GEV {distributions3} | R Documentation |

The GEV distribution arises from the Extremal Types Theorem, which is rather
like the Central Limit Theorem (see `\link{Normal}`

) but it relates to
the *maximum* of `n`

i.i.d. random variables rather than to the sum.
If, after a suitable linear rescaling, the distribution of this maximum
tends to a non-degenerate limit as `n`

tends to infinity then this limit
must be a GEV distribution. The requirement that the variables are independent
can be relaxed substantially. Therefore, the GEV distribution is often used
to model the maximum of a large number of random variables.

```
GEV(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 GEV random variable with location
parameter `mu`

= `\mu`

, scale parameter `sigma`

= `\sigma`

and
shape parameter `xi`

= `\xi`

.

**Support**:
`(-\infty, \mu - \sigma / \xi)`

for `\xi < 0`

;
`(\mu - \sigma / \xi, \infty)`

for `\xi > 0`

;
and `R`

, the set of all real numbers, for `\xi = 0`

.

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

for
`\xi < 1, \xi \neq 0`

;
`\mu + \sigma\gamma`

for `\xi = 0`

, where `\gamma`

is Euler's constant, approximately equal to 0.57722; undefined otherwise.

**Median**: `\mu + \sigma[(\ln 2) ^ {-\xi} - 1]/\xi`

for `\xi \neq 0`

;
`\mu - \sigma\ln(\ln 2)`

for `\xi = 0`

.

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

for `\xi < 1 / 2, \xi \neq 0`

;
`\sigma^2 \pi^2 / 6`

for `\xi = 0`

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

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

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

In the `\xi = 0`

(Gumbel) special case

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

for `x`

in `R`

, the set of all real numbers.

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

If `\xi \neq 0`

then

`F(x) = \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`

(Gumbel) special case

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

for `x`

in `R`

, the set of all real numbers.

A `GEV`

object.

Other continuous distributions:
`Beta()`

,
`Cauchy()`

,
`ChiSquare()`

,
`Erlang()`

,
`Exponential()`

,
`FisherF()`

,
`Frechet()`

,
`GP()`

,
`Gamma()`

,
`Gumbel()`

,
`LogNormal()`

,
`Logistic()`

,
`Normal()`

,
`RevWeibull()`

,
`StudentsT()`

,
`Tukey()`

,
`Uniform()`

,
`Weibull()`

```
set.seed(27)
X <- GEV(1, 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]