cdfgpa: Generalized Pareto distribution

In lmom: L-Moments

Generalized Pareto distribution

Description

Distribution function and quantile function of the generalized Pareto distribution.

Usage

cdfgpa(x, para = c(0, 1, 0))
quagpa(f, para = c(0, 1, 0))

Arguments

x	Vector of quantiles.
f	Vector of probabilities.
para	Numeric vector containing the parameters of the distribution, in the order \xi, \alpha, k (location, scale, shape).

Details

The generalized Pareto distribution with location parameter \xi, scale parameter \alpha and shape parameter k has distribution function

F(x)=1-\exp(-y)

where

y=-k^{-1}\log\lbrace1-k(x-\xi)/\alpha\rbrace,

with x bounded by \xi+\alpha/k from below if k<0 and from above if k>0, and quantile function

x(F)=\xi+{\alpha\over k}\lbrace 1-(1-F)^k\rbrace.

The exponential distribution is the special case k=0. The uniform distribution is the special case k=1.

Value

cdfgpa gives the distribution function; quagpa gives the quantile function.

Note

The functions expect the distribution parameters in a vector, rather than as separate arguments as in the standard R distribution functions pnorm, qnorm, etc.

Two parametrizations of the generalized Pareto distribution are in common use. When Jenkinson (1955) introduced the generalized extreme-value distribution he wrote the distribution function in the form

F(x) = \exp [ - \lbrace 1 - k ( x - \xi ) / \alpha) \rbrace^{1/k}].

Hosking and Wallis (1987) wrote the distribution function of the generalized Pareto distribution analogously as

F(x) = 1 - \lbrace 1 - k ( x - \xi ) / \alpha) \rbrace^{1/k}

and that is the form used in R package lmom. A slight inconvenience with it is that the skewness of the distribution is a decreasing function of the shape parameter k. Perhaps for this reason, authors of some other R packages prefer a form in which the sign of the shape parameter k is changed and the parameters are renamed:

F(x) = 1 - \lbrace 1 + \xi ( x - \mu ) / \sigma) \rbrace ^{-1/\xi}.

Users should be able to mix functions from packages that use either form; just be aware that the sign of the shape parameter will need to be changed when converting from one form to the other (and that \xi is a location parameter in one form and a shape parameter in the other).

References

Hosking, J. R. M., and Wallis, J. R. (1987). Parameter and quantile estimation for the generalized Pareto distribution. Technometrics, 29, 339-349.

Jenkinson, A. F. (1955). The frequency distribution of the annual maximum (or minimum) of meteorological elements. Quarterly Journal of the Royal Meteorological Society, 81, 158-171.

See Also

cdfexp for the exponential distribution.

cdfkap for the kappa distribution and cdfwak for the Wakeby distribution, which generalize the generalized Pareto distribution.

Examples

# Random sample from the generalized Pareto distribution
# with parameters xi=0, alpha=1, k=-0.5.
quagpa(runif(100), c(0,1,-0.5))

lmom documentation built on Oct. 1, 2024, 1:08 a.m.