Gumbel (extreme-value type I) distribution

Distribution function and quantile function of the Gumbel distribution.

cdfgum(x, para = c(0, 1))
quagum(f, para = c(0, 1))

Arguments

x

Vector of quantiles.

f

Vector of probabilities.

para

Numeric vector containing the parameters of the distribution, in the order \(\xi, \alpha\) (location, scale).

Details

The Gumbel distribution with location parameter \(\xi\) and scale parameter \(\alpha\) has distribution function $$F(x)=\exp[-\exp\lbrace-(x-\xi)/\alpha\rbrace]$$ and quantile function $$x(F)=\xi-\alpha\log(-\log F).$$

Value

cdfgum gives the distribution function; quagum gives the quantile function.

Note

The functions expect the distribution parameters in a vector, rather than as separate arguments as in the standard R distribution functions pnorm, qnorm, etc.

See also

cdfgev for the generalized extreme-value distribution, which generalizes the Gumbel distribution.

Examples

# Random sample from the Gumbel distribution with parameters xi=0, alpha=3.
quagum(runif(100), c(0,3))
#>   [1] 11.12980142 -2.33790195 10.50084763  2.50595186 -3.17451660  0.86957295
#>   [7] -0.60159393  2.11684216 -2.24157811  2.31033047  3.23872269  2.13461948
#>  [13] -3.64371368  2.69955771  2.55834074  0.69055611  1.24613095  2.81976017
#>  [19]  6.19605740  3.48438419  0.25096942  0.51321925 -1.96978596  0.68438850
#>  [25]  1.81085050 -2.92743376  3.63232228  1.53620467  2.36337450 -2.06514117
#>  [31]  7.87997810  2.53845168  1.35146629 -1.19862715 12.24581753 -2.38705705
#>  [37] -2.93502198  4.14112170  2.61638554 -0.90063538  7.38861601 -1.93463452
#>  [43]  2.42753050 -1.41612631  3.64992456  5.39993816  5.53072714  4.55760026
#>  [49] -3.95312255  4.42291001  6.35037926  2.01554646  0.53197735  7.46378640
#>  [55]  2.17400708  0.55624420  6.66688574  4.01083620 -2.16056631  1.44030441
#>  [61] -2.15032319  1.19492747  1.48688907 -1.05142809  4.55302327  8.78468135
#>  [67]  0.32257303  1.22803765 11.94884532 -0.11168800  0.72387260 -1.24429605
#>  [73] -2.75255922  0.01106311 -2.17712911  3.59917156  9.29414269  2.73049450
#>  [79]  0.82688075  1.43633851 -0.30164419  2.30277474  0.69635778  6.39178910
#>  [85]  4.24298456 -2.04939683 -0.71300184  3.17701859  0.60993972  2.83212105
#>  [91]  3.10139031 -2.96001117  9.48309153 -1.02780461  7.70844940  1.12334706
#>  [97]  8.12463922  0.58180156 -0.14188996  0.89813472