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