gumbelUC {VGAM} | R Documentation

## The Gumbel Distribution

### Description

Density, distribution function, quantile function and random generation for the Gumbel distribution with location parameter `location` and scale parameter `scale`.

### Usage

    dgumbel(x, location=0, scale=1, log=FALSE)
    pgumbel(q, location=0, scale=1)
    qgumbel(p, location=0, scale=1)
    rgumbel(n, location=0, scale=1)

### Arguments

`x, q` | vector of quantiles.
`p` | vector of probabilities.
`n` | number of observations. If `length(n) > 1` then the length is taken to be the number required.
`location` | the location parameter mu. This is not the mean of the Gumbel distribution (see Details below).
`scale` | the scale parameter sigma. This is not the standard deviation of the Gumbel distribution (see Details below).
`log` | Logical. If `log=TRUE` then the logarithm of the density is returned.

### Details

The Gumbel distribution is a special case of the generalized extreme value (GEV) distribution where the shape parameter xi = 0. The latter has 3 parameters, so the Gumbel distribution has two. The Gumbel distribution function is

G(y) = exp( -exp[ - (y-mu)/sigma ] )

where -Inf<y<Inf, -Inf<mu<Inf and sigma>0. Its mean is

mu - sigma * gamma

and its variance is

sigma^2 * pi^2 / 6

where gamma is Euler's constant (which can be obtained as `-digamma(1)`).

See `gumbel`, the VGAM family function for estimating the two parameters by maximum likelihood estimation, for formulae and other details. Apart from `n`, all the above arguments may be vectors and are recyled to the appropriate length if necessary.

### Value

`dgumbel` gives the density, `pgumbel` gives the distribution function, `qgumbel` gives the quantile function, and `rgumbel` generates random deviates.

### Note

The VGAM family function `gumbel` can estimate the parameters of a Gumbel distribution using maximum likelihood estimation.

### Author(s)

T. W. Yee

### References

Coles, S. (2001) An Introduction to Statistical Modeling of Extreme Values. London: Springer-Verlag.

### See Also

`gumbel`, `egumbel`, `gev`.

### Examples

    mu = 1
    sigma = 2
    y = rgumbel(n=100, loc=mu, scale=sigma)
    mean(y)
    mu - sigma * digamma(1)  # population mean
    var(y)
    sigma^2 * pi^2 / 6       # population variance

    ## Not run:
    x = seq(-2.5, 3.5, by=0.01)
    loc = 0; sigma = 1
    plot(x, dgumbel(x, loc, sigma), type="l", col="blue", ylim=c(0,1),
         main="Blue is density, red is cumulative distribution function",
         sub="Purple are 5,10,...,95 percentiles", ylab="", las=1)
    abline(h=0, col="blue", lty=2)
    lines(qgumbel(seq(0.05,0.95,by=0.05), loc, sigma),
          dgumbel(qgumbel(seq(0.05,0.95,by=0.05), loc, sigma), loc, sigma),
          col="purple", lty=3, type="h")
    lines(x, pgumbel(x, loc, sigma), type="l", col="red")
    abline(h=0, lty=2)
    ## End(Not run)

[Package VGAM version 0.7-9 Index]