GEV family distribution for fitting a GAMLSS

The functions GEVfisher() and GEVquasi() each define the generalized extreme value (GEV) family distribution, a three parameter distribution, for a gamlss.dist::gamlss.family() object to be used in GAMLSS fitting using the function gamlss::gamlss(). The only difference between GEVfisher() and GEVquasi() is the form of scoring method used to define the weights used in the fitting algorithm. Fisher's scoring, based on the expected Fisher information is used in GEVfisher(), whereas a quasi-Newton scoring, based on the cross products of the first derivatives of the log-likelihood, is used in GEVquasi(). The functions dGEV, pGEV, qGEV and rGEV define the density, distribution function, quantile function and random generation for the specific parameterization of the generalized extreme value distribution given in Details below.

Usage

GEVfisher(mu.link = "identity", sigma.link = "log", nu.link = "identity")

GEVquasi(mu.link = "identity", sigma.link = "log", nu.link = "identity")

dGEV(x, mu = 0, sigma = 1, nu = 0, log = FALSE)

pGEV(q, mu = 0, sigma = 1, nu = 0, lower.tail = TRUE, log.p = FALSE)

qGEV(p, mu = 0, sigma = 1, nu = 0, lower.tail = TRUE, log.p = FALSE)

rGEV(n, mu = 0, sigma = 1, nu = 0)

Arguments

mu.link: Defines the mu.link, with "identity" link as the default for the mu parameter.
sigma.link: Defines the sigma.link, with "log" link as the default for the sigma parameter.
nu.link: Defines the nu.link, with "identity" link as the default for the nu parameter.
x, q: Vector of quantiles.
mu, sigma, nu: Vectors of location, scale and shape parameter values.
log, log.p: Logical. If TRUE, probabilities eqn{p} are given as $\log(p)$.
lower.tail: Logical. If TRUE (the default), probabilities are $P[X \leq x]$, otherwise, $P[X > x]$.
p: Vector of probabilities.
n: Number of observations. If length(n) > 1, the length is taken to be the number required.

Value

GEVfisher() and GEVquasi() each return a gamlss.dist::gamlss.family() object which can be used to fit a regression model with a GEV response distribution using the gamlss::gamlss() function. dGEV() gives the density, pGEV() gives the distribution function, qGEV() gives the quantile function, and rGEV() generates random deviates.

Details

The distribution function of a GEV distribution with parameters loc = $\mu$, scale = $\sigma (> 0)$ and shape = $\xi$ ($= \nu$) is $$F(x) = P(X \leq x) = \exp\left\{ -\left[ 1+\xi\left(\frac{x-\mu}{\sigma}\right) \right]_+^{-1/\xi} \right\},$$ where $x_+ = \max(x, 0)$. 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 $\xi = 0$. See https://en.wikipedia.org/wiki/Generalized_extreme_value_distribution and/or Chapter 3 of Coles (2001) for further information.

For each observation in the data, the restriction that $\xi > -1/2$ is imposed, which is necessary for the usual asymptotic likelihood theory to be applicable.

Examples

See the examples in fitGEV().

References

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