R: The Gumbel Distribution

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, lower.tail = TRUE, log.p = FALSE)
    qgumbel(p, location = 0, scale = 1, lower.tail = TRUE, log.p = FALSE)
    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.
`lower.tail, log.p` | Same meaning as in `punif` or `qunif`.

### 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`, `gumbelff`, `gev`, `dgompertz`.

### Examples

    mu <- 1; sigma <- 2;
    y <- rgumbel(n = 100, loc = mu, scale = sigma)
    c(mean(y), mu - sigma * digamma(1))  # Sample and population means
    c(var(y), sigma^2 * pi^2 / 6)  # Sample and population variances

    ## 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 1.1-3 Index]