exdex
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). Two versions of this
type of approach are provided: the iterated weight least squares
approach of Süveges (2007)
and the K-gaps model of Süveges and Davison
(2010). For the K-gaps model 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
interexceedance times. A graphical diagnostic for the threshold level
and the runs parameter K is 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:
#> [1] 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.