Fitting a bivariate extreme value distribution to bivariate exceedances over thresholds using censored maximum likelihood procedure.
fitbvgpd(data, threshold, model = "log", start, ..., cscale = FALSE,
cshape = FALSE, std.err.type = "observed", corr = FALSE, warn.inf = TRUE,
method = "BFGS")The function returns an object of class c("bvpot","pot"). As
usual, one can extract several features using fitted (or
The maximum likelihood estimates of the bivariate extreme value distribution.
A vector containing the standard errors - only present when the observed information matrix is not singular.
The asymptotic variance covariance matrix - only presents when the observed information matrix is not singular.
The deviance.
The correlation matrix.
Informations taken from the
optim function.
The marginal thresholds.
The marginal proportion above the threshold.
The marginal number above the threshold.
The bivariate matrix of observations.
The marginal exceedances.
The call of the current function.
The model for the bivariate extreme value distribution.
The chi statistic of Coles (1999). A value near 1 (resp. 0) indicates perfect dependence (resp. independence).
A matrix with two columns which gives the observation
vector for margin 1 and 2 respectively. NA values are
considered to fall below the threshold.
A numeric vector for the threshold (of length 2).
A character string which specifies the model used. Must
be one of log (the default), alog, nlog,
anlog, mix and amix for the logistic, asymmetric
logistic, negative logistic, asymmetric negative logistic, mixed and
asymmetric mixed models.
Optional. A list for starting values in the fitting procedure.
Additional parameters to be passed to the
optim function or to the bivariate model. In
particular, parameter of the model can be hand fixed.
Logical. Should the two scale parameters be equal?
Logical. Should the two shape parameters be equal?
The type of the standard error. Currently, one
must specify "observed" for observed Fisher information
matrix or "none" for no computations of the standard
errors.
Logical. Should the correlation matrix be computed?
Logical. Should users be warned if likelihood is not finite at starting values?
The optimization method, see optim.
Because of numerical problems, their 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 \([0,\infty[\);
For the asymmetric negative logistic model: \(\alpha\) must lie in [0.2, 15] instead of \([0,\infty[\);
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. < 1e-5) 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.
Mathieu Ribatet
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).
Coles, S., Heffernan, J. and Tawn, J. (1999) Dependence Measure for Extreme Value Analyses. Extremes, 2:4 339--365.
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.
The following usual generic functions are available
print,
plot and
anova
as well as new generic functions
retlev and
convassess.
See also pickdep, specdens.
For optimization in R, see optim.
x <- rgpd(1000, 0, 1, 0.25)
y <- rgpd(1000, 3, 1, -0.25)
ind <- fitbvgpd(cbind(x, y), c(0, 3), "log")
ind
ind2 <- fitbvgpd(cbind(x, y), c(0, 3), "log", alpha = 1)
ind2
ind3 <- fitbvgpd(cbind(x, y), c(0, 3), cscale = TRUE)
ind3
##The mixed model can not deal with perfect dependent variables
##Thus, there is a substantial bias in GPD parameter estimates
dep <- fitbvgpd(cbind(x, x + 3), c(0, 3), "mix")
dep
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