Estimates the extremal index \(\theta\) using the iterated weighted least squares method of Suveges (2007). At the moment no estimates of uncertainty are provided.
iwls(data, u, maxit = 100)A numeric vector of raw data. No missing values are allowed.
A numeric scalar. Extreme value threshold applied to data.
A numeric scalar. The maximum number of iterations.
An object (a list) of class "iwls", "exdex" containing
theta The estimate of \(\theta\).
conv A convergence indicator: 0 indicates successful
convergence; 1 indicates that maxit has been reached.
niter The number of iterations performed.
n_gaps The number of time gaps between successive exceedances.
call The call to iwls.
The iterated weighted least squares algorithm on page 46 of
Suveges (2007) is used to estimate the value of the extremal index.
This approach uses the time gaps between successive exceedances
in the data data of the threshold u. The \(i\)th
gap is defined as \(T_i - 1\), where \(T_i\) is the difference in
the occurrence time of exceedance \(i\) and exceedance \(i + 1\).
Therefore, threshold exceedances at adjacent time points produce a gap
of zero.
The model underlying this approach is an exponential-point mas mixture
for scaled gaps, that is, gaps multiplied by the proportion of
values in data that exceed u. Under this model
scaled gaps are zero (`within-cluster' inter-exceedance times) with
probability \(1 - \theta\) and otherwise (`between-cluster'
inter-exceedance times) follow an exponential distribution with mean
\(1 / \theta\).
The estimation method is based on fitting the `broken stick' model of
Ferro (2003) to an exponential quantile-quantile plot of all of the
scaled gaps. Specifically, the broken stick is a horizontal line
and a line with gradient \(1 / \theta\) which intersect at
\((-\log\theta, 0)\). The algorithm on page 46 of
Suveges (2007) uses a weighted least squares minimization applied to
the exponential
part of this model to seek a compromise between the role of \(\theta\)
as the proportion of inter-exceedance times that are between-cluster
and the reciprocal of the mean of an exponential distribution for these
inter-exceedance times. The weights (see Ferro (2003)) are based on the
variances of order statistics of a standard exponential sample: larger
order statistics have larger sampling variabilities and therefore
receive smaller weight than smaller order statistics.
Note that in step (1) of the algorithm on page 46 of Suveges there is a typo: \(N_c + 1\) should be \(N\), where \(N\) is the number of threshold exceedances. Also, the gaps are scaled as detailed above, not by their mean.
Suveges, M. (2007) Likelihood estimation of the extremal index. Extremes, 10, 41-55. doi:10.1007/s10687-007-0034-2
Ferro, C.A.T. (2003) Statistical methods for clusters of extreme values. Ph.D. thesis, Lancaster University.
iwls_methods for S3 methods for "iwls"
objects.
### S&P 500 index
u <- quantile(sp500, probs = 0.60)
theta <- iwls(sp500, u)
theta
#>
#> Call:
#> iwls(data = sp500, u = u)
#>
#> Convergence (0 means success): 0
#>
#> Estimate of the extremal index theta:
#> [1] 0.7707
coef(theta)
#> [1] 0.770706
nobs(theta)
#> [1] 2899
### Newlyn sea surges
u <- quantile(newlyn, probs = 0.90)
theta <- iwls(newlyn, u)
theta
#>
#> Call:
#> iwls(data = newlyn, u = u)
#>
#> Convergence (0 means success): 0
#>
#> Estimate of the extremal index theta:
#> [1] 0.2514