The function POISXL() defines the Discrete Poisson XLindley distribution, one-parameter
discrete distribution, for a gamlss.family object to be used in GAMLSS fitting
using the function gamlss().
POISXL(mu.link = "log")Returns a gamlss.family object which can be used
to fit a Discrete Poisson XLindley distribution
in the gamlss() function.
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.
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.
# 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"))
#> (Intercept)
#> 0.9750579
# 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)
#> Warning: summary: vcov has failed, option qr is used instead
#> ******************************************************************
#> Family: c("POISXL", "Poisson-XLindley")
#>
#> Call:
#> gamlss(formula = y ~ x1, family = POISXL, data = dat, control = gamlss.control(n.cyc = 500,
#> trace = FALSE))
#>
#> Fitting method: RS()
#>
#> ------------------------------------------------------------------
#> Mu link function: log
#> Mu Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 1.5568 0.2296 6.779 1.73e-11 ***
#> x1 -3.6936 0.4536 -8.143 8.03e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> ------------------------------------------------------------------
#> No. of observations in the fit: 1500
#> Degrees of Freedom for the fit: 2
#> Residual Deg. of Freedom: 1498
#> at cycle: 3
#>
#> Global Deviance: 5424.381
#> AIC: 5428.381
#> SBC: 5439.007
#> ******************************************************************
# 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"))
#> (Intercept)
#> 0.8611103
# 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"))
#> (Intercept)
#> 2.046229