fitmcgpd  R Documentation 
Fitting a Markov chain to cluster exceedances using a bivariate extreme value distribution and a censored maximum likelihood procedure.
fitmcgpd(data, threshold, model = "log", start, ..., std.err.type = "observed", corr = FALSE, warn.inf = TRUE, method = "BFGS")
data 
A vector of observations. 
threshold 
The threshold value. 
model 
A character string which specifies the model used. Must
be one of 
start 
Optional. A list for starting values in the fitting procedure. 
... 
Additional parameters to be passed to the

std.err.type 
The type of the standard error. Currently, one
must specify 
corr 
Logical. Should the correlation matrix be computed? 
warn.inf 
Logical. Should users be warned if likelihood is not finite at starting values? 
method 
The optimization method, see 
The Markov Chain model is defined as follows:
f(x_1;theta_1) f(x_2  x_1;theta_1, theta_2) … f(x_n  x_{n1}; theta_1, theta_2)
As exceedances above a (high enough) threshold are of interest, it is assumed that the marginal are GPD distributed, while the joint distribution is represented by a bivariate extreme value distribution. Smith et al. (1997) present theoretical results about this Markov Chain model.
The bivariate exceedances are fitted using censored likelihood procedure. This methodology is fully described in Ledford (1996).
Most of models are described in Kluppelberg (2006).
The function returns an object of class c("mcpot", "uvpot",
"pot")
. As usual, one can extract several features using
fitted
(or fitted.values
),
deviance
, logLik
and AIC
functions.
fitted.values 
The maximum likelihood estimates of the Markov chain including estimated parameters of the bivariate extreme value distribution. 
std.err 
A vector containing the standard errors  only present when the observed information matrix is not singular. 
var.cov 
The asymptotic variance covariance matrix  only presents when the observed information matrix is not singular. 
deviance 
The deviance. 
corr 
The correlation matrix. 
convergence, counts, message 
Informations taken from the

threshold 
The threshold. 
pat 
The proportion above the threshold. 
nat 
The number above the threshold. 
data 
The observations. 
exceed 
The exceedances. 
call 
The call of the current function. 
model 
The model for the bivariate extreme value distribution. 
chi 
The chi statistic of Coles (1999). A value near 1 (resp. 0) indicates perfect dependence (resp. independence). 
Because of numerical problems, there exists artificial numerical constraints imposed on each model. These are:
For the logistic and asymmetric logistic models: alpha must lie in [0.05, 1] instead of [0,1];
For the negative logistic model: alpha must lie in [0.01, 15] instead of [0, infinity[;
For the asymmetric negative logistic model: alpha must lie in [0.2, 15] instead of [0, infinity[;
For the mixed and asymmetric mixed models: None artificial numerical constraints are imposed.
For this purpose, users must check if estimates are near these artificial numerical constraints. Such cases may lead to substantial biases on the GP parameter estimates. One way to detect quickly if estimates are near the border constraints is to look at the standard errors for the dependence parameters. Small values (i.e. < 1e5) often indicates that numerical constraints have been reached.
In addition, users must be aware that the mixed and asymmetric mixed models can not deal with perfect dependence.
Thus, user may want to plot the Pickands' dependence function to see
if variable are near independence or dependence cases using the
pickdep
function.
In addition, we recommend to fix the marginal parameters. Indeed, even this is a two steps optimization procedure, this avoid numerical troubles  the likelihood function for the Markov chain model seems to be problematic. Thus, estimates are often better using the two stages approach.
Mathieu Ribatet
Kl\"uppelberg, C., and May A. (2006) Bivariate extreme value distributions based on polynomial dependence functions. Mathematical Methods in the Applied Sciences, 29 1467–1480.
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
The following usual generic functions are available
print
,
plot
as well as new generic functions
retlev
and
convassess
.
See also pickdep
.
For optimization in R, see optim
.
mc < simmc(1000, alpha = 0.25) mc < qgpd(mc, 0, 1, 0.25) ##A first application when marginal parameter are estimated fitmcgpd(mc, 0) ##Another one where marginal parameters are fixed fmle < fitgpd(mc, 0) fitmcgpd(mc, 0, scale = fmle$param["scale"], shape = fmle$param["shape"])
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