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KumIW.R
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KumIW.R
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#' The Kumaraswamy Inverse Weibull family
#'
#' @author Johan David Marin Benjumea, \email{johand.marin@@udea.edu.co}
#'
#' @description
#' The Kumaraswamy Inverse 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{dKumIW}
#'
#' @details
#' The Kumaraswamy Inverse Weibull Distribution with parameters \code{mu},
#' \code{sigma} and \code{nu} has density given by
#'
#' \eqn{f(x)= \mu \sigma \nu x^{-\mu - 1} \exp{- \sigma x^{-\mu}} (1 - \exp{- \sigma x^{-\mu}})^{\nu - 1},}
#'
#' for \eqn{x > 0}, \eqn{\mu > 0}, \eqn{\sigma > 0} and \eqn{\nu > 0}.
#'
#' @returns Returns a gamlss.family object which can be used to fit a KumIW distribution in the \code{gamlss()} function.
#'
#' @example examples/examples_KumIW.R
#'
#' @references
#'\insertRef{almalki2014modifications}{RelDists}
#'
#'\insertRef{shahbaz2012kumaraswamy}{RelDists}
#'
#'@importFrom gamlss.dist checklink
#' @importFrom gamlss rqres.plot
#' @export
KumIW <- function (mu.link="log", sigma.link="log", nu.link="log"){
mstats <- checklink("mu.link", "Kumaraswamy Inverse-Weibull",
substitute(mu.link), c("log", "own"))
dstats <- checklink("sigma.link", "Kumaraswamy Inverse-Weibull",
substitute(sigma.link), c("log", "own"))
vstats <- checklink("nu.link", "Kumaraswamy Inverse-Weibull",
substitute(nu.link), c("log", "own"))
structure(list(family=c("KumIW", "Kumaraswamy Inverse-Weibull"),
parameters=list(mu=TRUE, sigma=TRUE, nu=TRUE),
nopar=4,
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,
dldm = function(y, mu, sigma, nu) {
exp1 <- sigma * y^(-mu)
exp2 <- 1- exp(-exp1)
exp3 <- (nu - 1) * (exp(-exp1)/(exp2))
dldm <- 1/mu - log(y) + exp1*log(y) - exp3*exp1*log(y)
dldm
},
dldd = function(y, mu, sigma, nu) {
exp1 <- sigma * y^(-mu)
exp2 <- 1- exp(-exp1)
exp3 <- (nu - 1) * (exp(-exp1)/(exp2))
dldd <- 1/sigma - y^(-mu) + exp3*y^(-mu)
dldd
},
dldv = function(y, mu, sigma, nu){
exp1 <- sigma * y^(-mu)
exp2 <- 1- exp(-exp1)
dldv <- 1/nu + log(exp2)
dldv
},
d2ldm2 = function(y, mu, sigma, nu) {
exp1 <- sigma * y^(-mu)
exp2 <- 1- exp(-exp1)
exp3 <- (nu - 1) * (exp(-exp1)/(exp2))
dldm <- 1/mu - log(y) + exp1*log(y) - exp3*exp1*log(y)
d2ldm2 <- -dldm * dldm
d2ldm2
},
d2ldmdd = function(y, mu, sigma, nu) {
exp1 <- sigma * y^(-mu)
exp2 <- 1- exp(-exp1)
exp3 <- (nu - 1) * (exp(-exp1)/(exp2))
dldm <- 1/mu - log(y) + exp1*log(y) - exp3*exp1*log(y)
dldd <- 1/sigma - y^(-mu) + exp3*y^(-mu)
d2ldmdd <- -dldm * dldd
d2ldmdd
},
d2ldmdv = function(y, mu, sigma, nu) {
exp1 <- sigma * y^(-mu)
exp2 <- 1- exp(-exp1)
exp3 <- (nu - 1) * (exp(-exp1)/(exp2))
dldm <- 1/mu - log(y) + exp1*log(y) - exp3*exp1*log(y)
dldv <- 1/nu + log(exp2)
d2ldmdv <- -dldm * dldv
d2ldmdv
},
d2ldd2 = function(y, mu, sigma, nu) {
exp1 <- sigma * y^(-mu)
exp2 <- 1- exp(-exp1)
exp3 <- (nu - 1) * (exp(-exp1)/(exp2))
dldd <- 1/sigma - y^(-mu) + exp3*y^(-mu)
d2ldd2 <- -dldd * dldd
d2ldd2
},
d2ldddv = function(y, mu, sigma, nu) {
exp1 <- sigma * y^(-mu)
exp2 <- 1- exp(-exp1)
exp3 <- (nu - 1) * (exp(-exp1)/(exp2))
dldd <- 1/sigma - y^(-mu) + exp3*y^(-mu)
dldv <- 1/nu + log(exp2)
d2ldddv <- -dldd * dldv
d2ldddv
},
d2ldv2 = function(y, mu, sigma, nu) {
exp1 <- sigma * y^(-mu)
exp2 <- 1- exp(-exp1)
dldv <- 1/nu + log(exp2)
d2ldv2 <- -dldv * dldv
d2ldv2
},
G.dev.incr = function(y, mu, sigma, nu, ...) -2*dKumIW(y, mu, sigma, nu, log=TRUE),
rqres = expression(rqres(pfun="pKumIW", 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"))
}