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The extremal index of a block-cluster process is the reciprocal cluster size

Source: R/ledger_scan2.R
morie_extremal_index_clusters.Rd

With n blocks of m days and cluster-start probability \(\tau/(nm)\), the no-exceedance probability is \((1-\tau/(nm))^n \to e^{-\tau/m}\) while the independent comparison gives \(e^{-\tau}\) (Research.P10.ExtremalIndex.cluster_tendsto, iid_tendsto), so \(\theta = 1/m\) (extremal_index).

Usage

morie_extremal_index_clusters(m, tau, n = c(10, 100, 1000))

Arguments

m

Cluster (block) length.

tau

Expected number of cluster starts.

n

Numbers of blocks at which to evaluate.

Value

A list: theta, limit_cluster, limit_iid and a data frame of finite-n survival probabilities.

References

Coles S (2001). An Introduction to Statistical Modeling of Extreme Values, Theorem 5.2. Springer.

Examples

morie_extremal_index_clusters(m = 4, tau = 2, n = c(10, 100, 1000))
#> $theta
#> [1] 0.25
#> 
#> $limit_cluster
#> [1] 0.6065307
#> 
#> $limit_iid
#> [1] 0.1353353
#> 
#> $table
#>      n   cluster       iid
#> 1   10 0.5987369 0.1285122
#> 2  100 0.6057704 0.1346580
#> 3 1000 0.6064548 0.1352676
#> 
#> attr(,"theorem")
#> [1] "Research.P10.ExtremalIndex.extremal_index"