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ExW.R
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ExW.R
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#' The Extended Weibull family
#'
#' @author Amylkar Urrea Montoya, \email{amylkar.urrea@@udea.edu.co}
#'
#' @description
#' The Extended Weibull family
#'
#' @param mu.link defines the mu.link, with "log" link as the default for the mu parameter.
#' @param sigma.link defines the sigma.link, with "log" link as the default for the sigma.
#' @param nu.link defines the nu.link, with "log" link as the default for the nu parameter.
#'
#' @seealso \link{dExW}
#'
#' @details
#' The Extended Weibull distribution with parameters \code{mu},
#' \code{sigma} and \code{nu} has density given by
#'
#' \eqn{f(x) = \frac{\mu \sigma \nu x^{\sigma -1} exp({-\mu x^{\sigma}})} {[1 -(1-\nu) exp({-\mu x^{\sigma}})]^2},}
#'
#' for x > 0.
#'
#' @returns Returns a gamlss.family object which can be used to fit a ExW distribution in the \code{gamlss()} function.
#'
#' @example examples/examples_ExW.R
#'
#' @references
#' \insertRef{almalki2014modifications}{RelDists}
#'
#' \insertRef{Zhang2007}{RelDists}
#'
#' @importFrom Rdpack reprompt
#' @importFrom gamlss.dist checklink
#' @importFrom gamlss rqres.plot
#' @export
ExW <- function (mu.link="log", sigma.link="log", nu.link="log") {
mstats <- checklink("mu.link", "Extended Weibull",
substitute(mu.link), c("log", "own"))
dstats <- checklink("sigma.link", "Extended Weibull",
substitute(sigma.link), c("log", "own"))
vstats <- checklink("nu.link", "Extended Weibull",
substitute(nu.link), c("log", "own"))
structure(list(family=c("ExW", "Extended Weibull"),
parameters=list(mu=TRUE, sigma=TRUE, nu=TRUE),
nopar=3,
type="Continuous",
mu.link = as.character(substitute(mu.link)),
sigma.link = as.character(substitute(sigma.link)),
nu.link = as.character(substitute(nu.link)),
mu.linkfun = mstats$linkfun,
sigma.linkfun = dstats$linkfun,
nu.linkfun = vstats$linkfun,
mu.linkinv = mstats$linkinv,
sigma.linkinv = dstats$linkinv,
nu.linkinv = vstats$linkinv,
mu.dr = mstats$mu.eta,
sigma.dr = dstats$mu.eta,
nu.dr = vstats$mu.eta,
# Primeras derivadas ---------------------------------
dldm = function(y, mu, sigma, nu) {
A <- y^(sigma) * exp(- mu * y^sigma)
B <- (1 - (1 - nu) * exp(- mu * y^sigma))
dldm <- 1 / mu - y^sigma + (2 * A) / B
dldm
},
dldd = function(y, mu, sigma, nu) {
A <- 1 / sigma + log(y) - mu * y^(sigma) * log(y)
B <- exp(- mu * y^sigma) * mu * y^(sigma) * log(y)
C <- (1 - (1 - nu) * exp(- mu * y^sigma))
dldd <- A + (2 * B) / C
dldd
},
dldv = function(y, mu, sigma, nu) {
A <- (1 - (1 - nu) * exp(- mu * y^sigma))
dldv <- 1 / nu - (2 * exp(- mu * y^sigma))/ A
dldv
},
# Segundas derivadas ---------------------------------
d2ldm2 = function(y, mu, sigma, nu) {
A <- y^(sigma) * exp(- mu * y^sigma)
B <- (1 - (1 - nu) * exp(- mu * y^sigma))
dldm <- 1 / mu - y^sigma + (2 * A) / B
d2ldm2 <- -dldm * dldm
d2ldm2
},
d2ldmdd = function(y, mu, sigma, nu) {
A <- y^(sigma) * exp(- mu * y^sigma)
B <- (1 - (1 - nu) * exp(- mu * y^sigma))
dldm <- 1 / mu - y^sigma + (2 * A) / B
C <- 1 / sigma + log(y) - mu * y^(sigma) * log(y)
D <- exp(- mu * y^sigma) * mu * y^(sigma) * log(y)
E <- (1 - (1 - nu) * exp(- mu * y^sigma))
dldd <- C + (2 * D) / E
d2ldmdd <- -dldm * dldd
d2ldmdd
},
d2ldmdv = function(y, mu, sigma, nu) {
A <- y^(sigma) * exp(- mu * y^sigma)
B <- (1 - (1 - nu) * exp(- mu * y^sigma))
dldm <- 1 / mu - y^sigma + (2 * A) / B
C <- (1 - (1 - nu) * exp(- mu * y^sigma))
dldv <- 1 / nu - (2 * exp(- mu * y^sigma))/ C
d2ldmdv <- -dldm * dldv
d2ldmdv
},
d2ldd2 = function(y, mu, sigma, nu) {
A <- 1 / sigma + log(y) - mu * y^(sigma) * log(y)
B <- exp(- mu * y^sigma) * mu * y^(sigma) * log(y)
C <- (1 - (1 - nu) * exp(- mu * y^sigma))
dldd <- A + (2 * B) / C
d2ldd2 <- -dldd * dldd
d2ldd2
},
d2ldddv = function(y, mu, sigma, nu) {
A <- 1 / sigma + log(y) - mu * y^(sigma) * log(y)
B <- exp(- mu * y^sigma) * mu * y^(sigma) * log(y)
C <- (1 - (1 - nu) * exp(- mu * y^sigma))
dldd <- A + (2 * B) / C
D <- (1 - (1 - nu) * exp(- mu * y^sigma))
dldv <- 1 / nu - (2 * exp(- mu * y^sigma))/ D
d2ldddv <- -dldd * dldv
d2ldddv
},
d2ldv2 = function(y, mu, sigma, nu) {
A <- (1 - (1 - nu) * exp(- mu * y^sigma))
dldv <- 1 / nu - (2 * exp(- mu * y^sigma))/ A
d2ldv2 <- -dldv * dldv
d2ldv2
},
G.dev.incr = function(y, mu, sigma, nu, ...) -2*dExW(y, mu, sigma, nu, log=TRUE),
rqres = expression(rqres(pfun="pExW", type="Continuous", y=y, mu=mu, sigma=sigma, nu=nu)),
mu.initial = expression(mu <- rep(1, length(y))),
sigma.initial = expression(sigma <- rep(1, length(y))),
nu.initial = expression(nu <- rep(1, length(y))),
mu.valid = function(mu) all(mu > 0),
sigma.valid = function(sigma) all(sigma > 0),
nu.valid = function(nu) all(nu > 0),
y.valid = function(y) all(y > 0)
),
class=c("gamlss.family", "family"))
}