Maxima of Two Gumbel Distributions

The extended Gumbel distribution models the maximum of two independent Gumbel-distributed random variables with potentially different location and scale parameters. It is parameterized by two pairs of location and scale parameters.

Usage

dgumbelx(x, loc1 = 0, scale1 = 1, loc2 = 0, scale2 = 1, log = FALSE)

pgumbelx(
  q,
  loc1 = 0,
  scale1 = 1,
  loc2 = 0,
  scale2 = 1,
  lower.tail = TRUE,
  log.p = FALSE
)

qgumbelx(
  p,
  loc1 = 0,
  scale1 = 1,
  loc2 = 0,
  scale2 = 1,
  lower.tail = TRUE,
  log.p = FALSE,
  interval = NULL,
  ...
)

rgumbelx(n, loc1 = 0, scale1 = 1, loc2 = 0, scale2 = 1)

Arguments

x, q

vector of quantiles.

loc1, scale1, loc2, scale2

location and scale parameters of the two Gumbel distributions. The distribution is symmetric in the two margins, so their order is immaterial.

log, log.p

logical; if TRUE, probabilities p are given as log(p) and the density is returned on the log scale.

lower.tail

logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x].

p

vector of probabilities.

interval

a length two vector containing the end-points of the interval to be searched for the quantiles, passed to uniroot(). By default a bracketing interval is derived from the quantiles of the two Gumbel margins.

...

other arguments passed to uniroot.

n

number of observations.

Value

dgumbelx() gives the density function, pgumbelx() gives the distribution function, qgumbelx() gives the quantile function, and rgumbelx() generates random deviates.

Details

Density function, distribution function, quantile function and random generation for the maxima of two Gumbel distributions, each with different location and scale parameters.

Note

Based on code by Alec Stephenson previously published in the evd package, adapted to conform to package standards.

See also

distributions-overview; uniroot(), which qgumbelx() uses for root finding

Examples

dgumbelx(2:4, 0, 1.1, 1, 0.5)
#> [1] 0.31056307 0.08836749 0.02808872
pgumbelx(2:4, 0, 1.1, 1, 0.5)
#> [1] 0.7425568 0.9196951 0.9715848
qgumbelx(seq(0.9, 0.6, -0.1), 0, 1.2, 2, 0.5)
#> [1] 3.489993 2.983368 2.692006 2.478481
rgumbelx(6, 0, 1.1, 1, 0.5)
#> [1] 0.8957238 1.3756205 2.2961160 0.8621185 2.7607791 1.4477668
p <- (1:9)/10
pgumbelx(qgumbelx(p, 0, 0.5, 1, 2), 0, 0.5, 1, 2)
#> [1] 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
## [1] 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9