. Error Bias adj.
#> N2015, sliding      0.2392    0.01990  0.003317
#> BB2018, sliding     0.3078    0.01642  0.003026
#> BB2018b, sliding    0.2578    0.01642  0.053030
#> N2015, disjoint     0.2350    0.02222  0.003726
#> BB2018, disjoint    0.3042    0.02101  0.003571
#> BB2018b, disjoint   0.2542    0.02101  0.053570

Now we estimate θ using the K-gaps model. The threshold u and runs parameter K were chosen using the graphical diagnostic provided by choose_uk().

u <- quantile(newlyn, probs = 0.60)
theta <- kgaps(newlyn, u, k = 2)
theta
#>
#> Call:
#> kgaps(data = newlyn, u = u, k = 2)
#>
#> Estimate of the extremal index theta:
#>  theta
#> 0.1758
summary(theta)
#>
#> Call:
#> kgaps(data = newlyn, u = u, k = 2)
#>
#>       Estimate Std. Error
#> theta   0.1758   0.009211

### Installation

To get the current released version from CRAN:

install.packages("exdex")

### Vignette

See vignette("exdex-vignette", package = "exdex") for an overview of the package.