# gumbelUC: The Gumbel Distribution In VGAM: Vector Generalized Linear and Additive Models

## Description

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

## Usage

 ```1 2 3 4``` ```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.

T. W. Yee

## References

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

`gumbel`, `gumbelff`, `gev`, `dgompertz`.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17``` ```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) ```