GP {distributions3} | R Documentation |

The GP distribution has a link to the `\link{GEV}`

distribution.
Suppose that the maximum of `n`

i.i.d. random variables has
approximately a GEV distribution. For a sufficiently large threshold
`u`

, the conditional distribution of the amount (the threshold
excess) by which a variable exceeds `u`

given that it exceeds `u`

has approximately a GP distribution. Therefore, the GP distribution is
often used to model the threshold excesses of a high threshold `u`

.
The requirement that the variables are independent can be relaxed
substantially, but then exceedances of `u`

may cluster.

```
GP(mu = 0, sigma = 1, xi = 0)
```

`mu` |
The location parameter, written |

`sigma` |
The scale parameter, written |

`xi` |
The shape parameter, written |

We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail and much greater clarity.

In the following, let `X`

be a GP random variable with location
parameter `mu`

= `\mu`

, scale parameter `sigma`

= `\sigma`

and
shape parameter `xi`

= `\xi`

.

**Support**:
`[\mu, \mu - \sigma / \xi]`

for `\xi < 0`

;
`[\mu, \infty)`

for `\xi \geq 0`

.

**Mean**: `\mu + \sigma/(1 - \xi)`

for
`\xi < 1`

; undefined otherwise.

**Median**: `\mu + \sigma[2 ^ \xi - 1]/\xi`

for `\xi \neq 0`

;
`\mu + \sigma\ln 2`

for `\xi = 0`

.

**Variance**:
`\sigma^2 / (1 - \xi)^2 (1 - 2\xi)`

for `\xi < 1 / 2`

; undefined otherwise.

**Probability density function (p.d.f)**:

If `\xi \neq 0`

then

`f(x) = \sigma^{-1} [1 + \xi (x - \mu) / \sigma] ^ {-(1 + 1/\xi)}`

for `1 + \xi (x - \mu) / \sigma > 0`

. The p.d.f. is 0 outside the
support.

In the `\xi = 0`

special case

`f(x) = \sigma ^ {-1} \exp[-(x - \mu) / \sigma]`

for `x`

in [`\mu, \infty`

). The p.d.f. is 0 outside the support.

**Cumulative distribution function (c.d.f)**:

If `\xi \neq 0`

then

`F(x) = 1 - \exp\{-[1 + \xi (x - \mu) / \sigma] ^ {-1/\xi} \}`

for `1 + \xi (x - \mu) / \sigma > 0`

. The c.d.f. is 0 below the
support and 1 above the support.

In the `\xi = 0`

special case

`F(x) = 1 - \exp[-(x - \mu) / \sigma] \}`

for `x`

in `R`

, the set of all real numbers.

A `GP`

object.

Other continuous distributions:
`Beta()`

,
`Cauchy()`

,
`ChiSquare()`

,
`Erlang()`

,
`Exponential()`

,
`FisherF()`

,
`Frechet()`

,
`GEV()`

,
`Gamma()`

,
`Gumbel()`

,
`LogNormal()`

,
`Logistic()`

,
`Normal()`

,
`RevWeibull()`

,
`StudentsT()`

,
`Tukey()`

,
`Uniform()`

,
`Weibull()`

```
set.seed(27)
X <- GP(0, 2, 0.1)
X
random(X, 10)
pdf(X, 0.7)
log_pdf(X, 0.7)
cdf(X, 0.7)
quantile(X, 0.7)
cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))
```

[Package *distributions3* version 0.2.1 Index]