Calculates sufficient statistics for the \(K\)-gaps model for the extremal index \(\theta\).
kgaps_stat(data, u, k = 1, inc_cens = FALSE)A numeric vector of raw data. No missing values are allowed.
A numeric scalar. Extreme value threshold applied to data.
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
\(S = \max(T - K, 0)\).
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.
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. \(q (S_0 + \cdots + S_N)\), where \(q\) is estimated by the proportion of threshold exceedances.
n_kgaps the number of \(K\)-gaps, including 2
censored \(K\)-gaps if inc_cens = TRUE.
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 $$l(\theta; S_0, \ldots, S_N) = N_0 \log(1 - \theta) + 2 N_1 \log \theta - \theta q (S_0 + \cdots + S_N),$$ 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).
If \(N_1 = 0\) then we are in the degenerate case where there is one cluster (all \(K\)-gaps are zero) and the likelihood is maximized at \(\theta = 0\).
If \(N_0 = 0\) then all exceedances occur singly (all \(K\)-gaps are positive) and the likelihood is maximized at \(\theta = 1\).
Suveges, M. and Davison, A. C. (2010) Model misspecification in peaks over threshold analysis, The Annals of Applied Statistics, 4(1), 203-221. https://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
kgaps for maximum likelihood estimation of the
extremal index \(\theta\) using the \(K\)-gaps model.
# NOT RUN {
u <- quantile(newlyn, probs = 0.90)
kgaps_stat(newlyn, u)
# }
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