GPD_par {AoE}R Documentation

Estimate GPD Parameters

Description

Computes estimates of the parameters (gamma, σ) of the generalized Pareto distribution fitted to excesses over a high threshold.

Usage

GPD_par(data, method = "ML", k = 5:(length(data) - 1))

Arguments

data A numeric vector.
method A character string determining which method will be used: "Hill", "ML", or "Moment".
k Integer vector. For each element of k, the parameter estimates will be computed based on the sample of excesses over the threshold u defined as the (k+1)th largest order statistic.

Details

Let X_{1:n} < ... < X_{n:n} be a the increasing order statistics of the sample. Let k = 1, ..., n-1. The function fits the generalized Pareto distribution

H(z) = 1 - (1 + gamma*z/σ)^(-1/gamma)

to the sample of excesses X_{n-k+i:n} - X_{n-k:n}, i = 1, ..., k over the threshold u = X_{n-k:n}.

In case method is "Hill" or "Moment", only those elements of k will be retained for which the corresponding order statistic is positive.

Value

A list with the class attribute "GPD_par", which is a list containing the following components:

gamma Numeric vector with the same length as k containing the estimates for gamma.
sigma Numeric vector with the same length as k containing the estimates for σ.
threshold Numeric vector of thresholds corresponding to k.
k Vector of k-values that have been used effectively.
n The sample size.

References

Dekkers, A.L.M., Einmahl, J.H.J. and de Haan, L. (1989). A moment estimator for the index of an extreme-value distribution. The Annals of Statistics 17, 1833-1855.

Hill, B.M. (1975). A simple general approach to inference about the tail of a distribution. The Annals of Statistics 3, 1163-1174.

Smith, R.L. (1987). Estimating tails of probability distributions. The Annals of Statistics 15, 1174-1207.

See Also

fitGPD, Hill, ML, Moment

Examples

# random sample of size 100 
# from the unit Frechet distribution:
x <- - 1/log(runif(100))
# fit GPD to sample of excesses over 21th largest observation:
out <- GPD_par(x) 
# plot estimates of gamma as a function of k (on logarithmic scale)
# together with the true gamma (= 1)
plot(out$k, out$gamma, type = "l", log = "x"); abline(h = 1)

[Package AoE version 1.0.1 Index]