dist.Generalized.Pareto {LaplacesDemon}

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 \mu. When \xi is non-negative, \mu must not be greater than \textbf{x}. When \xi is negative, \mu must be less than \textbf{x} + \sigma / \xi.

sigma

This is a positive-only scalar or vector of scale parameters \sigma.

xi

This is a scalar or vector of shape parameters \xi.

log

Logical. If log=TRUE, then the logarithm of the density is returned.

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 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.

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

dpareto

Examples

library(LaplacesDemon)
x <- dgpd(0,0,1,0,log=TRUE)
x <- rgpd(10,0,1,0)