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 3.1-4 Index]