Fit the GP Distribution {POT}R Documentation

Fitting a GPD to Peaks Over a Threshold

Description

Maximum Likelihood, Unbiased Probability Weighted Moments, Biased Probability Weighted Moments and Moments Estimator to fit Peaks Over a Threshold to a GP distribution.

Usage

fitgpd(data, threshold, method = "mle", ...)

Arguments

data A numeric vector.
threshold A numeric value giving the threshold for the GPD. 'mle' method allows varying threshold; so that threshold could be for this case a numeric vector. Be careful, varying thresholds are used cyclically if length doesn't match with data.
method A string giving the names of the estimator. It can be 'mle' (the default), 'moments', 'pwmu', 'pwmb', 'mdpd', 'med' and 'pickands' for the maximum likelihood, moments, unbiased probability weighted moments, biased probability weigthed moments, minimum density power divergence, medians and pickands' estimators respectively.
... Other optional arguments to be passed to the optim function and allow hand fixed parameters.

Value

This function returns a list with components:

fitted.values A vector containing the maximum likelihood estimates.
std.err A vector containing the standard errors.
fixed A vector containing the parameters of the model that have been held fixed.
param A vector containing all parameters (optimized and fixed).
deviance The deviance at the maximum likelihood estimates.
corr The correlation matrix - for the mle method.
convergence, counts, message Components taken from the list returned by optim - for the mle method.
threshold The threshold passed to argument threshold.
nat, pat The number and proportion of exceedances.
data The data passed to the argument data.
exceed The exceedances, or the maxima of the clusters of exceedances.
scale The scale parameter for the fitted generalized Pareto distribution.
std.err.type The standard error type - for 'mle' only. That is Observed or Expected Information matrix of Fisher.
var.thresh Logical. Specify if the threshold is a varying one - 'mle' only. For other methods, threshold is always constant i.e. var.thresh = FALSE.

Note

The Maximum Likelihood estimator is obtained through a slightly modified version of Alec Stephenson's fpot.norm function in the evd package.

Author(s)

Mathieu Ribatet

References

Coles, S. (2001) An Introduction to Statistical Modelling of Extreme Values. Springer Series in Statistics. London.

Hosking, J. and Wallis, J. (1987) Parameters and Quantile Estimation for the Generalized Pareto Distribution. Technometrics 29:339–349.

Juarez, S. and Schucany, W. (2004) Robust and Efficient Estimation for the Generalized Pareto Distribution. Extremes 7:237–251.

Peng, L. and Welsh, A. (2001) Robust Estimation of the Generalized Pareto Distribution. Extremes 4:53–65.

Embrechts, P and Klüppelberg, C. and Mikosch, T (1997) Modelling Extremal Events for Insurance and Finance. Springers.

Pickands, J. (1975) Statistical Inference Using Extreme Order Statistics. Annals of Statistics. 3:119–131.

Examples

x <- rgpd(200, 1, 2, 0.25)
mle <- fitgpd(x, 1, "mle")$param
pwmu <- fitgpd(x, 1, "pwmu")$param
pwmb <- fitgpd(x, 1, "pwmb")$param
pickands <- fitgpd(x, 1, "pickands")$param    ##Check if Pickands estimates
                                              ##are valid or not !!!
med <- fitgpd(x, 1, "med", start = mle)$param ##Sometimes the fitting algo is not
                                              ##accurate. So specify
                                              ##good starting values is
                                              ##a good idea.  
mdpd <- fitgpd(x, 1, "mdpd")$param

print(rbind(mle, pwmu, pwmb, pickands, med, mdpd))

[Package POT version 1.0-1 Index]