Distributions of Maxima and Minima

Density, distribution, quantile, and random generation functions for the maximum or minimum of a given number of independent and identically distributed random variables from any specified distribution, derived analytically from the underlying distribution function.

Usage

dextreme(x, dFun, pFun, ..., distn, mlen = 1, largest = TRUE, log = FALSE)

pextreme(
  q,
  pFun,
  ...,
  distn,
  mlen = 1,
  largest = TRUE,
  lower.tail = TRUE,
  log.p = FALSE
)

qextreme(
  p,
  qFun,
  ...,
  distn,
  mlen = 1,
  largest = TRUE,
  lower.tail = TRUE,
  log.p = FALSE
)

rextreme(n, qFun, ..., distn, mlen = 1, largest = TRUE)

Arguments

x, q: vector of quantiles.
dFun, pFun, qFun: density, distribution and quantile function of the specified distribution. The density function must have a log argument (a simple wrapper can always be constructed to achieve this).
...: parameters of the specified distribution.
distn: a character string, optionally given as an alternative to dFun, pFun and qFun such that the density, distribution and quantile functions are formed upon the addition of the prefixes d, p and q respectively.
mlen: the number of independent variables.
largest: logical; if TRUE (default) use maxima, otherwise minima.
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.
n: number of observations.

Value

dextreme() gives the density function, pextreme() gives the distribution function and qextreme() gives the quantile function of the maximum/minimum of mlen independent variables from a specified distribution. rextreme() generates random deviates.

Details

Density function, distribution function, quantile function and random generation for the maximum/minimum of a given number of independent variables from a specified distribution.

Note

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

Examples


dextreme(2:4, dnorm, pnorm, mean = 0.5, sd = 1.2, mlen = 5)
#> [1] 0.48689660 0.17602941 0.02346192
dextreme(2:4, distn = "norm", mean = 0.5, sd = 1.2, mlen = 5)
#> [1] 0.48689660 0.17602941 0.02346192
dextreme(2:4, distn = "exp", mlen = 2, largest = FALSE)
#> [1] 0.0366312778 0.0049575044 0.0006709253
pextreme(2:4, distn = "exp", rate = 1.2, mlen = 2)
#> [1] 0.8267938 0.9460991 0.9836082
qextreme(seq(0.9, 0.6, -0.1), distn = "exp", rate = 1.2, mlen = 2)
#> [1] 2.474783 1.873629 1.509935 1.241553
rextreme(5, qgamma, shape = 1, mlen = 10)
#> [1] 2.624579 4.592664 1.719339 1.870533 2.646311

p <- (1:9)/10
pexp(qextreme(p, distn = "exp", rate = 1.2, mlen = 1), rate = 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