What does exdex do?

The extremal indexθ is a measure of the degree of local dependence in the extremes of a stationary process. Theexdex package performs frequentist inference aboutθ 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, followingNorthrop, 2015 andBerghaus 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 ofSüveges (2007), theK-gaps model ofSüveges and Davison (2010) and a similar approach ofHolesovsky and Fusek (2020) that we refer to as D-gaps. For theK-gaps andD-gaps models theexdex 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 parametersK andD 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 bychoose_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  0.2542summary(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 estimateθ using theK-gaps model. The thresholdu and runs parameterK were chosen using the graphical diagnostic provided bychoose_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#> 0.1758summary(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

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