R/functions.R
In zmpldistribution/zmpl: Zero Modified Poisson-Lindley distribution
#'Zero Modified Poisson-Lindley (ZMPL) distribution
#'
#'@description Density, distribution function, quantile function and random generation for the ZMPL distribution.
#'
#'@usage
#'dzmpl(x, theta, p0 = 0, log = FALSE)
#'pzmpl(q, theta = 5, p0 = 0, lower.tail = TRUE, log.p = FALSE)
#'qzmpl(p, theta = 5, p0 = 0, lower.tail = TRUE, log.p = FALSE)
#'rzmpl(n, theta = 5, p0 = 0)
#'
#' @param x,q vector of observations/quantiles
#' @param theta vector of scale parameter values
#' @param p0 vector of shape parameter values
#' @param log, log.p logical; if TRUE, probabilities p are given as log(p).
#' @param lower.tail logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x]
#' @param p vector of probabilities.
#' @param n number of observations. If \code{length(n) > 1}, the length is taken to be the number required.
#'
#'
#' @details If \code{theta} or \code{p0} not specified they assume the default values of 5 and 0, respectively.
#'
#'The ZMPL distribution has density:
#'
#'\deqn{
#' Pr(X=k)=\left\{\begin{array}{ll} \pi + (1 - \pi)\frac{\theta^2(\theta + 2)}{(\theta + 1)^3},&k = 0, \\&\\(1 - \pi)\frac{\theta^2(k + \theta + 2)}{(\theta + 1)^{k+3}},&k \in \mathbb{N}^{ {\tiny +}}. \end{array}\right.
#' }
#'
#'@return returns a object of the ZMPL distribution.
#'
#'@note For the function ZMPL(), p0 is the zero-modification
#'parameter and different values lead to different
#'modifications of the ZMPL distribution.
#'
#'@references
#'Santos-Neto, M., Xavier, D., Bourguignon, M. and Tomazella, V. (2017) Zero-Modified Poisson-Lindley distribution with applications in inflated and deflated of zeros.
#'\emph{to appear.}
#'
#'@author
#'Manoel Santos-Neto \email{mn.neco@gmail.com}, Danillo Xavier \email{danilloxavier@gmail.com}, Marcelo Bourguignon \email{m.p.bourguignon@gmail.com} and Vera Tomazella \email{vera@ufscar.com}
#'
#'@examples
#'dat <- rzmpl(1000); plot(table(dat),type="h",xlab="x",ylab="PMF")
#'fit <- fitzmpl(dat)
#'
#'@export
#'
qzmpl <- function (p, theta = 5, p0 = 0, lower.tail = TRUE, log.p = FALSE)
{
if (log.p == TRUE)
p <- exp(p)
else p <- p
if (lower.tail == TRUE)
p <- p
else p <- 1 - p
if (any(p <= 0) | any(p > 1))
stop(paste("p must be between 0 and 1", "\n", ""))
p0plind <- ((theta^2)*(theta+2))/((theta+1)^3)
pnew0 <- ((p - p0)/(1 - p0))
pnew <- ifelse(pnew0 > 0, pnew0, 0)
q <- qpoislind(pnew, theta = theta)
q
}
#'Zero Modified Poisson-Lindley (ZMPL) distribution
#'
#'@description Density, distribution function, quantile function and random generation for the ZMPL distribution.
#'
#'@usage
#'dzmpl(x, theta, p0 = 0, log = FALSE)
#'pzmpl(q, theta = 5, p0 = 0, lower.tail = TRUE, log.p = FALSE)
#'qzmpl(p, theta = 5, p0 = 0, lower.tail = TRUE, log.p = FALSE)
#'rzmpl(n, theta = 5, p0 = 0)
#'
#' @param x,q vector of observations/quantiles
#' @param theta vector of scale parameter values
#' @param p0 vector of shape parameter values
#' @param log, log.p logical; if TRUE, probabilities p are given as log(p).
#' @param lower.tail logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x]
#' @param p vector of probabilities.
#' @param n number of observations. If \code{length(n) > 1}, the length is taken to be the number required.
#'
#'
#' @details If \code{theta} or \code{p0} not specified they assume the default values of 5 and 0, respectively.
#'
#'The ZMPL distribution has density:
#'
#'\deqn{
#' Pr(X=k)=\left\{\begin{array}{ll} \pi + (1 - \pi)\frac{\theta^2(\theta + 2)}{(\theta + 1)^3},&k = 0, \\&\\(1 - \pi)\frac{\theta^2(k + \theta + 2)}{(\theta + 1)^{k+3}},&k \in \mathbb{N}^{ {\tiny +}}. \end{array}\right.
#' }
#'
#'@return returns a object of the ZMPL distribution.
#'
#'@note For the function ZMPL(), p0 is the zero-modification
#'parameter and different values lead to different
#'modifications of the ZMPL distribution.
#'
#'@references
#'Santos-Neto, M., Xavier, D., Bourguignon, M. and Tomazella, V. (2017) Zero-Modified Poisson-Lindley distribution with applications in inflated and deflated of zeros.
#'\emph{to appear.}
#'
#'@author
#'Manoel Santos-Neto \email{mn.neco@gmail.com}, Danillo Xavier \email{danilloxavier@gmail.com}, Marcelo Bourguignon \email{m.p.bourguignon@gmail.com} and Vera Tomazella \email{vera@ufscar.com}
#'
#'@examples
#'dat <- rzmpl(1000); plot(table(dat),type="h",xlab="x",ylab="PMF")
#'fit <- fitzmpl(dat)
#'
#'@export
#'
pzmpl <- function (q, theta = 5, p0=0, lower.tail = TRUE, log.p = FALSE)
{
cdf <- rep(0, length(q))
cdf <- ppoislind(q, theta = theta, lower.tail = TRUE, log.p = FALSE)
cdf <- p0 + (1 - p0) * cdf
if (lower.tail == TRUE)
cdf <- cdf
else cdf <- 1 - cdf
if (log.p == FALSE)
cdf <- cdf
else cdf <- log(cdf)
cdf
}
#'Zero Modified Poisson-Lindley (ZMPL) distribution
#'
#'@description Density, distribution function, quantile function and random generation for the ZMPL distribution.
#'
#'@usage
#'dzmpl(x, theta, p0 = 0, log = FALSE)
#'pzmpl(q, theta = 5, p0 = 0, lower.tail = TRUE, log.p = FALSE)
#'qzmpl(p, theta = 5, p0 = 0, lower.tail = TRUE, log.p = FALSE)
#'rzmpl(n, theta = 5, p0 = 0)
#'
#' @param x,q vector of observations/quantiles
#' @param theta vector of scale parameter values
#' @param p0 vector of shape parameter values
#' @param log, log.p logical; if TRUE, probabilities p are given as log(p).
#' @param lower.tail logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x]
#' @param p vector of probabilities.
#' @param n number of observations. If \code{length(n) > 1}, the length is taken to be the number required.
#'
#'
#' @details If \code{theta} or \code{p0} not specified they assume the default values of 5 and 0, respectively.
#'
#'The ZMPL distribution has density:
#'
#'\deqn{
#' Pr(X=k)=\left\{\begin{array}{ll} \pi + (1 - \pi)\frac{\theta^2(\theta + 2)}{(\theta + 1)^3},&k = 0, \\&\\(1 - \pi)\frac{\theta^2(k + \theta + 2)}{(\theta + 1)^{k+3}},&k \in \mathbb{N}^{ {\tiny +}}. \end{array}\right.
#' }
#'
#'@return returns a object of the ZMPL distribution.
#'
#'@note For the function ZMPL(), p0 is the zero-modification
#'parameter and different values lead to different
#'modifications of the ZMPL distribution.
#'
#'@references
#'Santos-Neto, M., Xavier, D., Bourguignon, M. and Tomazella, V. (2017) Zero-Modified Poisson-Lindley distribution with applications in inflated and deflated of zeros.
#'\emph{to appear.}
#'
#'@author
#'Manoel Santos-Neto \email{mn.neco@gmail.com}, Danillo Xavier \email{danilloxavier@gmail.com}, Marcelo Bourguignon \email{m.p.bourguignon@gmail.com} and Vera Tomazella \email{vera@ufscar.com}
#'
#'@examples
#'dat <- rzmpl(1000); plot(table(dat),type="h",xlab="x",ylab="PMF")
#'fit <- fitzmpl(dat)
#'
#'@export
#'
dzmpl <- function (x, theta=5, p0 = 0, log = FALSE)
{
ans <- NULL
x1 <- min(x)
deflat.limit <- - ( (theta^2)*(theta+2)/( (theta^2) +3*theta+1) )
if(p0 < deflat.limit) ans <- NaN
if(p0 > 1) ans <- NaN
if(p0 == deflat.limit)
{
if(x1 == 0) stop(paste("The value of x must be >= 1", "\n", ""))
if(log == TRUE){
ans <- log1p(-p0) + dpoislind(x,theta, log = TRUE)
} else{
ans <- (1 - p0)*dpoislind(x, theta)
}
}
if(p0 >= deflat.limit )
{
if(log==TRUE){
if(x1 == 0)
{
if(length(x) > 1 )
{
ans[1] <- log(p0 + (1-p0)*dpoislind(0,theta))
ans[2:length(x)] <- log((1 - p0)) + dpoislind(x[x!=0],theta,log=TRUE)
} else{
ans <- log(p0 + (1-p0)*dpoislind(0,theta))
}
}
else{
ans <- log((1 - p0)) + dpoislind(x,theta,log=TRUE)
}
}
else{
if(x1 == 0)
{
if(length(x) > 1){
ans[1] <- p0 + (1-p0)*dpoislind(0,theta)
ans[2:length(x)] <- (1 - p0)*dpoislind(x[x!=0],theta)
} else{
ans <- p0 + (1-p0)*dpoislind(0,theta)
}
}
else{
ans <- (1 - p0)*dpoislind(x,theta)
}
}
}
ans
}
#'Zero Modified Poisson-Lindley (ZMPL) distribution
#'
#'@description Density, distribution function, quantile function and random generation for the ZMPL distribution.
#'
#'@usage
#'dzmpl(x, theta, p0 = 0, log = FALSE)
#'pzmpl(q, theta = 5, p0 = 0, lower.tail = TRUE, log.p = FALSE)
#'qzmpl(p, theta = 5, p0 = 0, lower.tail = TRUE, log.p = FALSE)
#'rzmpl(n, theta = 5, p0 = 0)
#'
#' @param x,q vector of observations/quantiles
#' @param theta vector of scale parameter values
#' @param p0 vector of shape parameter values
#' @param log, log.p logical; if TRUE, probabilities p are given as log(p).
#' @param lower.tail logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x]
#' @param p vector of probabilities.
#' @param n number of observations. If \code{length(n) > 1}, the length is taken to be the number required.
#'
#'
#' @details If \code{theta} or \code{p0} not specified they assume the default values of 5 and 0, respectively.
#'
#'The ZMPL distribution has density:
#'
#'\deqn{
#' Pr(X=k)=\left\{\begin{array}{ll} \pi + (1 - \pi)\frac{\theta^2(\theta + 2)}{(\theta + 1)^3},&k = 0, \\&\\(1 - \pi)\frac{\theta^2(k + \theta + 2)}{(\theta + 1)^{k+3}},&k \in \mathbb{N}^{ {\tiny +}}. \end{array}\right.
#' }
#'
#'@return returns a object of the ZMPL distribution.
#'
#'@note For the function ZMPL(), p0 is the zero-modification
#'parameter and different values lead to different
#'modifications of the ZMPL distribution.
#'
#'@references
#'Santos-Neto, M., Xavier, D., Bourguignon, M. and Tomazella, V. (2017) Zero-Modified Poisson-Lindley distribution with applications in inflated and deflated of zeros.
#'\emph{to appear.}
#'
#'@author
#'Manoel Santos-Neto \email{mn.neco@gmail.com}, Danillo Xavier \email{danilloxavier@gmail.com}, Marcelo Bourguignon \email{m.p.bourguignon@gmail.com} and Vera Tomazella \email{vera@ufscar.com}
#'
#'@examples
#'dat <- rzmpl(1000); plot(table(dat),type="h",xlab="x",ylab="PMF")
#'fit <- fitzmpl(dat)
#'
#'@export
#'
rzmpl <- function(n, theta=5, p0=0){
deflat.limit <- - ( ((theta^2)*(theta+2)) /( (theta^2) +3*theta+1) )
if(p0 == deflat.limit){
u <- runif(n)
p <- (theta*(theta+2))/( (theta^2) +3*theta+1 )
lambda <- NULL
x <- NULL
for(i in 1:n)
{
lambda[i] <- ifelse(u[i] <= p, rexp(1,theta), rgamma(1,shape = 2, rate=theta) )
x[i] <- 1 + rpois(1,lambda[i] )
}
x
}
else if(deflat.limit < p0 & p0 < 0){
x<- qZMPL(runif(n),theta=theta,p0=p0)
}
else if(0<= p0 & p0 <= 1){
x <- ifelse(rbinom(n, 1, p0), 0, rpoislind(n, theta))
}
else{x <- NaN}
x
}
#'Zero Modified Poisson-Lindley (ZMPL) distribution
#'
#'@description The fuction \code{ZMPL()} defines the ZMPL distribution, a two paramenter
#'distribution. The zero modified Poisson-Lindley distribution is similar
#'to the Poisson-Lindley distribution but allows zeros as y values. The extra parameter
#'models the probabilities at zero. The functions dZMPL, pZMPL, qZMPL and rZMPL define
#'the density, distribution function, quantile function and random generation for
#'the zero modified Poisson-Lindley distribution.
#'plotZMPL can be used to plot the distribution. meanZMPL calculates the expected
#'value of the response for a fitted model.
#'
#'@usage
#'dZMPL(x, mu = 1, sigma = 1, nu=0.1, log = FALSE)
#'pZMPL(q, mu = 1, sigma = 1, nu=0.1, lower.tail = TRUE, log.p = FALSE)
#'qZMPL(p, mu = 1, sigma = 1, nu=0.1, lower.tail = TRUE, log.p = FALSE)
#'rZMPL(n, mu = 1, sigma = 1)
#'plotZMPL(mu = .5, sigma = 1, nu=0.1, from = 0, to = 0.999, n = 101, ...)
#'meanZMPL(obj)
#'
#' @param x,q vector of observations/quantiles
#' @param mu vector of scale parameter values
#' @param sigma vector of shape parameter values
#' @param log, log.p logical; if TRUE, probabilities p are given as log(p).
#' @param lower.tail logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x]
#' @param p vector of probabilities.
#' @param n number of observations. If \code{length(n) > 1}, the length is taken to be the number required.
#' @param from where to start plotting the distribution from
#' @param to up to where to plot the distribution
#' @param ... other graphical parameters for plotting
#'
#'
#' @details The probability massa function of the is given by
#'
#'\deqn{ f_{Y}(y;\mu,\delta,p) =\frac{[1-p]\sqrt{\delta+1}}{4\,y^{3/2}\,\sqrt{\pi\mu}}\left[y+\frac{\delta\mu}{\delta+1} \right]\exp\left(-\frac{\delta}{4}\left[\frac{y[\delta+1]}{\delta\mu}+\frac{\delta\mu}{y[\delta+1]}-2\right]\right) I_{(0, \infty)}(y)+ pI_{\{0\}}(y).}
#'
#'@return returns a object of the ZMPL distribution.
#'
#'@note For the function ZMPL(), mu is the mean and sigma is the precision parameter and nu is the proportion of zeros of the zero adjusted Birnbaum-Saunders distribution.
#'
#'@references
#'Leiva, V., Santos-Neto, M., Cysneiros, F.J.A., Barros, M. (2015) A methodology for stochastic inventory models based on a zero-adjusted Birnbaum-Saunders distribution.
#'\emph{Applied Stochastic Models in Business and Industry.} \email{10.1002/asmb.2124}.
#'
#'@author
#'Manoel Santos-Neto \email{manoel.ferreira@ufcg.edu.br}, F.J.A. Cysneiros \email{cysneiros@de.ufpe.br}, Victor Leiva \email{victorleivasanchez@gmail.com} and Michelli Barros \email{michelli.karinne@gmail.com}
#'
#'@examples plotZARBS()
#'dat <- rZARBS(1000); hist(dat)
#'fit <- gamlss(dat~1,family=ZARBS(),method=CG())
#'meanZABS(fit)
#'
#'data(papatoes);
#'fit = gamlss(I(Demand/10000)~1,sigma.formula=~1, nu.formula=~1, family=ZARBS(mu.link="identity",sigma.link = "identity",nu.link = "identity"),method=CG(),data=papatoes)
#'summary(fit)
#'
#'data(oil)
#'fit1 = gamlss(Demand~1,sigma.formula=~1, nu.formula=~1, family=ZARBS(mu.link="identity",sigma.link = "identity",nu.link = "identity"),method=CG(),data=oil)
#'summary(fit1)
#'
#'@export
#'
mle <- function(x, family="zmpl")
{
x <- x
n <- length(x)
if(family=="zmpl")
{
s <- mean(x)
r <- sum(x^2)
thetamm <- (4*s)/(r-s)
PImm <- 1 - (thetamm*(thetamm+1)*s)/(thetamm+2)
fr <- function(vP){
theta <- vP[1]
PI <- vP[2]
l <- rep(NA, times = n)
for(i in 1:n)
{
if(x[i]==0) l[i] <- log(PI + (1 - PI)*( ((theta+2)*theta^2)/ ((theta+1)^3)) )
else l[i] <- log(1-PI) + 2*log(theta) - (x[i] + 3)*log(theta + 1) + log(x[i] + theta + 2)
}
return(sum(l))
}
vp <- maxLik(fr,start = c(theta1=thetamm,theta2=PImm),method = "BFGS")
}
else if(family == "poisson")
{
fr <- function(vp)
{
theta <- vp[1]
l<- dpois(x, lambda = theta, log=TRUE)
return(sum(l))
}
vp <- maxLik(fr,start = c(theta1=mean(x)),method = "BFGS")
}
else if(family == "zmp")
{
s <- mean(x)
r <- var(x)
a <- (r-s)/(s^2)
PImm <- a/(1+a)
thetamm <- s/(1-PImm)
fr <- function(vP){
theta <- vP[1]
PI <- vP[2]
# l <- dzmpois(x, lambda = theta, p0=PI, log=TRUE)
l <- rep(NA, times = n)
for(i in 1:n)
{
if(x[i]==0) l[i] <- log(PI + (1 - PI)*exp(-theta) )
else l[i] <- log(1-PI) + dpois(x[i],lambda=theta,log=TRUE)
}
return(sum(l))
}
vp <- maxLik(fr,start = c(theta1=thetamm,theta2=PImm),method = "BFGS")
}
else{
fr <- function(vp)
{
theta <- vp[1]
l<- dpoislind(x, theta = theta, log=TRUE)
return(sum(l))
}
s <- mean(x)
thetamm <- (-(s-1) + sqrt( ((s-1)^2) + 8*s))/(2*s)
vp <- maxLik(fr,start = c(theta1=thetamm),method = "BFGS")
}
vp
}
#'Zero Modified Poisson-Lindley (ZMPL) distribution
#'
#'@description The fuction \code{ZMPL()} defines the ZMPL distribution, a two paramenter
#'distribution. The zero modified Poisson-Lindley distribution is similar
#'to the Poisson-Lindley distribution but allows zeros as y values. The extra parameter
#'models the probabilities at zero. The functions dZMPL, pZMPL, qZMPL and rZMPL define
#'the density, distribution function, quantile function and random generation for
#'the zero modified Poisson-Lindley distribution.
#'plotZMPL can be used to plot the distribution. meanZMPL calculates the expected
#'value of the response for a fitted model.
#'
#'@usage
#'dZMPL(x, mu = 1, sigma = 1, nu=0.1, log = FALSE)
#'pZMPL(q, mu = 1, sigma = 1, nu=0.1, lower.tail = TRUE, log.p = FALSE)
#'qZMPL(p, mu = 1, sigma = 1, nu=0.1, lower.tail = TRUE, log.p = FALSE)
#'rZMPL(n, mu = 1, sigma = 1)
#'plotZMPL(mu = .5, sigma = 1, nu=0.1, from = 0, to = 0.999, n = 101, ...)
#'meanZMPL(obj)
#'
#' @param x,q vector of observations/quantiles
#' @param mu vector of scale parameter values
#' @param sigma vector of shape parameter values
#' @param log, log.p logical; if TRUE, probabilities p are given as log(p).
#' @param lower.tail logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x]
#' @param p vector of probabilities.
#' @param n number of observations. If \code{length(n) > 1}, the length is taken to be the number required.
#' @param from where to start plotting the distribution from
#' @param to up to where to plot the distribution
#' @param ... other graphical parameters for plotting
#'
#'
#' @details The probability massa function of the is given by
#'
#'\deqn{ f_{Y}(y;\mu,\delta,p) =\frac{[1-p]\sqrt{\delta+1}}{4\,y^{3/2}\,\sqrt{\pi\mu}}\left[y+\frac{\delta\mu}{\delta+1} \right]\exp\left(-\frac{\delta}{4}\left[\frac{y[\delta+1]}{\delta\mu}+\frac{\delta\mu}{y[\delta+1]}-2\right]\right) I_{(0, \infty)}(y)+ pI_{\{0\}}(y).}
#'
#'@return returns a object of the ZMPL distribution.
#'
#'@note For the function ZMPL(), mu is the mean and sigma is the precision parameter and nu is the proportion of zeros of the zero adjusted Birnbaum-Saunders distribution.
#'
#'@references
#'Leiva, V., Santos-Neto, M., Cysneiros, F.J.A., Barros, M. (2015) A methodology for stochastic inventory models based on a zero-adjusted Birnbaum-Saunders distribution.
#'\emph{Applied Stochastic Models in Business and Industry.} \email{10.1002/asmb.2124}.
#'
#'@author
#'Manoel Santos-Neto \email{manoel.ferreira@ufcg.edu.br}, F.J.A. Cysneiros \email{cysneiros@de.ufpe.br}, Victor Leiva \email{victorleivasanchez@gmail.com} and Michelli Barros \email{michelli.karinne@gmail.com}
#'
#'@examples plotZARBS()
#'dat <- rZARBS(1000); hist(dat)
#'fit <- gamlss(dat~1,family=ZARBS(),method=CG())
#'meanZABS(fit)
#'
#'data(papatoes);
#'fit = gamlss(I(Demand/10000)~1,sigma.formula=~1, nu.formula=~1, family=ZARBS(mu.link="identity",sigma.link = "identity",nu.link = "identity"),method=CG(),data=papatoes)
#'summary(fit)
#'
#'data(oil)
#'fit1 = gamlss(Demand~1,sigma.formula=~1, nu.formula=~1, family=ZARBS(mu.link="identity",sigma.link = "identity",nu.link = "identity"),method=CG(),data=oil)
#'summary(fit1)
#'
#'@export
descriptive.zmpl <- function(x)
{
barx <- mean(x)
varx <- var(x)
skwx <- skewness(x)
kurx <- kurtosis(x)
Findex <- var(x)/mean(x)
minx <- min(x)
maxx <- max(x)
out <- round(cbind(minx,maxx,barx,varx,skwx,kurx,Findex),2)
colnames(out) <- c("Min","Max","Mean","Var","Skew","Kurt","FI")
rownames(out) <- c("Values")
out
}
#'Zero Modified Poisson-Lindley (ZMPL) distribution
#'
#'@description The fuction \code{ZMPL()} defines the ZMPL distribution, a two paramenter
#'distribution. The zero modified Poisson-Lindley distribution is similar
#'to the Poisson-Lindley distribution but allows zeros as y values. The extra parameter
#'models the probabilities at zero. The functions dZMPL, pZMPL, qZMPL and rZMPL define
#'the density, distribution function, quantile function and random generation for
#'the zero modified Poisson-Lindley distribution.
#'plotZMPL can be used to plot the distribution. meanZMPL calculates the expected
#'value of the response for a fitted model.
#'
#'@usage
#'dZMPL(x, theta = 1, p0=0.1, log = FALSE)
#'pZMPL(q, theta = 1, p0=0.1, lower.tail = TRUE, log.p = FALSE)
#'qZMPL(p, theta = 1, p0=0.1, lower.tail = TRUE, log.p = FALSE)
#'rZMPL(n, theta = 1, p0=0.1)
#'plotZMPL(theta = 1, p0=0.1, from = 0, to = 0.999, n = 101, ...)
#'meanZMPL(obj)
#'
#' @param x,q vector of observations/quantiles
#' @param theta theta parameter values
#' @param p0 p0 parameter values
#' @param log, log.p logical; if TRUE, probabilities p are given as log(p).
#' @param lower.tail logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x]
#' @param p vector of probabilities.
#' @param n number of observations. If \code{length(n) > 1}, the length is taken to be the number required.
#' @param from where to start plotting the distribution from
#' @param to up to where to plot the distribution
#' @param ... other graphical parameters for plotting
#'
#'
#'@details The probability massa function of the is given by
#'
#'\deqn{ f_{X}(x;\theta,p0) = p0 + (1-p0)*P[Y=0] I(y=0) + (1-p0)*P[Y>0] I(y>0)}
#'where \deqn{P[Y=y]} is a probability massa function of the Poisson-Lindley distribution.
#'
#'@references
#'Bourguignon, M., Xavier, D., Santos-Neto, M., Tomazella, V. (2017) The Modified Poisson-Lindley distribution: A model for overdispersed and underdispersed count data.
#'\emph{manuscript.}
#'
#'@author
#'Manoel Santos-Neto \email{mn.neco@gmail.com}, Marcelo Bourguignon \email{m.p.bourguignon@gmail.com}, Danillo Xavier \email{danilloxavier@gmail.com} and Vera Tomazella \email{vera@ufscar.br}
#'
#'@examples FIZMPL()
#'theta <- seq(0.05,1,l=100)
#'aux0 <- - ((theta^2)*(theta+2)/( (theta^2) +3*theta+1) )
#'p0 <- seq(0,1,l=100)
#'g <- expand.grid(x = theta, y = p0)
#'z <- FIZMPL(theta,p0)
#'z[z<0] <- 0
#'g$z <- z
#'print(wireframe(z ~ x * y ,g, xlab=expression(theta), scales = list(arrows = FALSE), ylab=expression(pi), zlab = "FI(X)") )
#'
#'
#'theta <- seq(0.05,1,l=100)
#'aux0 <- - ((theta^2)*(theta+2)/( (theta^2) +3*theta+1) )
#'p0 <- seq(min(aux0),0,l=100)
#'g <- expand.grid(x = theta, y = p0)
#'z <- FIZMPL(theta,p0)
#'z[z<0] <- 0
#'g$z <- z
#'print(wireframe(z ~ x * y ,g, xlab=expression(theta), scales = list(arrows = FALSE), ylab=expression(pi), zlab = "FI(X)") )
#'@export
fi.zmpl <- function(theta, p0)
{
muPL <- (theta+2)/(theta*(theta+1))
varPL <- ((theta^3) + 4*(theta^2) + 6*theta +2)/(theta*theta*(theta+1)*(theta+1))
FIY <- varPL/mPL
FIX <- muPL*p0 + FIY
FIX
}
zmpldistribution/zmpl documentation built on May 11, 2019, 5:16 p.m.
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#'Zero Modified Poisson-Lindley (ZMPL) distribution
#'
#'@description Density, distribution function, quantile function and random generation for the ZMPL distribution.
#'
#'@usage
#'dzmpl(x, theta, p0 = 0, log = FALSE)
#'pzmpl(q, theta = 5, p0 = 0, lower.tail = TRUE, log.p = FALSE)
#'qzmpl(p, theta = 5, p0 = 0, lower.tail = TRUE, log.p = FALSE)
#'rzmpl(n, theta = 5, p0 = 0)
#'
#' @param x,q vector of observations/quantiles
#' @param theta vector of scale parameter values
#' @param p0 vector of shape parameter values
#' @param log, log.p logical; if TRUE, probabilities p are given as log(p).
#' @param lower.tail logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x]
#' @param p vector of probabilities.
#' @param n number of observations. If \code{length(n) > 1}, the length is taken to be the number required.
#'
#'
#' @details If \code{theta} or \code{p0} not specified they assume the default values of 5 and 0, respectively.
#'
#'The ZMPL distribution has density:
#'
#'\deqn{
#' Pr(X=k)=\left\{\begin{array}{ll} \pi + (1 - \pi)\frac{\theta^2(\theta + 2)}{(\theta + 1)^3},&k = 0, \\&\\(1 - \pi)\frac{\theta^2(k + \theta + 2)}{(\theta + 1)^{k+3}},&k \in \mathbb{N}^{ {\tiny +}}. \end{array}\right.
#' }
#'
#'@return returns a object of the ZMPL distribution.
#'
#'@note For the function ZMPL(), p0 is the zero-modification
#'parameter and different values lead to different
#'modifications of the ZMPL distribution.
#'
#'@references
#'Santos-Neto, M., Xavier, D., Bourguignon, M. and Tomazella, V. (2017) Zero-Modified Poisson-Lindley distribution with applications in inflated and deflated of zeros.
#'\emph{to appear.}
#'
#'@author
#'Manoel Santos-Neto \email{mn.neco@gmail.com}, Danillo Xavier \email{danilloxavier@gmail.com}, Marcelo Bourguignon \email{m.p.bourguignon@gmail.com} and Vera Tomazella \email{vera@ufscar.com}
#'
#'@examples
#'dat <- rzmpl(1000); plot(table(dat),type="h",xlab="x",ylab="PMF")
#'fit <- fitzmpl(dat)
#'
#'@export
#'
qzmpl <- function (p, theta = 5, p0 = 0, lower.tail = TRUE, log.p = FALSE)
{
if (log.p == TRUE)
p <- exp(p)
else p <- p
if (lower.tail == TRUE)
p <- p
else p <- 1 - p
if (any(p <= 0) | any(p > 1))
stop(paste("p must be between 0 and 1", "\n", ""))
p0plind <- ((theta^2)*(theta+2))/((theta+1)^3)
pnew0 <- ((p - p0)/(1 - p0))
pnew <- ifelse(pnew0 > 0, pnew0, 0)
q <- qpoislind(pnew, theta = theta)
q
}
#'Zero Modified Poisson-Lindley (ZMPL) distribution
#'
#'@description Density, distribution function, quantile function and random generation for the ZMPL distribution.
#'
#'@usage
#'dzmpl(x, theta, p0 = 0, log = FALSE)
#'pzmpl(q, theta = 5, p0 = 0, lower.tail = TRUE, log.p = FALSE)
#'qzmpl(p, theta = 5, p0 = 0, lower.tail = TRUE, log.p = FALSE)
#'rzmpl(n, theta = 5, p0 = 0)
#'
#' @param x,q vector of observations/quantiles
#' @param theta vector of scale parameter values
#' @param p0 vector of shape parameter values
#' @param log, log.p logical; if TRUE, probabilities p are given as log(p).
#' @param lower.tail logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x]
#' @param p vector of probabilities.
#' @param n number of observations. If \code{length(n) > 1}, the length is taken to be the number required.
#'
#'
#' @details If \code{theta} or \code{p0} not specified they assume the default values of 5 and 0, respectively.
#'
#'The ZMPL distribution has density:
#'
#'\deqn{
#' Pr(X=k)=\left\{\begin{array}{ll} \pi + (1 - \pi)\frac{\theta^2(\theta + 2)}{(\theta + 1)^3},&k = 0, \\&\\(1 - \pi)\frac{\theta^2(k + \theta + 2)}{(\theta + 1)^{k+3}},&k \in \mathbb{N}^{ {\tiny +}}. \end{array}\right.
#' }
#'
#'@return returns a object of the ZMPL distribution.
#'
#'@note For the function ZMPL(), p0 is the zero-modification
#'parameter and different values lead to different
#'modifications of the ZMPL distribution.
#'
#'@references
#'Santos-Neto, M., Xavier, D., Bourguignon, M. and Tomazella, V. (2017) Zero-Modified Poisson-Lindley distribution with applications in inflated and deflated of zeros.
#'\emph{to appear.}
#'
#'@author
#'Manoel Santos-Neto \email{mn.neco@gmail.com}, Danillo Xavier \email{danilloxavier@gmail.com}, Marcelo Bourguignon \email{m.p.bourguignon@gmail.com} and Vera Tomazella \email{vera@ufscar.com}
#'
#'@examples
#'dat <- rzmpl(1000); plot(table(dat),type="h",xlab="x",ylab="PMF")
#'fit <- fitzmpl(dat)
#'
#'@export
#'
pzmpl <- function (q, theta = 5, p0=0, lower.tail = TRUE, log.p = FALSE)
{
cdf <- rep(0, length(q))
cdf <- ppoislind(q, theta = theta, lower.tail = TRUE, log.p = FALSE)
cdf <- p0 + (1 - p0) * cdf
if (lower.tail == TRUE)
cdf <- cdf
else cdf <- 1 - cdf
if (log.p == FALSE)
cdf <- cdf
else cdf <- log(cdf)
cdf
}
#'Zero Modified Poisson-Lindley (ZMPL) distribution
#'
#'@description Density, distribution function, quantile function and random generation for the ZMPL distribution.
#'
#'@usage
#'dzmpl(x, theta, p0 = 0, log = FALSE)
#'pzmpl(q, theta = 5, p0 = 0, lower.tail = TRUE, log.p = FALSE)
#'qzmpl(p, theta = 5, p0 = 0, lower.tail = TRUE, log.p = FALSE)
#'rzmpl(n, theta = 5, p0 = 0)
#'
#' @param x,q vector of observations/quantiles
#' @param theta vector of scale parameter values
#' @param p0 vector of shape parameter values
#' @param log, log.p logical; if TRUE, probabilities p are given as log(p).
#' @param lower.tail logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x]
#' @param p vector of probabilities.
#' @param n number of observations. If \code{length(n) > 1}, the length is taken to be the number required.
#'
#'
#' @details If \code{theta} or \code{p0} not specified they assume the default values of 5 and 0, respectively.
#'
#'The ZMPL distribution has density:
#'
#'\deqn{
#' Pr(X=k)=\left\{\begin{array}{ll} \pi + (1 - \pi)\frac{\theta^2(\theta + 2)}{(\theta + 1)^3},&k = 0, \\&\\(1 - \pi)\frac{\theta^2(k + \theta + 2)}{(\theta + 1)^{k+3}},&k \in \mathbb{N}^{ {\tiny +}}. \end{array}\right.
#' }
#'
#'@return returns a object of the ZMPL distribution.
#'
#'@note For the function ZMPL(), p0 is the zero-modification
#'parameter and different values lead to different
#'modifications of the ZMPL distribution.
#'
#'@references
#'Santos-Neto, M., Xavier, D., Bourguignon, M. and Tomazella, V. (2017) Zero-Modified Poisson-Lindley distribution with applications in inflated and deflated of zeros.
#'\emph{to appear.}
#'
#'@author
#'Manoel Santos-Neto \email{mn.neco@gmail.com}, Danillo Xavier \email{danilloxavier@gmail.com}, Marcelo Bourguignon \email{m.p.bourguignon@gmail.com} and Vera Tomazella \email{vera@ufscar.com}
#'
#'@examples
#'dat <- rzmpl(1000); plot(table(dat),type="h",xlab="x",ylab="PMF")
#'fit <- fitzmpl(dat)
#'
#'@export
#'
dzmpl <- function (x, theta=5, p0 = 0, log = FALSE)
{
ans <- NULL
x1 <- min(x)
deflat.limit <- - ( (theta^2)*(theta+2)/( (theta^2) +3*theta+1) )
if(p0 < deflat.limit) ans <- NaN
if(p0 > 1) ans <- NaN
if(p0 == deflat.limit)
{
if(x1 == 0) stop(paste("The value of x must be >= 1", "\n", ""))
if(log == TRUE){
ans <- log1p(-p0) + dpoislind(x,theta, log = TRUE)
} else{
ans <- (1 - p0)*dpoislind(x, theta)
}
}
if(p0 >= deflat.limit )
{
if(log==TRUE){
if(x1 == 0)
{
if(length(x) > 1 )
{
ans[1] <- log(p0 + (1-p0)*dpoislind(0,theta))
ans[2:length(x)] <- log((1 - p0)) + dpoislind(x[x!=0],theta,log=TRUE)
} else{
ans <- log(p0 + (1-p0)*dpoislind(0,theta))
}
}
else{
ans <- log((1 - p0)) + dpoislind(x,theta,log=TRUE)
}
}
else{
if(x1 == 0)
{
if(length(x) > 1){
ans[1] <- p0 + (1-p0)*dpoislind(0,theta)
ans[2:length(x)] <- (1 - p0)*dpoislind(x[x!=0],theta)
} else{
ans <- p0 + (1-p0)*dpoislind(0,theta)
}
}
else{
ans <- (1 - p0)*dpoislind(x,theta)
}
}
}
ans
}
#'Zero Modified Poisson-Lindley (ZMPL) distribution
#'
#'@description Density, distribution function, quantile function and random generation for the ZMPL distribution.
#'
#'@usage
#'dzmpl(x, theta, p0 = 0, log = FALSE)
#'pzmpl(q, theta = 5, p0 = 0, lower.tail = TRUE, log.p = FALSE)
#'qzmpl(p, theta = 5, p0 = 0, lower.tail = TRUE, log.p = FALSE)
#'rzmpl(n, theta = 5, p0 = 0)
#'
#' @param x,q vector of observations/quantiles
#' @param theta vector of scale parameter values
#' @param p0 vector of shape parameter values
#' @param log, log.p logical; if TRUE, probabilities p are given as log(p).
#' @param lower.tail logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x]
#' @param p vector of probabilities.
#' @param n number of observations. If \code{length(n) > 1}, the length is taken to be the number required.
#'
#'
#' @details If \code{theta} or \code{p0} not specified they assume the default values of 5 and 0, respectively.
#'
#'The ZMPL distribution has density:
#'
#'\deqn{
#' Pr(X=k)=\left\{\begin{array}{ll} \pi + (1 - \pi)\frac{\theta^2(\theta + 2)}{(\theta + 1)^3},&k = 0, \\&\\(1 - \pi)\frac{\theta^2(k + \theta + 2)}{(\theta + 1)^{k+3}},&k \in \mathbb{N}^{ {\tiny +}}. \end{array}\right.
#' }
#'
#'@return returns a object of the ZMPL distribution.
#'
#'@note For the function ZMPL(), p0 is the zero-modification
#'parameter and different values lead to different
#'modifications of the ZMPL distribution.
#'
#'@references
#'Santos-Neto, M., Xavier, D., Bourguignon, M. and Tomazella, V. (2017) Zero-Modified Poisson-Lindley distribution with applications in inflated and deflated of zeros.
#'\emph{to appear.}
#'
#'@author
#'Manoel Santos-Neto \email{mn.neco@gmail.com}, Danillo Xavier \email{danilloxavier@gmail.com}, Marcelo Bourguignon \email{m.p.bourguignon@gmail.com} and Vera Tomazella \email{vera@ufscar.com}
#'
#'@examples
#'dat <- rzmpl(1000); plot(table(dat),type="h",xlab="x",ylab="PMF")
#'fit <- fitzmpl(dat)
#'
#'@export
#'
rzmpl <- function(n, theta=5, p0=0){
deflat.limit <- - ( ((theta^2)*(theta+2)) /( (theta^2) +3*theta+1) )
if(p0 == deflat.limit){
u <- runif(n)
p <- (theta*(theta+2))/( (theta^2) +3*theta+1 )
lambda <- NULL
x <- NULL
for(i in 1:n)
{
lambda[i] <- ifelse(u[i] <= p, rexp(1,theta), rgamma(1,shape = 2, rate=theta) )
x[i] <- 1 + rpois(1,lambda[i] )
}
x
}
else if(deflat.limit < p0 & p0 < 0){
x<- qZMPL(runif(n),theta=theta,p0=p0)
}
else if(0<= p0 & p0 <= 1){
x <- ifelse(rbinom(n, 1, p0), 0, rpoislind(n, theta))
}
else{x <- NaN}
x
}
#'Zero Modified Poisson-Lindley (ZMPL) distribution
#'
#'@description The fuction \code{ZMPL()} defines the ZMPL distribution, a two paramenter
#'distribution. The zero modified Poisson-Lindley distribution is similar
#'to the Poisson-Lindley distribution but allows zeros as y values. The extra parameter
#'models the probabilities at zero. The functions dZMPL, pZMPL, qZMPL and rZMPL define
#'the density, distribution function, quantile function and random generation for
#'the zero modified Poisson-Lindley distribution.
#'plotZMPL can be used to plot the distribution. meanZMPL calculates the expected
#'value of the response for a fitted model.
#'
#'@usage
#'dZMPL(x, mu = 1, sigma = 1, nu=0.1, log = FALSE)
#'pZMPL(q, mu = 1, sigma = 1, nu=0.1, lower.tail = TRUE, log.p = FALSE)
#'qZMPL(p, mu = 1, sigma = 1, nu=0.1, lower.tail = TRUE, log.p = FALSE)
#'rZMPL(n, mu = 1, sigma = 1)
#'plotZMPL(mu = .5, sigma = 1, nu=0.1, from = 0, to = 0.999, n = 101, ...)
#'meanZMPL(obj)
#'
#' @param x,q vector of observations/quantiles
#' @param mu vector of scale parameter values
#' @param sigma vector of shape parameter values
#' @param log, log.p logical; if TRUE, probabilities p are given as log(p).
#' @param lower.tail logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x]
#' @param p vector of probabilities.
#' @param n number of observations. If \code{length(n) > 1}, the length is taken to be the number required.
#' @param from where to start plotting the distribution from
#' @param to up to where to plot the distribution
#' @param ... other graphical parameters for plotting
#'
#'
#' @details The probability massa function of the is given by
#'
#'\deqn{ f_{Y}(y;\mu,\delta,p) =\frac{[1-p]\sqrt{\delta+1}}{4\,y^{3/2}\,\sqrt{\pi\mu}}\left[y+\frac{\delta\mu}{\delta+1} \right]\exp\left(-\frac{\delta}{4}\left[\frac{y[\delta+1]}{\delta\mu}+\frac{\delta\mu}{y[\delta+1]}-2\right]\right) I_{(0, \infty)}(y)+ pI_{\{0\}}(y).}
#'
#'@return returns a object of the ZMPL distribution.
#'
#'@note For the function ZMPL(), mu is the mean and sigma is the precision parameter and nu is the proportion of zeros of the zero adjusted Birnbaum-Saunders distribution.
#'
#'@references
#'Leiva, V., Santos-Neto, M., Cysneiros, F.J.A., Barros, M. (2015) A methodology for stochastic inventory models based on a zero-adjusted Birnbaum-Saunders distribution.
#'\emph{Applied Stochastic Models in Business and Industry.} \email{10.1002/asmb.2124}.
#'
#'@author
#'Manoel Santos-Neto \email{manoel.ferreira@ufcg.edu.br}, F.J.A. Cysneiros \email{cysneiros@de.ufpe.br}, Victor Leiva \email{victorleivasanchez@gmail.com} and Michelli Barros \email{michelli.karinne@gmail.com}
#'
#'@examples plotZARBS()
#'dat <- rZARBS(1000); hist(dat)
#'fit <- gamlss(dat~1,family=ZARBS(),method=CG())
#'meanZABS(fit)
#'
#'data(papatoes);
#'fit = gamlss(I(Demand/10000)~1,sigma.formula=~1, nu.formula=~1, family=ZARBS(mu.link="identity",sigma.link = "identity",nu.link = "identity"),method=CG(),data=papatoes)
#'summary(fit)
#'
#'data(oil)
#'fit1 = gamlss(Demand~1,sigma.formula=~1, nu.formula=~1, family=ZARBS(mu.link="identity",sigma.link = "identity",nu.link = "identity"),method=CG(),data=oil)
#'summary(fit1)
#'
#'@export
#'
mle <- function(x, family="zmpl")
{
x <- x
n <- length(x)
if(family=="zmpl")
{
s <- mean(x)
r <- sum(x^2)
thetamm <- (4*s)/(r-s)
PImm <- 1 - (thetamm*(thetamm+1)*s)/(thetamm+2)
fr <- function(vP){
theta <- vP[1]
PI <- vP[2]
l <- rep(NA, times = n)
for(i in 1:n)
{
if(x[i]==0) l[i] <- log(PI + (1 - PI)*( ((theta+2)*theta^2)/ ((theta+1)^3)) )
else l[i] <- log(1-PI) + 2*log(theta) - (x[i] + 3)*log(theta + 1) + log(x[i] + theta + 2)
}
return(sum(l))
}
vp <- maxLik(fr,start = c(theta1=thetamm,theta2=PImm),method = "BFGS")
}
else if(family == "poisson")
{
fr <- function(vp)
{
theta <- vp[1]
l<- dpois(x, lambda = theta, log=TRUE)
return(sum(l))
}
vp <- maxLik(fr,start = c(theta1=mean(x)),method = "BFGS")
}
else if(family == "zmp")
{
s <- mean(x)
r <- var(x)
a <- (r-s)/(s^2)
PImm <- a/(1+a)
thetamm <- s/(1-PImm)
fr <- function(vP){
theta <- vP[1]
PI <- vP[2]
# l <- dzmpois(x, lambda = theta, p0=PI, log=TRUE)
l <- rep(NA, times = n)
for(i in 1:n)
{
if(x[i]==0) l[i] <- log(PI + (1 - PI)*exp(-theta) )
else l[i] <- log(1-PI) + dpois(x[i],lambda=theta,log=TRUE)
}
return(sum(l))
}
vp <- maxLik(fr,start = c(theta1=thetamm,theta2=PImm),method = "BFGS")
}
else{
fr <- function(vp)
{
theta <- vp[1]
l<- dpoislind(x, theta = theta, log=TRUE)
return(sum(l))
}
s <- mean(x)
thetamm <- (-(s-1) + sqrt( ((s-1)^2) + 8*s))/(2*s)
vp <- maxLik(fr,start = c(theta1=thetamm),method = "BFGS")
}
vp
}
#'Zero Modified Poisson-Lindley (ZMPL) distribution
#'
#'@description The fuction \code{ZMPL()} defines the ZMPL distribution, a two paramenter
#'distribution. The zero modified Poisson-Lindley distribution is similar
#'to the Poisson-Lindley distribution but allows zeros as y values. The extra parameter
#'models the probabilities at zero. The functions dZMPL, pZMPL, qZMPL and rZMPL define
#'the density, distribution function, quantile function and random generation for
#'the zero modified Poisson-Lindley distribution.
#'plotZMPL can be used to plot the distribution. meanZMPL calculates the expected
#'value of the response for a fitted model.
#'
#'@usage
#'dZMPL(x, mu = 1, sigma = 1, nu=0.1, log = FALSE)
#'pZMPL(q, mu = 1, sigma = 1, nu=0.1, lower.tail = TRUE, log.p = FALSE)
#'qZMPL(p, mu = 1, sigma = 1, nu=0.1, lower.tail = TRUE, log.p = FALSE)
#'rZMPL(n, mu = 1, sigma = 1)
#'plotZMPL(mu = .5, sigma = 1, nu=0.1, from = 0, to = 0.999, n = 101, ...)
#'meanZMPL(obj)
#'
#' @param x,q vector of observations/quantiles
#' @param mu vector of scale parameter values
#' @param sigma vector of shape parameter values
#' @param log, log.p logical; if TRUE, probabilities p are given as log(p).
#' @param lower.tail logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x]
#' @param p vector of probabilities.
#' @param n number of observations. If \code{length(n) > 1}, the length is taken to be the number required.
#' @param from where to start plotting the distribution from
#' @param to up to where to plot the distribution
#' @param ... other graphical parameters for plotting
#'
#'
#' @details The probability massa function of the is given by
#'
#'\deqn{ f_{Y}(y;\mu,\delta,p) =\frac{[1-p]\sqrt{\delta+1}}{4\,y^{3/2}\,\sqrt{\pi\mu}}\left[y+\frac{\delta\mu}{\delta+1} \right]\exp\left(-\frac{\delta}{4}\left[\frac{y[\delta+1]}{\delta\mu}+\frac{\delta\mu}{y[\delta+1]}-2\right]\right) I_{(0, \infty)}(y)+ pI_{\{0\}}(y).}
#'
#'@return returns a object of the ZMPL distribution.
#'
#'@note For the function ZMPL(), mu is the mean and sigma is the precision parameter and nu is the proportion of zeros of the zero adjusted Birnbaum-Saunders distribution.
#'
#'@references
#'Leiva, V., Santos-Neto, M., Cysneiros, F.J.A., Barros, M. (2015) A methodology for stochastic inventory models based on a zero-adjusted Birnbaum-Saunders distribution.
#'\emph{Applied Stochastic Models in Business and Industry.} \email{10.1002/asmb.2124}.
#'
#'@author
#'Manoel Santos-Neto \email{manoel.ferreira@ufcg.edu.br}, F.J.A. Cysneiros \email{cysneiros@de.ufpe.br}, Victor Leiva \email{victorleivasanchez@gmail.com} and Michelli Barros \email{michelli.karinne@gmail.com}
#'
#'@examples plotZARBS()
#'dat <- rZARBS(1000); hist(dat)
#'fit <- gamlss(dat~1,family=ZARBS(),method=CG())
#'meanZABS(fit)
#'
#'data(papatoes);
#'fit = gamlss(I(Demand/10000)~1,sigma.formula=~1, nu.formula=~1, family=ZARBS(mu.link="identity",sigma.link = "identity",nu.link = "identity"),method=CG(),data=papatoes)
#'summary(fit)
#'
#'data(oil)
#'fit1 = gamlss(Demand~1,sigma.formula=~1, nu.formula=~1, family=ZARBS(mu.link="identity",sigma.link = "identity",nu.link = "identity"),method=CG(),data=oil)
#'summary(fit1)
#'
#'@export
descriptive.zmpl <- function(x)
{
barx <- mean(x)
varx <- var(x)
skwx <- skewness(x)
kurx <- kurtosis(x)
Findex <- var(x)/mean(x)
minx <- min(x)
maxx <- max(x)
out <- round(cbind(minx,maxx,barx,varx,skwx,kurx,Findex),2)
colnames(out) <- c("Min","Max","Mean","Var","Skew","Kurt","FI")
rownames(out) <- c("Values")
out
}
#'Zero Modified Poisson-Lindley (ZMPL) distribution
#'
#'@description The fuction \code{ZMPL()} defines the ZMPL distribution, a two paramenter
#'distribution. The zero modified Poisson-Lindley distribution is similar
#'to the Poisson-Lindley distribution but allows zeros as y values. The extra parameter
#'models the probabilities at zero. The functions dZMPL, pZMPL, qZMPL and rZMPL define
#'the density, distribution function, quantile function and random generation for
#'the zero modified Poisson-Lindley distribution.
#'plotZMPL can be used to plot the distribution. meanZMPL calculates the expected
#'value of the response for a fitted model.
#'
#'@usage
#'dZMPL(x, theta = 1, p0=0.1, log = FALSE)
#'pZMPL(q, theta = 1, p0=0.1, lower.tail = TRUE, log.p = FALSE)
#'qZMPL(p, theta = 1, p0=0.1, lower.tail = TRUE, log.p = FALSE)
#'rZMPL(n, theta = 1, p0=0.1)
#'plotZMPL(theta = 1, p0=0.1, from = 0, to = 0.999, n = 101, ...)
#'meanZMPL(obj)
#'
#' @param x,q vector of observations/quantiles
#' @param theta theta parameter values
#' @param p0 p0 parameter values
#' @param log, log.p logical; if TRUE, probabilities p are given as log(p).
#' @param lower.tail logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x]
#' @param p vector of probabilities.
#' @param n number of observations. If \code{length(n) > 1}, the length is taken to be the number required.
#' @param from where to start plotting the distribution from
#' @param to up to where to plot the distribution
#' @param ... other graphical parameters for plotting
#'
#'
#'@details The probability massa function of the is given by
#'
#'\deqn{ f_{X}(x;\theta,p0) = p0 + (1-p0)*P[Y=0] I(y=0) + (1-p0)*P[Y>0] I(y>0)}
#'where \deqn{P[Y=y]} is a probability massa function of the Poisson-Lindley distribution.
#'
#'@references
#'Bourguignon, M., Xavier, D., Santos-Neto, M., Tomazella, V. (2017) The Modified Poisson-Lindley distribution: A model for overdispersed and underdispersed count data.
#'\emph{manuscript.}
#'
#'@author
#'Manoel Santos-Neto \email{mn.neco@gmail.com}, Marcelo Bourguignon \email{m.p.bourguignon@gmail.com}, Danillo Xavier \email{danilloxavier@gmail.com} and Vera Tomazella \email{vera@ufscar.br}
#'
#'@examples FIZMPL()
#'theta <- seq(0.05,1,l=100)
#'aux0 <- - ((theta^2)*(theta+2)/( (theta^2) +3*theta+1) )
#'p0 <- seq(0,1,l=100)
#'g <- expand.grid(x = theta, y = p0)
#'z <- FIZMPL(theta,p0)
#'z[z<0] <- 0
#'g$z <- z
#'print(wireframe(z ~ x * y ,g, xlab=expression(theta), scales = list(arrows = FALSE), ylab=expression(pi), zlab = "FI(X)") )
#'
#'
#'theta <- seq(0.05,1,l=100)
#'aux0 <- - ((theta^2)*(theta+2)/( (theta^2) +3*theta+1) )
#'p0 <- seq(min(aux0),0,l=100)
#'g <- expand.grid(x = theta, y = p0)
#'z <- FIZMPL(theta,p0)
#'z[z<0] <- 0
#'g$z <- z
#'print(wireframe(z ~ x * y ,g, xlab=expression(theta), scales = list(arrows = FALSE), ylab=expression(pi), zlab = "FI(X)") )
#'@export
fi.zmpl <- function(theta, p0)
{
muPL <- (theta+2)/(theta*(theta+1))
varPL <- ((theta^3) + 4*(theta^2) + 6*theta +2)/(theta*theta*(theta+1)*(theta+1))
FIY <- varPL/mPL
FIX <- muPL*p0 + FIY
FIX
}
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