kgaps_stat {exdex} | R Documentation |
Sufficient statistics for the
-gaps model
Description
Calculates sufficient statistics for the -gaps model for the extremal
index
. Called by
kgaps
.
Usage
kgaps_stat(data, u, q_u, k = 1, inc_cens = TRUE)
Arguments
data |
A numeric vector of raw data. |
u |
A numeric scalar. Extreme value threshold applied to data. |
q_u |
A numeric scalar. An estimate of the probability with which
the threshold |
k |
A numeric scalar. Run parameter |
inc_cens |
A logical scalar indicating whether or not to include contributions from right-censored inter-exceedance times relating to the first and last observation. It is known that these times are greater than or equal to the time observed. See Attalides (2015) for details. |
Details
The sample -gaps are
,
where
are uncensored and
and
are right-censored. Under the assumption that the
-gaps are independent, the log-likelihood of the
-gaps
model is given by
where
-
is the threshold exceedance probability, estimated by the proportion of threshold exceedances,
-
is the number of uncensored sample
-gaps that are equal to zero,
(apart from an adjustment for the contributions of
and
)
is the number of positive sample
-gaps,
specifically, if
inc_cens = TRUE
thenis equal to the number of
that are positive plus
, where
if
is greater than zero and
otherwise, and similarly for
.
The differing treatment of uncensored and right-censored -gaps
reflects differing contributions to the likelihood. Right-censored
-gaps that are equal to zero add no information to the likelihood.
For full details see Suveges and Davison (2010) and Attalides (2015).
If then we are in the degenerate case where there is one
cluster (all
-gaps are zero) and the likelihood is maximized at
.
If then all exceedances occur singly (all
-gaps are
positive) and the likelihood is maximized at
.
Value
A list containing the sufficient statistics, with components
N0 |
the number of zero |
N1 |
contribution from non-zero |
sum_qs |
the sum of the (scaled) |
n_kgaps |
the number of |
References
Suveges, M. and Davison, A. C. (2010) Model misspecification in peaks over threshold analysis, Annals of Applied Statistics, 4(1), 203-221. doi:10.1214/09-AOAS292
Attalides, N. (2015) Threshold-based extreme value modelling, PhD thesis, University College London. https://discovery.ucl.ac.uk/1471121/1/Nicolas_Attalides_Thesis.pdf
See Also
kgaps
for maximum likelihood estimation of the
extremal index using the
-gaps model.
Examples
u <- quantile(newlyn, probs = 0.90)
kgaps_stat(newlyn, u)