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 \(\mu\). 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.

`dgpd`

gives the density, and
`rgpd`

generates random deviates.

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.

Pickands J. (1975). "Statistical Inference Using Extreme Order
Statistics". *The Annals of Statistics*, 3, p. 119--131.

```
# NOT RUN {
library(LaplacesDemon)
x <- dgpd(0,0,1,0,log=TRUE)
x <- rgpd(10,0,1,0)
# }
```

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