Peaks-Over-Threshold Modelling using the Generalized Pareto Distribution

Description

Numerical optimization of the Generalized Pareto distribution over a high threshold.

Usage

gp.fit(xdat, threshold, method = c("Grimshaw", "nlm", "optim", "ismev", "zs",
  "zhang"), show = TRUE, MCMC = NULL)

Arguments

Details

The default method is "Grimshaw", consisting in maximization of the profile likelihood for the scale. Other options for maximization of the profile likelihood are nlm and optim, which use respectively nlm and optim. Method "ismev" is the two-dimensional optimization routine gpd.fit from the ismev library, with in addition the algebraic gradient. The approximate Bayesian methods ("zs" and "zhang") are extracted respectively from Zhang and Stephens (2009) and Zhang (2010) and consists of a approximate posterior mean calculated via importance sampling assuming a GPD prior is placed on the parameter of the profile likelihood.

Value

If method is neither "zs" nor "zhang", a list containing the following components:

• estimate a vector containing all parameters (optimized and fixed).

• std.err a vector containing the standard errors.

• var.cov the variance covariance matrix, obtained as the numerical inverse of the observed information matrix.

• threshold the threshold.

• method the method used to fit the parameter. See details.

• deviance the deviance at the maximum likelihood estimates.

• nat number of points lying above the threshold.

• pat proportion of points lying above the threshold.

• convergence components taken from the list returned by optim. Values other than 0 indicate that the algorithm likely did not converge (in particular 1 and 50).

• counts components taken from the list returned by optim.

Otherwise, a list containing

• threshold the threshold.

• method the method used to fit the parameter. See Details.

• nat number of points lying above the threshold.

• pat proportion of points lying above the threshold.

• approx.mean a vector containing containing the approximate posterior mean estimates.

and in addition if MCMC is neither FALSE, nor NULL

• post.mean a vector containing the posterior mean estimates.

• post.se a vector containing the posterior standard error estimates.

• accept.rate proportion of points lying above the threshold.

• niter length of resulting Markov Chain

• burnin amount of discarded iterations at start, capped at 10000.

• thin thinning integer parameter describing

Note

Some of the internal functions (which are hidden from the user) allow for modelling of the parameters using covariates. This is not currently implemented within gp.fit, but users can call internal functions should they wish to use these features.

Author(s)

Paul J. Northrop and Claire L. Coleman for the frequentist functions. Zhang and Stephens (2009) and Zhang (2010) for the zs and zhang approximate methods and L. Belzile for the wrapper and MCMC samplers.

References

Davison, A.C. (1984). Modelling excesses over high thresholds, with an application, in Statistical extremes and applications, J. Tiago de Oliveira (editor), D. Reidel Publishing Co., 461–482.

Grimshaw, S.D. (1993). Computing Maximum Likelihood Estimates for the Generalized Pareto Distribution, Technometrics, 35(2), 185–191.

Northrop, P.J. and C. L. Coleman (2014). Improved threshold diagnostic plots for extreme value analyses, Extremes, 17(2), 289–303.

Zhang, J. (2010). Improving on estimation for the generalized Pareto distribution, Technometrics 52(3), 335–339.

Zhang, J. and M.A. Stephens (2009). A new and efficient estimation method for the generalized Pareto distribution. Technometrics 51(3), 316–325.

See Also

fpot and gpd.fit

Examples

library(ismev)
data(rain)
threshold <- quantile(rain,0.9)
gp.fit(rain, threshold, method="Grimshaw")
gp.fit(rain, threshold, method="zs")

Results