kgaps_stats: Sufficient statistics for the K-gaps model

In ConstantinosChr/exdex: Estimation of the Extremal Index

Description

Calculates sufficient statistics for the K-gaps model for the extremal index θ.

Usage

kgaps_stats(data, thresh, k = 1, inc_cens = FALSE)

Arguments

data	A numeric vector of raw data. No missing values are allowed.
thresh	A numeric scalar. Extreme value threshold applied to data.
k	A numeric scalar. Run parameter K, as defined in Suveges and Davison (2010). Threshold inter-exceedances times that are not larger than k units are assigned to the same cluster, resulting in a K-gap equal to zero. Specifically, the K-gap S corresponding to an inter-exceedance time of T is given by S = max(T - K, 0).
inc_cens	A logical scalar indicating whether or not to include contributions from censored inter-exceedance times relating to the first and last observation. See Attalides (2015) for details.

Details

The sample K-gaps are S_0, S_1, ..., S_(N-1), S_N, where S_1, ..., S_(N-1) are uncensored and S_0 and S_N are censored. Under the assumption that the K-gaps are independent, the log-likelihood of the K-gaps model is given by

l(θ; S_0, ..., S_N) = N_0 log(1 - θ) + 2 N_1 log θ - θ q (S_0 + ... + S_N),

where q is the threshold exceedance probability, N_0 is the number of sample K-gaps that are equal to zero and (apart from an adjustment for the contributions of S_0 and S_N) N_1 is the number of positive sample K-gaps. Specifically, N_1 is equal to the number of S_1, ..., S_(N-1) that are positive plus (I_0 + I_N) / 2, where I_0 = 1 if S_0 is greater than zero and similarly for I_N. The differing treatment of uncensored and censored K-gaps reflects differing contributions to the likelihood. For full details see Suveges and Davison (2010) and Attalides (2015).

Value

A list containing the sufficient statistics, with components

• N0 : the number of zero K-gaps

• N1 : contribution from non-zero K-gaps (see Details)

• sum_qs : the sum of the (scaled) K-gaps, i.e. q (S_0 + ... + S_N), where q is estimated by the proportion of threshold exceedances.

References

Suveges, M. and Davison, A. C. (2010) Model misspecification in peaks over threshold analysis, The Annals of Applied Statistics, 4(1), 203-221. http://dx.doi.org/10.1214/09-AOAS292"

Attalides, N. (2015) Threshold-based extreme value modelling, PhD thesis, University College London. http://discovery.ucl.ac.uk/1471121/1/Nicolas_Attalides_Thesis.pdf

See Also

kgaps_mle for maximum likelihood estimation of the extremal index θ using the K-gaps model.

Examples

u <- quantile(newlyn, probs = 0.90)
kgaps_stats(newlyn, u)

ConstantinosChr/exdex documentation built on May 14, 2019, 4:16 p.m.