# Gumbel: The Gumbel Distribution In actuar: Actuarial Functions and Heavy Tailed Distributions

 Gumbel R Documentation

## The Gumbel Distribution

### Description

Density function, distribution function, quantile function, random generation and raw moments for the Gumbel extreme value distribution with parameters alpha and scale.

### Usage

dgumbel(x, alpha, scale, log = FALSE)
pgumbel(q, alpha, scale, lower.tail = TRUE, log.p = FALSE)
qgumbel(p, alpha, scale, lower.tail = TRUE, log.p = FALSE)
rgumbel(n, alpha, scale)
mgumbel(order, alpha, scale)
mgfgumbel(t, alpha, scale, log = FALSE)


### Arguments

 x, q vector of quantiles. p vector of probabilities. n number of observations. If length(n) > 1, the length is taken to be the number required. alpha location parameter. scale parameter. Must be strictly positive. log, log.p logical; if TRUE, probabilities/densities p are returned as \log(p). lower.tail logical; if TRUE (default), probabilities are P[X \le x], otherwise, P[X > x]. order order of the moment. Only values 1 and 2 are supported. t numeric vector.

### Details

The Gumbel distribution with parameters alpha = \alpha and scale = \theta has distribution function:

F(x) = \exp[-\exp(-(x - \alpha)/\theta)]

for -\infty < x < \infty, -\infty < a < \infty and \theta > 0.

The mode of the distribution is in \alpha, the mean is \alpha + \gamma\theta, where \gamma = 0.57721566 is the Euler-Mascheroni constant, and the variance is \pi^2 \theta^2/6.

### Value

dgumbel gives the density, pgumbel gives the distribution function, qgumbel gives the quantile function, rgumbel generates random deviates, mgumbel gives the kth raw moment, k = 1, 2, and mgfgamma gives the moment generating function in t.

Invalid arguments will result in return value NaN, with a warning.

### Note

Distribution also knonw as the generalized extreme value distribution Type-I.

The "distributions" package vignette provides the interrelations between the continuous size distributions in actuar and the complete formulas underlying the above functions.

### Author(s)

Vincent Goulet vincent.goulet@act.ulaval.ca

### References

Klugman, S. A., Panjer, H. H. and Willmot, G. E. (2012), Loss Models, From Data to Decisions, Fourth Edition, Wiley.

### Examples

dgumbel(c(-5, 0, 10, 20), 0.5, 2)

p <- (1:10)/10
pgumbel(qgumbel(p, 2, 3), 2, 3)

curve(pgumbel(x, 0.5, 2), from = -5, to = 20, col = "red")
curve(pgumbel(x, 1.0, 2), add = TRUE, col = "green")
curve(pgumbel(x, 1.5, 3), add = TRUE, col = "blue")
curve(pgumbel(x, 3.0, 4), add = TRUE, col = "cyan")

a <- 3; s <- 4
mgumbel(1, a, s)                        # mean
a - s * digamma(1)                      # same

mgumbel(2, a, s) - mgumbel(1, a, s)^2   # variance
(pi * s)^2/6                            # same


actuar documentation built on Nov. 8, 2023, 9:06 a.m.