The Generalised Extreme Value Distribution

Density function, distribution function, quantile function and random generation for the generalised extreme value (GEV) distribution.

Usage

dgev(x, loc = 0, scale = 1, shape = 0, log = FALSE, m = 1)

pgev(q, loc = 0, scale = 1, shape = 0, lower.tail = TRUE, log.p = FALSE, m = 1)

qgev(p, loc = 0, scale = 1, shape = 0, lower.tail = TRUE, log.p = FALSE, m = 1)

rgev(n, loc = 0, scale = 1, shape = 0, m = 1)

Arguments

x, q: Numeric vectors of quantiles.
loc, scale, shape: Numeric vectors. Location, scale and shape parameters. All elements of scale must be positive.
log, log.p: A logical scalar; if TRUE, probabilities p are given as log(p).
m: A numeric scalar. The distribution is reparameterised by working with the GEV(loc, scale, shape) distribution function raised to the power m. See Details.
lower.tail: A logical scalar. If TRUE (default), probabilities are $P[X \leq x]$, otherwise, $P[X > x]$.
p: A numeric vector of probabilities in [0,1].
n: Numeric scalar. The number of observations to be simulated. If length(n) > 1 then length(n) is taken to be the number required.

Value

dgev gives the density function, pgev gives the distribution function, qgev gives the quantile function, and rgev generates random deviates.

The length of the result is determined by n for rgev, and is the maximum of the lengths of the numerical arguments for the other functions.

The numerical arguments other than n are recycled to the length of the result.

Details

The distribution function of a GEV distribution with parameters loc = $\mu$, scale = $\sigma (> 0)$ and shape = $\xi$ is $$F(x) = \exp\{-[1 + \xi (x - \mu) / \sigma] ^ {-1/\xi} \}$$ for $1 + \xi (x - \mu) / \sigma > 0$. If $\xi = 0$ the distribution function is defined as the limit as $\xi$ tends to zero. The support of the distribution depends on $\xi$: it is $x \leq \mu - \sigma / \xi$ for $\xi < 0$; $x \geq \mu - \sigma / \xi$ for $\xi > 0$; and $x$ is unbounded for $\xi = 0$. Note that if $\xi < -1$ the GEV density function becomes infinite as $x$ approaches $\mu -\sigma / \xi$ from below.

If lower.tail = TRUE then if p = 0 (p = 1) then the lower (upper) limit of the distribution is returned, which is -Inf or Inf in some cases. Similarly, but reversed, if lower.tail = FALSE.

See https://en.wikipedia.org/wiki/Generalized_extreme_value_distribution for further information.

The effect of m is to change the location, scale and shape parameters to $(\mu + \sigma \log m, \sigma, \xi)$ if $\xi = 0$ and $(\mu + \sigma (m ^ \xi - 1) / \xi, \sigma m ^ \xi, \xi)$. For integer m we can think of this as working with the maximum of m independent copies of the original GEV(loc, scale, shape) variable.

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. doi:10.1002/qj.49708134804

Coles, S. G. (2001) An Introduction to Statistical Modeling of Extreme Values, Springer-Verlag, London. Chapter 3: doi:10.1007/978-1-4471-3675-0_3

Examples

dgev(-1:4, 1, 0.5, 0.8)
#> [1] 0.00000000 0.00000000 0.73575888 0.17210639 0.06706381 0.03428205
dgev(1:6, 1, 0.5, -0.2, log = TRUE)
#> [1] -0.3068528 -1.4279153 -5.7449245       -Inf       -Inf       -Inf
dgev(1, shape = c(-0.2, 0.4))
#> [1] 0.2951551 0.2001168

pgev(-1:4, 1, 0.5, 0.8)
#> [1] 0.0000000 0.0000000 0.3678794 0.7386812 0.8467772 0.8948490
pgev(1:6, 1, 0.5, -0.2)
#> [1] 0.3678794 0.9251864 0.9996801 1.0000000 1.0000000 1.0000000
pgev(1, c(1, 2), c(1, 2), c(-0.2, 0.4))
#> [1] 0.3678794 0.1743086
pgev(-3, c(1, 2), c(1, 2), c(-0.2, 0.4))
#> [1] 6.218855e-09 0.000000e+00
pgev(7, 1, 1, c(-0.2, 0.4))
#> [1] 1.0000000 0.9541694

qgev((1:9)/10, 2, 0.5, 0.8)
#> [1] 1.695706 1.802111 1.913749 2.045276 2.212953 2.444700 2.800811 3.449973
#> [9] 5.157141
qgev(0.5, c(1,2), c(0.5, 1), c(-0.5, 0.5))
#> [1] 1.167445 2.402245

p <- (1:9)/10
pgev(qgev(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

rgev(6, 1, 0.5, 0.8)
#> [1] 0.8606835 1.2189038 1.0191857 0.9256591 1.5097553 0.8640248