# The Gumbel Distribution Density, distribution function, quantile function and random generation for the Gumbel distribution with location parameter `location` and scale parameter `scale`. ```r 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 $$ G(y) = \exp \left( - \exp \left[ - \frac{y-\mu}{\sigma} \right]\right)%G(y) = exp( -exp[ - (y-mu)/sigma ] ) $$ where $-Inf0$. Its mean is $$ \mu - \sigma * \gamma%mu - sigma * gamma $$ and its variance is $$ \sigma^2 * \pi^2 / 6%sigma^2 * pi^2 / 6 $$ 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. ## Returns `dgumbel` gives the density, `pgumbel` gives the distribution function, `qgumbel` gives the quantile function, and `rgumbel` generates random deviates. ## References Coles, S. (2001). **An Introduction to Statistical Modeling of Extreme Values**. London: Springer-Verlag. ## Author(s) T. W. Yee ## Note The `VGAM` family function `gumbel` can estimate the parameters of a Gumbel distribution using maximum likelihood estimation. ## See Also `gumbel`, `gumbelff`, `gev`, `dgompertz`. ## Examples ```r 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", main = "Blue is density, red is the CDF", ylim = c(0, 1), 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) ```

gumbelUC function