GUMBEL {nsRFA} | R Documentation

## Two parameter Gumbel distribution and L-moments

### Description

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

### Usage

    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): xi (location), α (scale).

Range of x: -infty < x < infty.

Probability density function:

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

Cumulative distribution function:

F(x) = exp[-exp(-(x-xi)/α)]

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

L-moments

λ_1 = xi + α gamma

λ_2 = α log 2

tau_3 = 0.1699 = log(9/8)/ log 2

tau_4 = 0.1504 = (16 log 2 - 10 log 3)/ log 2

Here gamma is Euler's constant, 0.5772...

Parameters

α=λ_2 / log 2

xi = λ_1 - gamma α

`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.

### 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, tau_3, tau_4)), `par.gumb` gives the parameters (`xi`, `alfa`), and `rand.gumb` generates random deviates.

### Note

For information on the package and the Author, and for all the references, see `nsRFA`.

### See Also

`rnorm`, `runif`, `EXP`, `GENLOGIS`, `GENPAR`, `GEV`, `KAPPA`, `LOGNORM`, `P3`; `DISTPLOTS`, `GOFmontecarlo`, `Lmoments`.

### Examples

    data(hydroSIMN)
    annualflows[1:10,]
    summary(annualflows)
    x <- annualflows["dato"][,]
    fac <- factor(annualflows["cod"][,])
    split(x,fac)

    camp <- split(x,fac)$"45"
    ll <- Lmoments(camp)
    parameters <- par.gumb(ll[1],ll[2])
    f.gumb(1800,parameters$xi,parameters$alfa)
    F.gumb(1800,parameters$xi,parameters$alfa)
    invF.gumb(0.7686843,parameters$xi,parameters$alfa)
    Lmom.gumb(parameters$xi,parameters$alfa)
    rand.gumb(100,parameters$xi,parameters$alfa)

    Rll <- regionalLmoments(x,fac); Rll
    parameters <- par.gumb(Rll[1],Rll[2])
    Lmom.gumb(parameters$xi,parameters$alfa)

[Package nsRFA version 0.6-9 Index]