dist.Generalized.Pareto | R Documentation |
These are the density and random generation functions for the generalized Pareto distribution.
dgpd(x, mu, sigma, xi, log=FALSE)
rgpd(n, mu, sigma, xi)
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 positive-only scalar or vector of scale
parameters |
xi |
This is a scalar or vector of shape parameters
|
log |
Logical. If |
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
tail-behavior. 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.
dgpd
gives the density, and
rgpd
generates random deviates.
Pickands J. (1975). "Statistical Inference Using Extreme Order Statistics". The Annals of Statistics, 3, p. 119–131.
dpareto
library(LaplacesDemon)
x <- dgpd(0,0,1,0,log=TRUE)
x <- rgpd(10,0,1,0)
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