ExtremalIndex {AoE}R Documentation

Intervals Estimator for the Extremal Index

Description

This function is an implementation of the intervals estimator of Ferro and Segers (2003).

Usage

ExtremalIndex(data, threshold = NULL, k = NULL, plot = TRUE)

Arguments

data A numeric vector containing the data.
threshold The threshold above which excesses will be counted.
k Alternatively, the threshold may be specified as the (k+1)th largest order statistic, so that in the abscence of ties there are exactly k excesses. If threshold is provided, then k is ignored.
plot If TRUE (the default), the result will be plotted.

Details

The extremal index 0 <= theta <= 1 of a strictly stationary time series is a measure for tendency of high-threshold excesses to appear in clusters (Leadbetter, 1988). The extremal index can be thought of as the reciprocal of the mean of the number of excesses in such a cluster. In particular, theta = 1 corresponds to no clustering, that is, asymptotic independence.

For a time series X_1, ..., X_n and a threshold u, let

1 <= S_1 < ... < S_N <= n

be the ordered collection of random time instants t = 1, ..., n such that X_t > u. The inter-arrival times are defined as

T_i = S_{i+1} - S_i

for i = 1, ..., N-1. Provided max_i T_i >= 3, the intervals estimator for the extremal index is defined as the minimum of 1 and

2 * (sum_i (T_i-1))^2 / sum_i ((T_i - 1) * (T_i - 2))

If the assumptions motivating the estimator are fulfilled, then the distribution of the inter-arrival times is a mixture of a point mass at zero and an exponential distribution. In an exponential quantile-quantile plot, the interarrival times should follow a broken-stick model, the location of the knot being determined by the extremal index. All inter-arrival times to the left of this knot correspond to intra-cluster inter-arrival times, that is, inter-arrival times within clusters of excesses; similarly, all inter-arrival times to the right of this knot correspond to inter-cluster inter-arrival times, that is, inter-arrival times between cluster of excesses. This heuristic also gives an automated way of partitioning high-threshold excesses into clusters.

If plot = TRUE, then two plots are being shown:

Value

The extremal index estimate.

References

Ferro, C.A.T. and Segers, J. (2003). Inference for clusters of extreme values. Journal of the Royal Statistical Society, Series B 65, 545-556.

Leadbetter, M.R. (1983). Extremes and local dependence in stationary sequences. Zeitschrift fuer Wahrscheinlichkeitstheorie und Verwandte Gebiete 65, 291-306.

Examples

data(ABN)
ExtremalIndex(-ABN[2000:2500], threshold = 0.05, plot = TRUE)

[Package AoE version 1.0.1 Index]