New Poisson-generalised Lindley distribution

The function NPGL() defines the Poisson-generalised Lindley distribution, a two parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss().

NPGL(mu.link = "log", sigma.link = "log")

Arguments

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

Value

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

Details

The Poisson-generalised Lindley distribution with parameters \(\mu\) and \(\sigma\) has support \(x = 0, 1, 2, \ldots\) and probability mass function given by

\(f(x \mid \mu, \sigma)=\frac{\mu^2+\frac{\mu^{\sigma}(\mu+1)^{1-\sigma}\Gamma(x+\sigma)}{\Gamma(\sigma)\Gamma(x+1)}}{(\mu+1)^{x+2}}\)

with \(\mu > 0\) and \(\sigma > 0\).

This distribution is useful for modeling over-dispersed count data.

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

References

Altun, E. A new two-parameter discrete poisson-generalized Lindley distribution with properties and applications to healthcare data sets. Comput Stat 36, 2841–2861 (2021). https://doi.org/10.1007/s00180-021-01097-0

Author

Tomás Mesa, tomas.mesaz@udea.edu.co

Examples

# Example 1
# Generating some random values with
# known mu and sigma

set.seed(123)
y <- rNPGL(n=100, mu=20, sigma=2)

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

# Extracting the fitted values for mu and sigma
exp(coef(mod1, what="mu"))
#> (Intercept) 
#>    17.14864 
exp(coef(mod1, what="sigma"))
#> (Intercept) 
#>    1.528697 

# Example 2
# Generating random values under some model

# A function to simulate a data set with Y ~ NPGL
gendat <- function(n) {
  x1 <- runif(n)
  x2 <- runif(n)
  mu    <- exp(1.7 - 2.8 * x1) # Approx 1.35
  sigma <- exp(0.73 + 1 * x2)  # Approx 3.42
  y <- rNPGL(n=n, mu=mu, sigma=sigma)
  data.frame(y=y, x1=x1, x2=x2)
}

set.seed(1234)
datos <- gendat(n=200)

mod2 <- gamlss(y~x1, sigma.fo=~x2, family=NPGL, data=datos,
               control=gamlss.control(n.cyc=800, trace=FALSE))

summary(mod2)
#> Warning: summary: vcov has failed, option qr is used instead
#> ******************************************************************
#> Family:  c("NPGL", "Poisson-generalised Lindley") 
#> 
#> Call:  gamlss(formula = y ~ x1, sigma.formula = ~x2, family = NPGL,  
#>     data = datos, control = gamlss.control(n.cyc = 800, trace = FALSE)) 
#> 
#> Fitting method: RS() 
#> 
#> ------------------------------------------------------------------
#> Mu link function:  log
#> Mu Coefficients:
#>             Estimate Std. Error t value Pr(>|t|)    
#> (Intercept)   1.7977     0.1780   10.10   <2e-16 ***
#> x1           -3.3376     0.2565  -13.01   <2e-16 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> ------------------------------------------------------------------
#> Sigma link function:  log
#> Sigma Coefficients:
#>             Estimate Std. Error t value Pr(>|t|)  
#> (Intercept)   0.5040     0.2235   2.255   0.0252 *
#> x2            0.7650     0.3499   2.187   0.0299 *
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> ------------------------------------------------------------------
#> No. of observations in the fit:  200 
#> Degrees of Freedom for the fit:  4
#>       Residual Deg. of Freedom:  196 
#>                       at cycle:  88 
#>  
#> Global Deviance:     685.502 
#>             AIC:     693.502 
#>             SBC:     706.6952 
#> ******************************************************************