fitmcgpd {POT}

Fitting Markov Chain Models to Peaks Over a Threshold

Description

Fitting a Markov chain to cluster exceedances using a bivariate extreme value distribution and a censored maximum likelihood procedure.

Usage

fitmcgpd(data, threshold, model = "log", start, ..., std.err.type =
"observed", corr = FALSE, warn.inf = TRUE, method = "BFGS")

Arguments

data

A vector of observations.

threshold

The threshold value.

model

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.

start

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.

std.err.type

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.

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 optim.

Details

The Markov Chain model is defined as follows:

L\left(y;\theta_1,\theta_2\right) = f\left(x_1; \theta_1\right) \prod_{i=2}^n f\left(y_i | y_{i-1};\theta_1,\theta_2\right)

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).

Value

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 optim function.

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).

Warnings

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,\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.

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.

Author(s)

Mathieu Ribatet

References

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

See Also

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.

Examples

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"])