fitGPD {AoE}R Documentation

Maximum likelihood estimation of the parameters of the (generalized) Pareto distribution

Description

Given a sample of positive observations, the functions fitGPD and fitPareto compute the maximum likelihood estimators of the parameters of the (generalized) Pareto distributions.

Usage

fitGPD(z)
fitPareto(z)

Arguments

z A numeric vector with positive elements.

Details

The generalized Pareto distribution with shape parameter gamma and scale parameter sigma > 0 is defined by its distribution function

F(z) = 1 - (1 + gamma*z/sigma)^(-1/gamma)

for all z >= 0 such that sigma + gamma*z > 0.

The Pareto distribution with shape parameter alpha > 0 and scale parameter tau > 0 is defined by its distribution function

F(z) = 1 - (1 + z/tau)^(-alpha)

for all z >= 0. It is a reparametrization of the generalized Pareto distribution with positive shape parameter gamma = 1/alpha and scale parameter sigma = tau/alpha.

Value

For fitGPD, a list with components gamma and sigma. For fitPareto, a list with components alpha and tau.

Note

The function fitGPD is called within the function GPD_par if its argument method is set to "ML".

References

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

See Also

GPD_par, ML

Examples

# order statistics of random sample of size 100 
# from the unit Frechet distribution:
x <- sort(- 1/log(runif(100)), decreasing = TRUE)
# fit GPD to sample of excesses over 21th largest observation:
fitGPD(x[1:20] - x[21])

[Package AoE version 1.0.1 Index]