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 \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.

dpareto
library(LaplacesDemon)