# dist.Generalized.Pareto: Generalized Pareto Distribution In LaplacesDemon: Complete Environment for Bayesian Inference

## Description

These are the density and random generation functions for the generalized Pareto distribution.

## Usage

 ```1 2``` ```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 x. When xi is negative, mu must be less than 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: 1/sigma (1 + xi z)^(-1/xi + 1) where z = (theta - mu)/sigma

• Inventor: Pickands (1975)

• Notation 1: theta ~ GPD(mu, sigma, xi)

• Notation 2: p(theta) ~ GPD(theta | mu, sigma, xi)

• Parameter 1: location mu, where mu <= theta when xi >= 0, and mu >= theta + sigma/xi when xi < 0

• Parameter 2: scale sigma > 0

• Parameter 3: shape xi

• Mean: mu + sigma / (1 - xi) when xi < 1

• Variance: 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.

`dpareto`
 ```1 2 3``` ```library(LaplacesDemon) x <- dgpd(0,0,1,0,log=TRUE) x <- rgpd(10,0,1,0) ```