dist.Generalized.Pareto {LaplacesDemon}  R Documentation 
Generalized Pareto Distribution
Description
These are the density and random generation functions for the generalized Pareto distribution.
Usage
dgpd(x, mu, sigma, xi, log=FALSE)
rgpd(n, mu, sigma, xi)
Arguments
x 
This is a vector of data. 
n 
This is a positive scalar integer, and is the number of observations to generate randomly. 
mu 
This is a scalar or vector location parameter

sigma 
This is a positiveonly scalar or vector of scale
parameters 
xi 
This is a scalar or vector of shape parameters

log 
Logical. If 
Details
Application: Continuous Univariate
Density:
p(\theta) = \frac{1}{\sigma}(1 + \xi\textbf{z})^(1/\xi + 1)
where\textbf{z} = \frac{\theta  \mu}{\sigma}
Inventor: Pickands (1975)
Notation 1:
\theta \sim \mathcal{GPD}(\mu, \sigma, \xi)
Notation 2:
p(\theta) \sim \mathcal{GPD}(\theta  \mu, \sigma, \xi)
Parameter 1: location
\mu
, where\mu \le \theta
when\xi \ge 0
, and\mu \ge \theta + \sigma / \xi
when\xi < 0
Parameter 2: scale
\sigma > 0
Parameter 3: shape
\xi
Mean:
\mu + \frac{\sigma}{1  \xi}
when\xi < 1
Variance:
\frac{\sigma^2}{(1  \xi)^2 (1  2\xi)}
when\xi < 0.5
Mode:
The generalized Pareto distribution (GPD) is a more flexible extension
of the Pareto (dpareto
) distribution. It is equivalent to
the exponential distribution when both \mu = 0
and
\xi = 0
, and it is equivalent to the Pareto
distribution when \mu = \sigma / \xi
and
\xi > 0
.
The GPD is often used to model the tails of another distribution, and
the shape parameter \xi
relates to
tailbehavior. Distributions with tails that decrease exponentially are
modeled with shape \xi = 0
. Distributions with tails that
decrease as a polynomial are modeled with a positive shape
parameter. Distributions with finite tails are modeled with a negative
shape parameter.
Value
dgpd
gives the density, and
rgpd
generates random deviates.
References
Pickands J. (1975). "Statistical Inference Using Extreme Order Statistics". The Annals of Statistics, 3, p. 119–131.
See Also
Examples
library(LaplacesDemon)
x < dgpd(0,0,1,0,log=TRUE)
x < rgpd(10,0,1,0)