The Generalized Extreme Value Distribution

Density function, distribution function, quantile function and random generation for the generalized Extreme value (GenExtrVal) distribution with location, scale and shape parameters.

dGenExtrVal(x, loc=0, scale=1, shape=0, log = FALSE)
pGenExtrVal(q, loc=0, scale=1, shape=0, lower.tail = TRUE)
qGenExtrVal(p, loc=0, scale=1, shape=0, lower.tail = TRUE)
rGenExtrVal(n, loc=0, scale=1, shape=0)

Arguments

x, q: Vector of quantiles.
p: Vector of probabilities.
n: Number of observations.
loc, scale, shape: Location, scale and shape parameters; the shape argument cannot be a vector (must have length one).
log: Logical; if TRUE, the log density is returned.
lower.tail: Logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x]

Details

The GenExtrVal distribution function with parameters $loc = a$, $scale = b$ and $shape = s$ is $$G(z) = \exp\left[-\{1+s(z-a)/b\}^{-1/s}\right]$$ for $1+s(z-a)/b > 0$, where $b > 0$. If $s = 0$ the distribution is defined by continuity. If $1+s(z-a)/b \leq 0$, the value $z$ is either greater than the upper end point (if $s < 0$), or less than the lower end point (if $s > 0$).

The parametric form of the GenExtrVal encompasses that of the Gumbel, Frechet and reverse Weibull distributions, which are obtained for $s = 0$, $s > 0$ and $s < 0$ respectively. It was first introduced by Jenkinson (1955).

Value

dGenExtrVal gives the density function, pGenExtrVal gives the distribution function, qGenExtrVal gives the quantile function, and rGenExtrVal generates random deviates.

References

Jenkinson, A. F. (1955) The frequency distribution of the annual maximum (or minimum) of meteorological elements. Quart. J. R. Met. Soc., 81, 158--171.

Author

Alec Stephenson <alec_stephenson@hotmail.com>

Examples

dGenExtrVal(2:4, 1, 0.5, 0.8)
#> [1] 0.17210639 0.06706381 0.03428205
pGenExtrVal(2:4, 1, 0.5, 0.8)
#> [1] 0.7386812 0.8467772 0.8948490
qGenExtrVal(seq(0.9, 0.6, -0.1), 2, 0.5, 0.8)
#> [1] 5.157141 3.449973 2.800811 2.444700
rGenExtrVal(6, 1, 0.5, 0.8)
#> [1] 0.6603152 1.8191045 0.6002917 0.7090576 0.6405541 4.3192024
p <- (1:9)/10
pGenExtrVal(qGenExtrVal(p, 1, 2, 0.8), 1, 2, 0.8)
#> [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