Description Usage Arguments Details Value Note Author(s) References See Also Examples

Density, distribution function, quantile function, and random generation for the (largest) extreme value distribution.

1 2 3 4 |

`x` |
vector of quantiles. |

`q` |
vector of quantiles. |

`p` |
vector of probabilities between 0 and 1. |

`n` |
sample size. If |

`location` |
vector of location parameters. |

`scale` |
vector of positive scale parameters. |

Let *X* be an extreme value random variable with parameters
`location=`

*η* and `scale=`

*θ*.
The density function of *X* is given by:

*f(x; η, θ) = \frac{1}{θ} e^{-(x-η)/θ} exp[-e^{-(x-η)/θ}]*

where *-∞ < x, η < ∞* and *θ > 0*.

The cumulative distribution function of *X* is given by:

*F(x; η, θ) = exp[-e^{-(x-η)/θ}]*

The *p^{th}* quantile of *X* is given by:

*x_{p} = η - θ log[-log(p)]*

The mode, mean, variance, skew, and kurtosis of *X* are given by:

*Mode(X) = η*

*E(X) = η + ε θ*

*Var(X) = θ^2 π^2 / 6*

*Skew(X) = √{β_1} = 1.139547*

*Kurtosis(X) = β_2 = 5.4*

where *ε* denotes Euler's constant,
which is equivalent to `-digamma(1)`

.

density (`devd`

), probability (`pevd`

), quantile (`qevd`

), or
random sample (`revd`

) for the extreme value distribution with
location parameter(s) determined by `location`

and scale
parameter(s) determined by `scale`

.

There are three families of extreme value distributions. The one
described here is the Type I, also called the Gumbel extreme value
distribution or simply Gumbel distribution. The name
“extreme value” comes from the fact that this distribution is
the limiting distribution (as *n* approaches infinity) of the
greatest value among *n* independent random variables each
having the same continuous distribution.

The Gumbel extreme value distribution is related to the
exponential distribution as follows.
Let *Y* be an exponential random variable
with parameter `rate=`

*λ*. Then *X = η - log(Y)*
has an extreme value distribution with parameters
`location=`

*η* and `scale=`

*1/λ*.

The distribution described above and used by `devd`

, `pevd`

,
`qevd`

, and `revd`

is the *largest* extreme value
distribution. The smallest extreme value distribution is the limiting
distribution (as *n* approaches infinity) of the smallest value among
*n* independent random variables each having the same continuous distribution.
If *X* has a largest extreme value distribution with parameters

`location=`

*η* and `scale=`

*θ*, then
*Y = -X* has a smallest extreme value distribution with parameters
`location=`

*-η* and `scale=`

*θ*. The smallest
extreme value distribution is related to the
Weibull distribution as follows.
Let *Y* be a Weibull random variable with parameters
`shape=`

*β* and `scale=`

*α*. Then *X = log(Y)*
has a smallest extreme value distribution with parameters `location=`

*log(α)*
and `scale=`

*1/β*.

The extreme value distribution has been used extensively to model the distribution of streamflow, flooding, rainfall, temperature, wind speed, and other meteorological variables, as well as material strength and life data.

Steven P. Millard ([email protected])

Forbes, C., M. Evans, N. Hastings, and B. Peacock. (2011). Statistical Distributions. Fourth Edition. John Wiley and Sons, Hoboken, NJ.

Johnson, N. L., S. Kotz, and N. Balakrishnan. (1995).
*Continuous Univariate Distributions, Volume 2*.
Second Edition. John Wiley and Sons, New York.

`eevd`

, `GEVD`

,
Probability Distributions and Random Numbers.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 | ```
# Density of an extreme value distribution with location=0, scale=1,
# evaluated at 0.5:
devd(.5)
#[1] 0.3307043
#----------
# The cdf of an extreme value distribution with location=1, scale=2,
# evaluated at 0.5:
pevd(.5, 1, 2)
#[1] 0.2769203
#----------
# The 25'th percentile of an extreme value distribution with
# location=-2, scale=0.5:
qevd(.25, -2, 0.5)
#[1] -2.163317
#----------
# Random sample of 4 observations from an extreme value distribution with
# location=5, scale=2.
# (Note: the call to set.seed simply allows you to reproduce this example.)
set.seed(20)
revd(4, 5, 2)
#[1] 9.070406 7.669139 4.511481 5.903675
``` |

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