Density, distribution function, quantile function and random generation for
the blended generalised extreme value distribution with parameters
location, scale, shape, pa, pb,
alpha and beta.
dbgev(
x,
location,
scale,
shape,
pa = 0.05,
pb = 0.2,
alpha = 0.5,
beta = 0.5,
log = FALSE
)pbgev(
q,
location,
scale,
shape,
pa = 0.05,
pb = 0.2,
alpha = 0.5,
beta = 0.5,
lower.tail = TRUE,
log.p = FALSE
)
qbgev(
p,
location,
scale,
shape,
pa = 0.05,
pb = 0.2,
alpha = 0.5,
beta = 0.5,
lower.tail = TRUE,
log.p = FALSE
)
rbgev(n, location, scale, shape, pa = 0.05, pb = 0.2, alpha = 0.5, beta = 0.5)
dbgev gives the density,
pbgev gives the distribution function,
qbgev gives the quantile function, and
rbgev generates random deviates.
vector of quantiles.
location, scale and shape parameters.
Gumbel to GEV mixing parameters; see Details.
logical; if TRUE, probabilities p are given as log(p).
logical; if TRUE (default), probabilities are \(P[X \le x]\), otherwise, \(P[X > x]\).
vector of probabilities.
number of observations.
The blended generalised extreme value distribution with location
parameter \(q_\alpha\), scale parameter \(s_\beta\) and
shape parameter \(\xi\) has cumulative distribution function
given by
$$H(x \mid q_\alpha, s_\beta, \xi) = F(x \mid q_\alpha, s_\beta, \xi)^{p(x')} G(x \mid \tilde q_\alpha \tilde s_\beta)^{1 - p(x')}$$
where
$$F(x \mid q_\alpha, s_\beta, \xi) = \exp \left\{ - \left[ \dfrac{x - q_\alpha}{s_\beta(\ell_{1 - \beta / 2, \xi} - \ell_{\beta / 2, \xi})^{-1}} + \ell_{\alpha, \xi} \right]^{-1/\xi}_+ \right\}$$
with \(\ell_{a, \xi} = (-\log a)^{-\xi}\), \([x]_+ = \max(0, x)\),
alpha and beta parameters \(\alpha\) and \(\beta\),
respectively,
$$G(x \mid q_{\tilde \alpha}, s_{\tilde \beta}) = \exp\left\{ -\exp\left[- \left\{\dfrac{x - \tilde q_\alpha}{\tilde s_\beta(\ell_{1 - \beta / 2} - \ell_{\beta / 2})^{-1}} + \ell_{\alpha} \right\}\right]\right\}$$
with \(\ell_a = \log(-\log a)\),
$$\tilde q_\alpha = a - \dfrac{(b - a)(\ell_\alpha - \ell_{p_a}))}{\ell_{p_a} - \ell_{p_b}}~~\text{and}~~\tilde s_\beta = \dfrac{(b - a)(\ell_{\beta / 2} - \ell_{1 - \beta/2})}{\ell_{p_a} - \ell_{p_b}},$$
with \(a = F^{-1}(p_a \mid q_\alpha, s_\beta, \xi)\),
\(b = F^{-1}(p_b \mid q_\alpha, s_\beta, \xi)\), with pa and pb
parameters \(p_a\) and \(p_b\), respectively, and where
\(x' = (x - a) / (b - a)\) and \(p(x)\) denotes the cumulative distribution
of the Beta(5, 5) distribution.
Default values for pa, pb, alpha and beta
are taken from Castro-Camilo et al. (2022).
Castro-Camilo, D., Huser, R., & Rue, H. (2022). Practical strategies for generalized extreme value-based regression models for extremes. Environmetrics, 33(6), e2742. tools:::Rd_expr_doi("10.1002/env.2742")
predict.evgam