The Gumbel-II Distribution

Density, cumulative distribution function, quantile function and random generation for the Gumbel-II distribution.

dgumbelII(x, scale = 1, shape, log = FALSE)
pgumbelII(q, scale = 1, shape, lower.tail = TRUE, log.p = FALSE)
qgumbelII(p, scale = 1, shape, lower.tail = TRUE, log.p = FALSE)
rgumbelII(n, scale = 1, shape)

Arguments

x, q

vector of quantiles.

p

vector of probabilities.

n

number of observations. Same as in runif.

log

Logical. If log = TRUE then the logarithm of the density is returned.

lower.tail, log.p

Same meaning as in pnorm or qnorm.

shape, scale

positive shape and scale parameters.

Value

dgumbelII gives the density, pgumbelII gives the cumulative distribution function, qgumbelII gives the quantile function, and rgumbelII generates random deviates.

Author

T. W. Yee and Kai Huang

Details

See gumbelII for details.

See also

gumbelII, dgumbel.

Examples

probs <- seq(0.01, 0.99, by = 0.01)
Scale <- exp(1); Shape <- exp( 0.5);
max(abs(pgumbelII(qgumbelII(p = probs, shape = Shape, Scale),
                  shape = Shape, Scale) - probs))  # Should be 0
#> [1] 1.110223e-16

if (FALSE)  x <- seq(-0.1, 10, by = 0.01);
plot(x, dgumbelII(x, shape = Shape, Scale), type = "l", col = "blue",
     main = "Blue is density, orange is the CDF", las = 1,
     sub = "Red lines are the 10,20,...,90 percentiles",
     ylab = "", ylim = 0:1)
#> Error in h(simpleError(msg, call)): error in evaluating the argument 'x' in selecting a method for function 'plot': object 'x' not found
abline(h = 0, col = "blue", lty = 2)
#> Error in int_abline(a = a, b = b, h = h, v = v, untf = untf, ...): plot.new has not been called yet
lines(x, pgumbelII(x, shape = Shape, Scale), col = "orange")
#> Error: object 'x' not found
probs <- seq(0.1, 0.9, by = 0.1)
Q <- qgumbelII(probs, shape = Shape, Scale)
lines(Q, dgumbelII(Q, Scale, Shape), col = "red", lty = 3, type = "h")
#> Error in plot.xy(xy.coords(x, y), type = type, ...): plot.new has not been called yet
pgumbelII(Q, shape = Shape, Scale) - probs # Should be all zero
#> [1]  1.387779e-17  0.000000e+00  5.551115e-17 -5.551115e-17  0.000000e+00
#> [6]  0.000000e+00  0.000000e+00  0.000000e+00  0.000000e+00
abline(h = probs, col = "red", lty = 3)  # \dontrun{}
#> Error in int_abline(a = a, b = b, h = h, v = v, untf = untf, ...): plot.new has not been called yet