Compute the density of the extremal index using simulations from a fitted markov chain model.
dexi(x, n.sim = 1000, n.mc = length(x$data), plot = TRUE, ...)The function returns a optionally plot of the kernel density estimate of the extremal index. In addition, the vector of extremal index estimations is returned invisibly.
A object of class 'mcpot' - most often the returned
object of the fitmcgpd function.
The number of simulation of Markov chains.
The length of the simulated Markov chains.
Logical. If TRUE (default), the density of the
extremal index is plotted.
Optional parameters to be passed to the
plot function.
Mathieu Ribatet
The Markov chains are simulated using the simmc
function to obtained dependent realisations \(u_i\) of standard
uniform realizations. Then, they are transformed to correspond to the
parameter of the fitted markov chain model. Thus, if \(u, \sigma,
\xi\) is the location, scale and shape parameters ; and
\(\lambda\) is the probability of exceedance of \(u\),
then by defining :
$$\sigma_* = \xi \times \frac{u}{\lambda^{-\xi} - 1}$$
the realizations \(y_i = qgpd(u_i, 0, \sigma_*, \xi)\) are distributed such as the probability of exceedance of \(u\) is equal to \(\lambda\).
At last, the extremal index for each generated Markov chain is estimated using the estimator of Ferro and Segers (2003) (and thus avoid any declusterization).
Fawcett L., and Walshaw D. (2006) Markov chain models for extreme wind speed. Environmetrics, 17:(8) 795--809.
Ferro, C. and Segers, J. (2003) Inference for clusters of extreme values. Journal of the Royal Statistical Society. Series B 65:(2) 545--556.
Ledford A., and Tawn, J. (1996) Statistics for near Independence in Multivariate Extreme Values. Biometrika, 83 169--187.
Smith, R., and Tawn, J., and Coles, S. (1997) Markov chain models for threshold exceedances. Biometrika, 84 249--268.
simmc, fitmcgpd, fitexi
mc <- simmc(100, alpha = 0.25)
mc <- qgpd(mc, 0, 1, 0.25)
fgpd1 <- fitmcgpd(mc, 2, shape = 0.25, scale = 1)
dexi(fgpd1, n.sim = 100)
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