# GUMBEL: Two parameter Gumbel distribution and L-moments In homtest: Homogeneity tests for Regional Frequency Analysis

## Description

`GUMBEL` provides the link between L-moments of a sample and the two parameter Gumbel distribution.

## Usage

 ```1 2 3 4 5 6``` ```f.gumb (x, xi, alfa) F.gumb (x, xi, alfa) invF.gumb (F, xi, alfa) Lmom.gumb (xi, alfa) par.gumb (lambda1, lambda2) rand.gumb (numerosita, xi, alfa) ```

## Arguments

 `x` vector of quantiles `xi` vector of gumb location parameters `alfa` vector of gumb scale parameters `F` vector of probabilities `lambda1` vector of sample means `lambda2` vector of L-variances `numerosita` numeric value indicating the length of the vector to be generated

## Details

See http://en.wikipedia.org/wiki/Fisher-Tippett_distribution for an introduction to the Gumbel distribution.

Definition

Parameters (2): ξ (location), α (scale).

Range of x: -∞ < x < ∞.

Probability density function:

f(x) = α^{-1} \exp[-(x-ξ)/α] \exp\{- \exp[-(x-ξ)/α]\}

Cumulative distribution function:

F(x) = \exp[-\exp(-(x-ξ)/α)]

Quantile function: x(F) = ξ - α \log(-\log F).

L-moments

λ_1 = ξ + α γ

λ_2 = α \log 2

τ_3 = 0.1699 = \log(9/8)/ \log 2

τ_4 = 0.1504 = (16 \log 2 - 10 \log 3)/ \log 2

Here γ is Euler's constant, 0.5772...

Parameters

α=λ_2 / \log 2

ξ = λ_1 - γ α

## Value

`f.gumb` gives the density f, `F.gumb` gives the distribution function F, `invF.gumb` gives the quantile function x, `Lmom.gumb` gives the L-moments (λ_1, λ_2, τ_3, τ_4)), `par.gumb` gives the parameters (`xi`, `alfa`), and `rand.gumb` generates random deviates.

## Note

`Lmom.gumb` and `par.gumb` accept input as vectors of equal length. In `f.gumb`, `F.gumb`, `invF.gumb` and `rand.gumb` parameters (`xi`, `alfa`) must be atomic.

## Author(s)

Alberto Viglione, e-mail: alviglio@tiscali.it.

## References

Hosking, J.R.M. and Wallis, J.R. (1997) Regional Frequency Analysis: an approach based on L-moments, Cambridge University Press, Cambridge, UK.

`rnorm`, `runif`, `GEV`, `Lmoments`.