| cdfgpa {lmom} | R Documentation |
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 |
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))