R: The Gumbel Distribution

Gumbel {actuar} | 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 <= 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` = a and `scale` = s has distribution function:

F(x) = exp[-exp(-(x - a)/s)],

for -Inf < x < Inf, -Inf < a < Inf and s > 0.

The mode of the distribution is in a, the mean is a + g * s, where g = 0.57721566 is the Euler-Mascheroni constant, and the variance is (pi * s)^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

[Package actuar version 3.1-2 Index]