Calculates sufficient statistics for the K-gaps model for the extremal index \(\theta\).
kgaps_stats(data, thresh, 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.
The sample K-gaps are \(S_0, S_1, \ldots, S_{N-1}, S_N\), where \(S_1, \ldots, S_{N-1}\) 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 \(S_1, \ldots, S_{N-1}\) 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).
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.
kgaps_mle for maximum likelihood estimation of the
extremal index \(\theta\) using the K-gaps model.
# NOT RUN {
u <- quantile(newlyn, probs = 0.90)
kgaps_stats(newlyn, u)
# }
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