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 ,
scale parameter
and
shape parameter
has distribution function
where
with bounded by
from below if
and from above if
,
and quantile function
The exponential distribution is the special case .
The uniform distribution is the special case
.
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
Hosking and Wallis (1987) wrote the distribution function of the generalized Pareto distribution analogously as
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 .
Perhaps for this reason, authors of some other R packages prefer a form in which
the sign of the shape parameter
is changed and the parameters are renamed:
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 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))