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))
Value
cdfgpa gives the distribution function;
quagpa gives the quantile function.
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 \(k=1\).
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.