DiscreteDists (version 1.1.0)

POISXL: The Discrete Poisson XLindley

Description

The function POISXL() defines the Discrete Poisson XLindley distribution a single parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss().

Usage

POISXL(mu.link = "log")

Value

Returns a gamlss.family object which can be used to fit a Discrete Poisson XLindley distribution in the gamlss() function.

Arguments

mu.link

defines the mu.link, with "log" link as the default for the mu parameter.

Author

Mariana Blandon Mejia,

Details

The Discrete Poisson XLindley distribution with parameters \(\mu\) has a support 0, 1, 2, ... and mass function given by

\(f(x | \mu) = \frac{\mu^2(x+\mu^2+3(1+\mu))}{(1+\mu)^{4+x}}\); with \(\mu>0\).

Note: in this implementation we changed the original parameters \(\alpha\) for \(\mu\), we did it to implement this distribution within gamlss framework.

References

Ahsan-ul-Haq, M., Al-Bossly, A., El-Morshedy, M., & Eliwa, M. S. (2022). Poisson XLindley distribution for count data: statistical and reliability properties with estimation techniques and inference. Computational Intelligence and neuroscience, 2022(1), 6503670.

See Also

dPOISXL.

Examples

# Example 1
# Generating some random values with
# known mu
y <- rPOISXL(n=1000, mu=1)

# Fitting the model
library(gamlss)
mod1 <- gamlss(y~1, family=POISXL,
               control=gamlss.control(n.cyc=500, trace=FALSE))

# Extracting the fitted values for mu
# using the inverse link function
exp(coef(mod1, what="mu"))

# Example 2
# Generating random values under some model

# A function to simulate a data set with Y ~ POISXL
gendat <- function(n) {
  x1 <- runif(n, min=0.4, max=0.6)
  mu <- exp(1.21 - 3 * x1) # 0.75 approximately
  y <- rPOISXL(n=n, mu=mu)
  data.frame(y=y, x1=x1)
}

dat <- gendat(n=1500)

# Fitting the model
mod2 <- NULL
mod2 <- gamlss(y~x1, family=POISXL, data=dat,
               control=gamlss.control(n.cyc=500, trace=FALSE))

summary(mod2)

# Example 3
# The counts the number of borers per hill of corn in an
# experiment conducted randomly on 8 hills in 15 replications.
# Taken from Ahsan-ul-Haq et al (2022) page 10.

y <- rep(x=0:8, times=c(43, 35, 17, 11, 5, 4, 1, 2, 2))

mod3 <- gamlss(y~1, family=POISXL,
               control=gamlss.control(n.cyc=500, trace=FALSE))

# Extracting the fitted values for mu
exp(coef(mod3, what="mu"))

# Example 4
# The number of mammalian cytogenetic dosimetry lesions produced
# by streptogramin exposure in rabbit.
# Taken from Ahsan-ul-Haq et al (2022) page 10.

y <- rep(x=0:4, times=c(200, 57, 30, 7, 6))

mod4 <- gamlss(y~1, family=POISXL,
               control=gamlss.control(n.cyc=500, trace=FALSE))

# Extracting the fitted values for mu
exp(coef(mod4, what="mu"))